定态Schr（？）dinger方程正反散射问题的数值解法研究
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摘要
本文研究了定态Schr(?)dinger方程的某些正反散射问题的数值计算方法.文章中对于这些算法给出了一些理论分析,并通过数值实验检验了算法的可行性.全文的具体内容如下:
第一章将简单介绍定态Schr(?)dinger方程正反散射问题的研究背景以及研究现状.第二章给出本文研究需要的某些预备知识,对于PML方法及因子分解法的研究背景及进展给予介绍,并分别以一种典型的问题为例来阐述各方法的基本原理.之后的两章是本文的主要工作.
第三章研究具有短程位势的二维散射问题的数值计算.针对一类特殊的短程位势,提出了一种PML方法.首先根据复化极径的思想,得到了极坐标下的PML方程,然后在关于吸收参数所做的假设下,证明了PML问题解的存在唯一性,并通过数值实验表明该方法具有一定的可行性.此外,本章还求出了具一类特殊位势的Schr(?)dinger方程的散射解在圆域外的级数表达式.
第四章考虑具紧支集位势的Schr(?)dinger方程反散射问题,利用因子分解法来重构位势的支集.首先推导出了散射振幅算子的分解公式;之后考虑内传播问题,研究了这一算子为单射的条件;根据其为单射、正规算子,可以通过散射振幅算子的谱数据来判断抽样点是否位于位势支集内.本章对于三维情况做了详细的理论分析,最后说明该方法可以用于二维问题,并针对二维情况给出了数值算例.
In recent years, the field of differential equations has come to distinguish between two different types of problems: the direct and the inverse. The main difference beween these two problems is that an inverse problem is often ill-posed and thus more complicated than the direct one. Scattering describing the processes of collision among particles is a class of basic problems in quantum mechanics. One can detect the structures of particles and promote the developements of theories by the analysis for results of scattering, so it's very necessary to study the direct and inverse scattering problems for Schr(?)dinger equations in mathematics. In this thesis, we consider the model independent of time, i.e., in stationary state.
Nowadays, the numerical computation has been the most important means for solving the Schr(?)dinger equations. Solving numerically a direct scattering problem is related to the computation in unbounded domain, so we have to truncate it into a bounded one, and present an absorbing boundary condition or set an absorbing layer out of the domain such that the error between the numerical and the exact solution isn't large. The studies on the inverse scattering problems for Schrodinger equations have a long history, including the work on the existence and uniqueness of the potential when scattering data known, the reconstruction of the potential from exact or noisy data and the stability analysis, etc., but there are still many problems open. In this thesis, we study the numerical computation on the direct and inverse scattering problems for stationary Schr(?)dinger equations as follows:
Ⅰ. The PML method for the 2D scattering problem with short range potential
1. The mathematical desciption for scattering problem
We consider the following Schr(?)dinger equation(?). (1) Given a plane wave u~i as the incident wave, i.e., u~i(x) = e~(ikω·x), where i = (?), k∈R is the wave number,ω∈S~1 is the direction of incidence, S~1 denotes the unit circle, V(x) denotes the potential. The incidence of u~i gives rise to the scattered wave u~s such that the total wave u =u~i + u~s satisfies the Schrodinger equation (1), and for the require of physical background, u~s must satisfies the Sommerfeld boundness and radiation conditions at infinity:(?). (2)By the stationary phase lemma, u~s satisfies the asymptotic behaviour uniformly with respect to (?), whereα((?),ω,κ) is called scattering amplitude. The scattering problem can be stated as follows: Given the potential V(x) and the incident wave u~i, find the total wave u = u~i + u~s satisfying the Schrodinger equation (1) such that the scattered wave u~s satisfies the condition (2).
Let r> 0 and the potential V(x) satisfy the following conditions: V(x)∈L~∞(B_(r_0)) and V(x) > 0 when r < r_0 and V(x) = b/r~δwhen r≥r_0, where B_(r_0) denotes the circle centered at origin and of radius r_0, both b > 0 andδ> 1 are constants, r= |x|. Since we consider the problem in unbounded domain, the assumption on the potential when r≥r_0 is essential and the refinement on V(x) when r < r_0 can be relaxed and even allowed not to be continuous on r = r_0.
Substitute u~i = e~(ikω·x) into (1), then u~s satisfies the equation(?). (3)A solution of equation (3) satisfying the condition (2) is called outgoing solution, the existence and uniqueness of such solutions are given by the following theorem:
Theorem 1.1 Except for a possibly discrete set composed of countable number of k, the equation (3) has a unique outgoing solution for every k∈R.
2. The presentation of PML method
We set a perfectly matched layer B~(PML)= {x∈R~2,R < |x| <ρ} outside B_R, where R > r_0. The fictitious medium in this layer can absorb the scattered wave very well such that the reflection on the artificial boundary very small or vanishing at all, so the solution in B_R can not be " polluted " heavily.
We set up a polar coordinates system by taking the origin and the positive direction of x-axis in rectangular coordinates system as the origin and the direction of polar-axis respectively. Assume that the argument of the incident directionωisθ_1, then the equation (3) can be written in polar form as(?). (4)By the idea of radius complexed, letα(r) = 1+ iσ(r), whereσ(r) is the absorbing factor satisfying thatσ∈C(R),σ≥0 andσ= 0 when r≤R. Let (?) denote the complex radius defined as follows:then(?). (5) Replace the r in (4) by (?) and using (5) again, we get(?). (6) where (?)(r,θ) = V((?),θ), The matrix A = A(x) satisfyingand its exact form is:By fomula (6), the PML problem corresponding to the problem (1)-(2) can be defined as the following boundary value problem: find (?), such that(?), in B_ρ, (7)(?)= 0, onΓ_ρ. (8)The aim of the PML method is to solve numerically the problem (7)-(8).
Let (?): H~1(B_ρ)×H~1(B_ρ)→C be a sesquilinear form: the weak formulation of the problem (7)-(8) is: findφ∈H~1(B_ρ), such that(?). (9)where (·,·)_(B_ρ) denotes the L~2 inner product in B_ρ.
For the absorbing factorσ, we assume as follows:(H) For a constantσ_0 > 0 and an integer m≥1,σ=σ_0 (?) ,R≤r≤ρ.
Theorem 1.2 Except for a possibly discrete set composed of countable number of k, the equation (9) has a unique solution for every k∈R.
In the numerical experiments, we give several examples to observe the effects of the position of absorbing layer placed、the thickness of the layer and the damping rate of potential to the scattered wave. The results show that solving 2D Schrodinger equation with short range potential by PML method is feasible to a certain extent.
3. The series expression of scattered wave outside some cicular domains
Let V(r) = b/r~2 oustide B_(r_0), r_1 > r_0, and the incident wave u~i = e~(iκω·x) we want to derive the expression of the scattered wave u~8 in r≥r_1. Assume that the incident directionωis the negtive direction of y-axis(The results have no essential differences when the directionωchanges), we take the origin and the positive direction of x-axis in rectangular coordinates system as origin and the direction of polar-axis in polar coordinates frame. Using the expression e~(iκω·x)=(?) in 2D and the geometrical relations betweenφand the argumentθ, we know that(?), (10) whereφis the angle between x andω, J_m denotes the mth order Bessel function. By the symmetry of the potential, we consider the variable seperated solution u(r,θ) =(?) of the equation (1) in polar form, then R_m(kr) satisfies theνth order Bessel equation:(?) (11)whereν~2 = m~2+ b. Since the fundamental solution of Bessel function are J_ν(kr) and H _ν~((1))(kr), the solution of equation (11) can be written as(?). (12) By the asymptotic behaviour of J_ν(kr) and H_ν~((1))(kr) when r→∞and the elastic scattering doesn't change the amplitude of each partial wave(Assume that only elastic scattering occurs), we getα_ν= e~(2iη_ν)- 1 = 2isinη_νe~(iη_ν), whereη_νis called phase shift in physics. The radiation condition for outgoing solution leads to b_m =(?). Substitute a_νand b_m into fomula (12), thenso the scattered wave u~s(r,θ) has the following series expression in r≥r_1:(?). (13)
Ⅱ. The factorization method for the inverse scattering problem with compact support potential
We extend the factorization method to the inverse scattering problem for Schr(?)dinger equation to reconstruct the support of the potential. Similar conclusions as the cases of inverse obstacle and medium scattering problems in acoustics can be obtained. We make the analysis carefully for the three-dimensioal case and state at last that this method can also be applied to the two-dimensional case.
1. The existence and uniqueness of the solutions of direct potential scattering problems
Let D (?) R~3 be a bounded domain with C~2 boundaryαD,R~3 \ (?) connected and the potential q(x) satisfy
q≥0, supp q = (?), there exists some a∈(0,1), such that q∈C~(0,α)(R~3), (14)where C~(0,α)(R~3) denotes the Holder continuous function space with H(?)lder indexα. Let u~i = e~(ik(?)·x) Denotesαplane wave in R~3 of direction (?)∈S~2:where S~2 denotes the unit sphere in R~3 and k > 0 denotes the wave number, we consider the following scattering problem for Schrodinger equation:(?). (15)(?), (16)(?), (17) where u~s denotes the scattering wave.
We denote byΦ{x,y) the fundamental solution of the Helmholtz equation in i.e.In fact,Φ(x, y) is also dependent of k and thus should be written asΦ_k(x, y), in this thesis we omit the index k most of the time except for several places. The main properties of the volume potential are summarized in the following lemma.
Lemma 2.1 We define the volume potentialωwith densityφ∈L~2(D) bythen the following holds:(a)ωis continuous in R~3, analytic in the exterior R~3 \ (?) of D, and satisfies the Helmholtz equation△ω+ k~2ω= 0 in R~3\ (?) and the Sommerfeld radiation condition (17).(b) Ifφ∈C{?), thenω∈C~(1,α)(D) for everyα∈(0,1).(c) If evenφ∈C~(0,α)(D) for someα∈(0,1], thenω∈C~(2,α)(D) and△ω+ k~2ω= -φin D. Furthemore, ifφ∈C~(0,α)(D) is of compact support, thenω∈C~(2,α)(R~3). and△ω+ k~2ω= -φin R~3; whereφis extended by zero into the whole of R~3.
In order to obtain the existence and uniquness of the solutions of scattering problem (15)-(17), we need the following several lemmas.
Lemma 2.2(Unique Continuation Principle) Let q(x) satisfy (14) and u∈C~2(R~3) be a solution of Schr(?)dinger equation△u + (k~2 - q(x))u = 0 in R~3 such that u(x) = 0 for all b≥αfor some |x|≥b, where (?) {x : |x|≤α}, then u has to vanish in all of R~3.
Lemma 2.3(Uniqueness) Let q(x) satisfy (14). The problem (15)-(17) has at most one solution, i.e., if u is a solution corresponding to u~i = 0, then u = 0.
Lemma 2.4(Equivalence) Let q(x) satisfy (14), then(a) If u∈C~2(R~3) is a solution of the scattering problem (15)-(17), then u∈C(?) solves the Lippmann-Schwinger integral equation(?), x∈R~3. (18)(b) If, on the other hand, u∈C(?) is a solution of the integral equation (18), then u can be extended by the right-hand side of (18) to a solution u∈C~2(R~3) of the scattering problem (15)-(17).
We define the operator T : L~2(D)→L~2(D) by(?), x∈D. Equation (18) is then written in short form as(?). (19)
Theorem 2.5 Under the assumptions (14) on q(x) , there exists a unique solution u of the scattering problem (15)-(17) or, equivalently, the Lippmann-Schwinger equation (18).
Lemma 2.6 If q(x) satisfies (14), the operator I + T is an isomorphism from L~2(D) onto itself.
2. The factorization of the scattering amplitude operator
The Lippmann-Schwinger equation (19) leads to a concept of weak solution as follows.
Definition 2.7 For h∈L~2(D), the solutionν∈L~2(D) of the integral equationν+ Tν= Th (20)is called the weak solution of the direct potential scattering problem.
Under the assumption (14) on q(x), the scattering waveν~s of the scattering problem (15)-(17) satisfies (?), (21)uniformly with respect to (?), whereν_α(?) is called scattering amplitude,(?) denotes the direction in which the incident wave is scattered. The inverse potential scattering problem is to reconstruct the potential q(x) by the given dataν_α(?). The uniqueness of this problem for fixed k~2 > 0 and all (?), (?)∈S~2 has been proved. For weak solutionν∈L~2(D) of the problem (20) we conclude from the asymptoticbehavioruniformly with respect toν∈(?) and (?), that the scattering amplitude pattern ofν~s is given by(?). (21) As a special case, the scattering amplitude of the scattering problem (15)—(17) takes the form(?). (22)
Now we define the scattering amplitude operator A : L~2(S~2)→L~2(S~2), i.e.(?). (23)If q(x) is of compact support, then the operator A is normal, and the scattering operator S = (?)A is unitary in L~2(S~2), i.e., SS~* = S~*S = I.
We define the operator G : L~2(D)→L~2(S~2) which maps h∈L~2(D) into the scattering amplitudeν_αof the weak solution v of the problem (20), i.e., Gh =ν_α. (24)
We will work with the weighted space L~2(D, q) which is defined with respect to the inner product (?). (25)In the following lemma we collect properties of the operators T and G in L~2(D, q).
Lemma 2.8 (a) The operator T is well defined and compact from L~2(D, q) into inself. (b) The adjoint T~# of T in L~2(D,q) with respect to〈·,·〉_q is given by(c) Both I+ T and I + T~# are isomorphisms from L~2(D,q) onto itself.(d) The operator G is also well defined and compact from L~2(D,q) into L~2(S~2).
Theorem 2.9 The scattering amplitude operator A = -4πG(I + T~#)G~#, whereG~# and T~# are the adjoints of G and T respectively.
3. An interior transmission problem
First, we formulate the interior transmssion problem in spaces of smooth functions. Given z∈D andα∈C, findν,ω∈C~2(D)∩C~1(?) such that(?), (26)(?). (27) We will show at the end of this section that there exist at most a countable number of k∈E_+ for which the homogeneous problem (26)-(27), i.e. forα= 0, admits nontrivial solutions under more assumptions on q(x).
We define two closed subspaces of L~2(D, q):(?), (28)(?). (29)
Lemma 2.10 If q(x) satisfies the condition (14), then the follwing conclusions hold:(a) Both I+ T and I + T~# are isomorphisms from H_q onto H_1.(b) The space (?) of Herglotz wavefunctions is dense in H_1.
Next we introduce the concept of weak solutions of the interior transmission problem (26)-(27).Definition 2.11 We fixed z∈D andα∈R,(a ) A pair (ν,ω)∈L~2(D,q)×L~2(D,q) is called the weak solution of (26)-(27), if(?). (30)(b) k~2 is called an interior transmission eigenvalue if there exists a weak solution (ν,ω)≠(0,0) of the homogeneous interior transmission problem (30), i.e. forα= 0.
The equivalence of (26)-(27) and (30) is shown in the following lemma.
Lemma 2.12 Fix z∈D andα∈C, letν,ω∈C~2(?) satisfy△ν+ k~2ν= 0 and△ω+ (k~2 - q(x))ω= 0 in D. The boundary conditionsω-ν=αΦ(·,z) and (?) onαD (31)hold if and only ifν=ω+Tωand (?). (32)
We construct a projection operator P from L~2(D, q) into the subspace:(?). (33) Note that H is the orthogonal complement of H_1 with respect to the sesquilinear form By Lax-Milgram theorem, for anyφ∈L~2(D,q) there exists a unique (?)∈H with (?) for all (?). We define P : L~2(D,q)→H by (?) and note that P is linear and bounded.
We will transform the interior transmission problem into a equivalent equation in the following.
Lemma 2.13 Fix z∈D andα∈C, we define d := inf{|x - z| : x∈αD}and chooseφ∈C~∞(R) withφ(t) = 0 for t≤(?) andφ(t) = 1 for t≥(?). Set(?) and (?), Note that (?) coincide withΦ(·, z)in a neighbourhood ofαD and therefore f_z is well defined in D, then the pair (ν,ω)∈H_1×H_q is a weak solution of (30) if and only ifω∈L~2(D, q) satisfies(?). (34)
In order to show that A is injective, we need the following lemma.
Lemma 2.14 LetΩ(?) R~3 be a domain that decomposed into two disjoint sub-domains: (?) such that (?). Let the boundariesαΩ_1 andαΩ_2 be smooth(i.e.,C~2). Let u_j∈C~2(Ω_j)∩C~1(?) for j = 1,2 be solutions of the Schr(?)dinger equation△u_1 + (k~2 - q(x))u_1 = 0 inΩ_1 and the Helmholtz equation△u_2 + k~2u_2 = 0 inΩ_2; where q(x) satisfies (14). Furthermore, let u_1 = u_2 and (?) onΓ, whereΓdenotes the common boundaryΓ:=αΩ_1∩αΩ_2. Then the function u, defined bycan be extended to a function in C~2(Ω) that satisfies the equation△u+(k~2 - q(x))u = 0
By the Lemma 2.14, we can prove the following lemma.
Lemma 2.15 g∈L~2(S~2) is a solution of the homogeneous integral equation(?), (35)if and only if there existsω∈C~2(D)∩C~1(?) such that (ν,ω) solve (26)-(27) forα= 0, whereνis the Herglotz function defined by(?). (36)
Next we give the most important theorem of this section.
Theorem 2.16 If k~2 is not an interior transmission eigenvalue then for every z∈D andα∈C, there exists a unique weak solution of the interior transmission problem (30). In this case, A is injective.
Now we give further assumptions on q(x) as follows:(?), (37)and there exist some k_0∈R_+ such that(?). (38)whereΦ_(k_0) (x, y) denotes the fundamental solution of Helmholtz equation for k = k_0. The following conclusion holds:
Theorem 2.17 If q(x) satisfies the conditions (14), (37) and (38), then except for at most a countable number of k, k~2 is not an interior transmission eigenvalue for every k∈R_+.
Remark 2.18 Assume that D contains in a sphere with radiusα, V and M denote the volume of D and upper bound of q(x) respectively, if(?). (39) then the condition (38) hold. The condition (39) can be checked easterly compared with (38).
4. The range of G
We denote byλ_j∈C andφ_j∈L~2(S~2) the eigenvalues and eigenfunctions. respectively, of A. Their existence is assured by the general spectral theory for compact normal operators. Furthermore, {φ_j ,j∈N} is a complete orthogonal system in L~2(S~2) provided A is injective. Letσ_j = |λ_j|, (?) , then Aφ_j = (?), (?), thus(?) is a singular system for A.
Defineφ_j∈L~2(D,q) by(?), (40)where we take the branch of the square root with Im(?) > 0. Note that from theunitary of S = (?)A the eigenvaluesλ_j lie on the circle of radius (?) with center (?), i.e. in particular Im(λ_j) > 0. Then, since A = -4πG(I + T~#)G~#,(?). (41)We will prove that {φ_j : j∈N} forms a Riesz basis in H_q = R(G~#) = N(G)~(?) (?) L~2(D,q).
Theorem 2.19 Assume that k~2 is not an interior transmission eigenvalue. Then the functions {φ_j : j∈N} from (40) formαRiesz basis in H_q = N(G)~(?) (?) L~2(D, q),i.e. every elementφ∈H_q has a unique representation in the form (?) with(?), and every sequence {α_j} (?) C with (?) generates an element (?), Furthermore, there exists c > 1 with (?), and every sequence {α_j} (?) C with (?) generates an element (?). (42)
Theorem 2.20 Assume that k~2 is not an interior transmission eigenvalue, then we have(?), (43)whereλ_j,φ_j are the eigenvalues and eigenfunctions, respectively, of A. Here G|_(H_1) is the restriction of G to the closed subspace H_1 of L~2(D,q).
As an application we give a simple characterization of the support of q(x) as follows.
Theorem 2.21 Assume that k~2 is not an interior transmission eigenvalue. Define the function r_z∈L~2(S~2) by r_z(?) := (?), (?)∈S~2. Then z∈D if and only if r_z∈R(G|_(H_1)), i.e. by the previous theorem(?) , (44)whereρ_j~((z))∈C are the expansion coefficients of r_z with respect toφ_j.
We make some detailed analysis for three-dimensional case, but all of the results also hold for the two-dimensional case with possibly different constants, the numerical experiments can show this very well.
第一章将简单介绍定态Schr(?)dinger方程正反散射问题的研究背景以及研究现状.第二章给出本文研究需要的某些预备知识,对于PML方法及因子分解法的研究背景及进展给予介绍,并分别以一种典型的问题为例来阐述各方法的基本原理.之后的两章是本文的主要工作.
第三章研究具有短程位势的二维散射问题的数值计算.针对一类特殊的短程位势,提出了一种PML方法.首先根据复化极径的思想,得到了极坐标下的PML方程,然后在关于吸收参数所做的假设下,证明了PML问题解的存在唯一性,并通过数值实验表明该方法具有一定的可行性.此外,本章还求出了具一类特殊位势的Schr(?)dinger方程的散射解在圆域外的级数表达式.
第四章考虑具紧支集位势的Schr(?)dinger方程反散射问题,利用因子分解法来重构位势的支集.首先推导出了散射振幅算子的分解公式;之后考虑内传播问题,研究了这一算子为单射的条件;根据其为单射、正规算子,可以通过散射振幅算子的谱数据来判断抽样点是否位于位势支集内.本章对于三维情况做了详细的理论分析,最后说明该方法可以用于二维问题,并针对二维情况给出了数值算例.
In recent years, the field of differential equations has come to distinguish between two different types of problems: the direct and the inverse. The main difference beween these two problems is that an inverse problem is often ill-posed and thus more complicated than the direct one. Scattering describing the processes of collision among particles is a class of basic problems in quantum mechanics. One can detect the structures of particles and promote the developements of theories by the analysis for results of scattering, so it's very necessary to study the direct and inverse scattering problems for Schr(?)dinger equations in mathematics. In this thesis, we consider the model independent of time, i.e., in stationary state.
Nowadays, the numerical computation has been the most important means for solving the Schr(?)dinger equations. Solving numerically a direct scattering problem is related to the computation in unbounded domain, so we have to truncate it into a bounded one, and present an absorbing boundary condition or set an absorbing layer out of the domain such that the error between the numerical and the exact solution isn't large. The studies on the inverse scattering problems for Schrodinger equations have a long history, including the work on the existence and uniqueness of the potential when scattering data known, the reconstruction of the potential from exact or noisy data and the stability analysis, etc., but there are still many problems open. In this thesis, we study the numerical computation on the direct and inverse scattering problems for stationary Schr(?)dinger equations as follows:
Ⅰ. The PML method for the 2D scattering problem with short range potential
1. The mathematical desciption for scattering problem
We consider the following Schr(?)dinger equation(?). (1) Given a plane wave u~i as the incident wave, i.e., u~i(x) = e~(ikω·x), where i = (?), k∈R is the wave number,ω∈S~1 is the direction of incidence, S~1 denotes the unit circle, V(x) denotes the potential. The incidence of u~i gives rise to the scattered wave u~s such that the total wave u =u~i + u~s satisfies the Schrodinger equation (1), and for the require of physical background, u~s must satisfies the Sommerfeld boundness and radiation conditions at infinity:(?). (2)By the stationary phase lemma, u~s satisfies the asymptotic behaviour uniformly with respect to (?), whereα((?),ω,κ) is called scattering amplitude. The scattering problem can be stated as follows: Given the potential V(x) and the incident wave u~i, find the total wave u = u~i + u~s satisfying the Schrodinger equation (1) such that the scattered wave u~s satisfies the condition (2).
Let r> 0 and the potential V(x) satisfy the following conditions: V(x)∈L~∞(B_(r_0)) and V(x) > 0 when r < r_0 and V(x) = b/r~δwhen r≥r_0, where B_(r_0) denotes the circle centered at origin and of radius r_0, both b > 0 andδ> 1 are constants, r= |x|. Since we consider the problem in unbounded domain, the assumption on the potential when r≥r_0 is essential and the refinement on V(x) when r < r_0 can be relaxed and even allowed not to be continuous on r = r_0.
Substitute u~i = e~(ikω·x) into (1), then u~s satisfies the equation(?). (3)A solution of equation (3) satisfying the condition (2) is called outgoing solution, the existence and uniqueness of such solutions are given by the following theorem:
Theorem 1.1 Except for a possibly discrete set composed of countable number of k, the equation (3) has a unique outgoing solution for every k∈R.
2. The presentation of PML method
We set a perfectly matched layer B~(PML)= {x∈R~2,R < |x| <ρ} outside B_R, where R > r_0. The fictitious medium in this layer can absorb the scattered wave very well such that the reflection on the artificial boundary very small or vanishing at all, so the solution in B_R can not be " polluted " heavily.
We set up a polar coordinates system by taking the origin and the positive direction of x-axis in rectangular coordinates system as the origin and the direction of polar-axis respectively. Assume that the argument of the incident directionωisθ_1, then the equation (3) can be written in polar form as(?). (4)By the idea of radius complexed, letα(r) = 1+ iσ(r), whereσ(r) is the absorbing factor satisfying thatσ∈C(R),σ≥0 andσ= 0 when r≤R. Let (?) denote the complex radius defined as follows:then(?). (5) Replace the r in (4) by (?) and using (5) again, we get(?). (6) where (?)(r,θ) = V((?),θ), The matrix A = A(x) satisfyingand its exact form is:By fomula (6), the PML problem corresponding to the problem (1)-(2) can be defined as the following boundary value problem: find (?), such that(?), in B_ρ, (7)(?)= 0, onΓ_ρ. (8)The aim of the PML method is to solve numerically the problem (7)-(8).
Let (?): H~1(B_ρ)×H~1(B_ρ)→C be a sesquilinear form: the weak formulation of the problem (7)-(8) is: findφ∈H~1(B_ρ), such that(?). (9)where (·,·)_(B_ρ) denotes the L~2 inner product in B_ρ.
For the absorbing factorσ, we assume as follows:(H) For a constantσ_0 > 0 and an integer m≥1,σ=σ_0 (?) ,R≤r≤ρ.
Theorem 1.2 Except for a possibly discrete set composed of countable number of k, the equation (9) has a unique solution for every k∈R.
In the numerical experiments, we give several examples to observe the effects of the position of absorbing layer placed、the thickness of the layer and the damping rate of potential to the scattered wave. The results show that solving 2D Schrodinger equation with short range potential by PML method is feasible to a certain extent.
3. The series expression of scattered wave outside some cicular domains
Let V(r) = b/r~2 oustide B_(r_0), r_1 > r_0, and the incident wave u~i = e~(iκω·x) we want to derive the expression of the scattered wave u~8 in r≥r_1. Assume that the incident directionωis the negtive direction of y-axis(The results have no essential differences when the directionωchanges), we take the origin and the positive direction of x-axis in rectangular coordinates system as origin and the direction of polar-axis in polar coordinates frame. Using the expression e~(iκω·x)=(?) in 2D and the geometrical relations betweenφand the argumentθ, we know that(?), (10) whereφis the angle between x andω, J_m denotes the mth order Bessel function. By the symmetry of the potential, we consider the variable seperated solution u(r,θ) =(?) of the equation (1) in polar form, then R_m(kr) satisfies theνth order Bessel equation:(?) (11)whereν~2 = m~2+ b. Since the fundamental solution of Bessel function are J_ν(kr) and H _ν~((1))(kr), the solution of equation (11) can be written as(?). (12) By the asymptotic behaviour of J_ν(kr) and H_ν~((1))(kr) when r→∞and the elastic scattering doesn't change the amplitude of each partial wave(Assume that only elastic scattering occurs), we getα_ν= e~(2iη_ν)- 1 = 2isinη_νe~(iη_ν), whereη_νis called phase shift in physics. The radiation condition for outgoing solution leads to b_m =(?). Substitute a_νand b_m into fomula (12), thenso the scattered wave u~s(r,θ) has the following series expression in r≥r_1:(?). (13)
Ⅱ. The factorization method for the inverse scattering problem with compact support potential
We extend the factorization method to the inverse scattering problem for Schr(?)dinger equation to reconstruct the support of the potential. Similar conclusions as the cases of inverse obstacle and medium scattering problems in acoustics can be obtained. We make the analysis carefully for the three-dimensioal case and state at last that this method can also be applied to the two-dimensional case.
1. The existence and uniqueness of the solutions of direct potential scattering problems
Let D (?) R~3 be a bounded domain with C~2 boundaryαD,R~3 \ (?) connected and the potential q(x) satisfy
q≥0, supp q = (?), there exists some a∈(0,1), such that q∈C~(0,α)(R~3), (14)where C~(0,α)(R~3) denotes the Holder continuous function space with H(?)lder indexα. Let u~i = e~(ik(?)·x) Denotesαplane wave in R~3 of direction (?)∈S~2:where S~2 denotes the unit sphere in R~3 and k > 0 denotes the wave number, we consider the following scattering problem for Schrodinger equation:(?). (15)(?), (16)(?), (17) where u~s denotes the scattering wave.
We denote byΦ{x,y) the fundamental solution of the Helmholtz equation in i.e.In fact,Φ(x, y) is also dependent of k and thus should be written asΦ_k(x, y), in this thesis we omit the index k most of the time except for several places. The main properties of the volume potential are summarized in the following lemma.
Lemma 2.1 We define the volume potentialωwith densityφ∈L~2(D) bythen the following holds:(a)ωis continuous in R~3, analytic in the exterior R~3 \ (?) of D, and satisfies the Helmholtz equation△ω+ k~2ω= 0 in R~3\ (?) and the Sommerfeld radiation condition (17).(b) Ifφ∈C{?), thenω∈C~(1,α)(D) for everyα∈(0,1).(c) If evenφ∈C~(0,α)(D) for someα∈(0,1], thenω∈C~(2,α)(D) and△ω+ k~2ω= -φin D. Furthemore, ifφ∈C~(0,α)(D) is of compact support, thenω∈C~(2,α)(R~3). and△ω+ k~2ω= -φin R~3; whereφis extended by zero into the whole of R~3.
In order to obtain the existence and uniquness of the solutions of scattering problem (15)-(17), we need the following several lemmas.
Lemma 2.2(Unique Continuation Principle) Let q(x) satisfy (14) and u∈C~2(R~3) be a solution of Schr(?)dinger equation△u + (k~2 - q(x))u = 0 in R~3 such that u(x) = 0 for all b≥αfor some |x|≥b, where (?) {x : |x|≤α}, then u has to vanish in all of R~3.
Lemma 2.3(Uniqueness) Let q(x) satisfy (14). The problem (15)-(17) has at most one solution, i.e., if u is a solution corresponding to u~i = 0, then u = 0.
Lemma 2.4(Equivalence) Let q(x) satisfy (14), then(a) If u∈C~2(R~3) is a solution of the scattering problem (15)-(17), then u∈C(?) solves the Lippmann-Schwinger integral equation(?), x∈R~3. (18)(b) If, on the other hand, u∈C(?) is a solution of the integral equation (18), then u can be extended by the right-hand side of (18) to a solution u∈C~2(R~3) of the scattering problem (15)-(17).
We define the operator T : L~2(D)→L~2(D) by(?), x∈D. Equation (18) is then written in short form as(?). (19)
Theorem 2.5 Under the assumptions (14) on q(x) , there exists a unique solution u of the scattering problem (15)-(17) or, equivalently, the Lippmann-Schwinger equation (18).
Lemma 2.6 If q(x) satisfies (14), the operator I + T is an isomorphism from L~2(D) onto itself.
2. The factorization of the scattering amplitude operator
The Lippmann-Schwinger equation (19) leads to a concept of weak solution as follows.
Definition 2.7 For h∈L~2(D), the solutionν∈L~2(D) of the integral equationν+ Tν= Th (20)is called the weak solution of the direct potential scattering problem.
Under the assumption (14) on q(x), the scattering waveν~s of the scattering problem (15)-(17) satisfies (?), (21)uniformly with respect to (?), whereν_α(?) is called scattering amplitude,(?) denotes the direction in which the incident wave is scattered. The inverse potential scattering problem is to reconstruct the potential q(x) by the given dataν_α(?). The uniqueness of this problem for fixed k~2 > 0 and all (?), (?)∈S~2 has been proved. For weak solutionν∈L~2(D) of the problem (20) we conclude from the asymptoticbehavioruniformly with respect toν∈(?) and (?), that the scattering amplitude pattern ofν~s is given by(?). (21) As a special case, the scattering amplitude of the scattering problem (15)—(17) takes the form(?). (22)
Now we define the scattering amplitude operator A : L~2(S~2)→L~2(S~2), i.e.(?). (23)If q(x) is of compact support, then the operator A is normal, and the scattering operator S = (?)A is unitary in L~2(S~2), i.e., SS~* = S~*S = I.
We define the operator G : L~2(D)→L~2(S~2) which maps h∈L~2(D) into the scattering amplitudeν_αof the weak solution v of the problem (20), i.e., Gh =ν_α. (24)
We will work with the weighted space L~2(D, q) which is defined with respect to the inner product (?). (25)In the following lemma we collect properties of the operators T and G in L~2(D, q).
Lemma 2.8 (a) The operator T is well defined and compact from L~2(D, q) into inself. (b) The adjoint T~# of T in L~2(D,q) with respect to〈·,·〉_q is given by(c) Both I+ T and I + T~# are isomorphisms from L~2(D,q) onto itself.(d) The operator G is also well defined and compact from L~2(D,q) into L~2(S~2).
Theorem 2.9 The scattering amplitude operator A = -4πG(I + T~#)G~#, whereG~# and T~# are the adjoints of G and T respectively.
3. An interior transmission problem
First, we formulate the interior transmssion problem in spaces of smooth functions. Given z∈D andα∈C, findν,ω∈C~2(D)∩C~1(?) such that(?), (26)(?). (27) We will show at the end of this section that there exist at most a countable number of k∈E_+ for which the homogeneous problem (26)-(27), i.e. forα= 0, admits nontrivial solutions under more assumptions on q(x).
We define two closed subspaces of L~2(D, q):(?), (28)(?). (29)
Lemma 2.10 If q(x) satisfies the condition (14), then the follwing conclusions hold:(a) Both I+ T and I + T~# are isomorphisms from H_q onto H_1.(b) The space (?) of Herglotz wavefunctions is dense in H_1.
Next we introduce the concept of weak solutions of the interior transmission problem (26)-(27).Definition 2.11 We fixed z∈D andα∈R,(a ) A pair (ν,ω)∈L~2(D,q)×L~2(D,q) is called the weak solution of (26)-(27), if(?). (30)(b) k~2 is called an interior transmission eigenvalue if there exists a weak solution (ν,ω)≠(0,0) of the homogeneous interior transmission problem (30), i.e. forα= 0.
The equivalence of (26)-(27) and (30) is shown in the following lemma.
Lemma 2.12 Fix z∈D andα∈C, letν,ω∈C~2(?) satisfy△ν+ k~2ν= 0 and△ω+ (k~2 - q(x))ω= 0 in D. The boundary conditionsω-ν=αΦ(·,z) and (?) onαD (31)hold if and only ifν=ω+Tωand (?). (32)
We construct a projection operator P from L~2(D, q) into the subspace:(?). (33) Note that H is the orthogonal complement of H_1 with respect to the sesquilinear form By Lax-Milgram theorem, for anyφ∈L~2(D,q) there exists a unique (?)∈H with (?) for all (?). We define P : L~2(D,q)→H by (?) and note that P is linear and bounded.
We will transform the interior transmission problem into a equivalent equation in the following.
Lemma 2.13 Fix z∈D andα∈C, we define d := inf{|x - z| : x∈αD}and chooseφ∈C~∞(R) withφ(t) = 0 for t≤(?) andφ(t) = 1 for t≥(?). Set(?) and (?), Note that (?) coincide withΦ(·, z)in a neighbourhood ofαD and therefore f_z is well defined in D, then the pair (ν,ω)∈H_1×H_q is a weak solution of (30) if and only ifω∈L~2(D, q) satisfies(?). (34)
In order to show that A is injective, we need the following lemma.
Lemma 2.14 LetΩ(?) R~3 be a domain that decomposed into two disjoint sub-domains: (?) such that (?). Let the boundariesαΩ_1 andαΩ_2 be smooth(i.e.,C~2). Let u_j∈C~2(Ω_j)∩C~1(?) for j = 1,2 be solutions of the Schr(?)dinger equation△u_1 + (k~2 - q(x))u_1 = 0 inΩ_1 and the Helmholtz equation△u_2 + k~2u_2 = 0 inΩ_2; where q(x) satisfies (14). Furthermore, let u_1 = u_2 and (?) onΓ, whereΓdenotes the common boundaryΓ:=αΩ_1∩αΩ_2. Then the function u, defined bycan be extended to a function in C~2(Ω) that satisfies the equation△u+(k~2 - q(x))u = 0
By the Lemma 2.14, we can prove the following lemma.
Lemma 2.15 g∈L~2(S~2) is a solution of the homogeneous integral equation(?), (35)if and only if there existsω∈C~2(D)∩C~1(?) such that (ν,ω) solve (26)-(27) forα= 0, whereνis the Herglotz function defined by(?). (36)
Next we give the most important theorem of this section.
Theorem 2.16 If k~2 is not an interior transmission eigenvalue then for every z∈D andα∈C, there exists a unique weak solution of the interior transmission problem (30). In this case, A is injective.
Now we give further assumptions on q(x) as follows:(?), (37)and there exist some k_0∈R_+ such that(?). (38)whereΦ_(k_0) (x, y) denotes the fundamental solution of Helmholtz equation for k = k_0. The following conclusion holds:
Theorem 2.17 If q(x) satisfies the conditions (14), (37) and (38), then except for at most a countable number of k, k~2 is not an interior transmission eigenvalue for every k∈R_+.
Remark 2.18 Assume that D contains in a sphere with radiusα, V and M denote the volume of D and upper bound of q(x) respectively, if(?). (39) then the condition (38) hold. The condition (39) can be checked easterly compared with (38).
4. The range of G
We denote byλ_j∈C andφ_j∈L~2(S~2) the eigenvalues and eigenfunctions. respectively, of A. Their existence is assured by the general spectral theory for compact normal operators. Furthermore, {φ_j ,j∈N} is a complete orthogonal system in L~2(S~2) provided A is injective. Letσ_j = |λ_j|, (?) , then Aφ_j = (?), (?), thus(?) is a singular system for A.
Defineφ_j∈L~2(D,q) by(?), (40)where we take the branch of the square root with Im(?) > 0. Note that from theunitary of S = (?)A the eigenvaluesλ_j lie on the circle of radius (?) with center (?), i.e. in particular Im(λ_j) > 0. Then, since A = -4πG(I + T~#)G~#,(?). (41)We will prove that {φ_j : j∈N} forms a Riesz basis in H_q = R(G~#) = N(G)~(?) (?) L~2(D,q).
Theorem 2.19 Assume that k~2 is not an interior transmission eigenvalue. Then the functions {φ_j : j∈N} from (40) formαRiesz basis in H_q = N(G)~(?) (?) L~2(D, q),i.e. every elementφ∈H_q has a unique representation in the form (?) with(?), and every sequence {α_j} (?) C with (?) generates an element (?), Furthermore, there exists c > 1 with (?), and every sequence {α_j} (?) C with (?) generates an element (?). (42)
Theorem 2.20 Assume that k~2 is not an interior transmission eigenvalue, then we have(?), (43)whereλ_j,φ_j are the eigenvalues and eigenfunctions, respectively, of A. Here G|_(H_1) is the restriction of G to the closed subspace H_1 of L~2(D,q).
As an application we give a simple characterization of the support of q(x) as follows.
Theorem 2.21 Assume that k~2 is not an interior transmission eigenvalue. Define the function r_z∈L~2(S~2) by r_z(?) := (?), (?)∈S~2. Then z∈D if and only if r_z∈R(G|_(H_1)), i.e. by the previous theorem(?) , (44)whereρ_j~((z))∈C are the expansion coefficients of r_z with respect toφ_j.
We make some detailed analysis for three-dimensional case, but all of the results also hold for the two-dimensional case with possibly different constants, the numerical experiments can show this very well.
引文
[1] R. G. Newton. Scattering Theory of Waves and Particles. Springer, New York, Second Edition, 1982.
[2] M. Read and B. Simon. Methods of Modern Mathematical Physics Ⅲ: Scattering Theory. Academic, New York, 1979.
[3] R. Milson. Liouville transformation and exactly solvable Schrodinger equations. Int. J. Theor. Phys., 37(6): 1735-1752, 1998.
[4] C. Lin. C. W. Wang and X. L. Zhang. Seried solution for the Schrodinger equation with a long-range spherically symmetric potential. Proceedings: Mathematical, Physical and Engineering Sciences, 458:1285-1290, 2002.
[5] K. Chadan and R. Kobayashi. New classes of potentials for which the radial Schrodinger equation can be solved at zero energy. J. Phys. A: Math. Gen., 39:3381-3396, 2006.
[6] K. Chadan and R. Kobayashi. New classes of potentials for which the radial Schrodinger equation can be solved at zero energy: Ⅱ. J. Phys. A: Math. Gen., 39:13691-13699, 2006.
[7] T. Fevens and H. Jiang. Absorbing boundary conditions for the Schrodinger equations. SIAM J. Sci. Comput, 21(1):255-282, 1999.
[8] M. Ehrhardt. Discrete transparent boundary conditions for Schroodinger-type equations for non-compactly supported initial data. Appl. Numer. Math., 58(5):660-673, 2008.
[9] M. Ehrhardt and A. Arnold. Discrete transparent boundary conditions for the Schro(?)dinger equations. Riv. Mat. Univ. Parma., 6:57-108, 2001.
[10] A. Schadle. Non-reflecting boundary conditions for the two-dimensional Schrodinger equation. Wave Motion., 35:181-188, 2002.
[11] X. Antoine, A. Arnold, C. Besse. M. Ehrhardt and A. Schadle. A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrodinger equations. Commun. Comput. Phys., 4(4):729-796, 2008.
[12] T. Monovasilis, Z. Kalogiratou and T. E. Simos. Numerical solution of the two-dimensional time independent Schr(?)dinger equation by symplectic schemes. Appl. Num. Anal. Comp. Math., 1(1):195-204, 2004.
[13] Z. Kalogiratou, T. Monovasilis and T. E. Simos. Numerical solution of the two-dimensional time independent Schrodinger equation with Numerov-type methods. J. Math. Chem., 37(3):271-279, 2005.
[14] A. S. Kadyrov, I. Bray. A. T. Stelbovics and B. Saha. Direct solution of the three-dimensional Lippmann-Schwinger equation. J. Phys. B: At. Mol. Opt. Phys., 38:509-515, 2005.
[15] V. Marchenko. Sturm-Liouville Operators and Applications. Birkhauser. Boston, 1986.
[16] B. Levitan. Inverse Strum-Liouville Problems. VNU Press, Utrecht, 1987.
[17] T. Aktosun and R. Weder. Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrodinger equation. Inverse Problems, 22:89-114, 2006.
[18] L. Amour and T. Raoux. Inverse spectral results for Schr(?)dinger operators on the unit interval with potential in L~p spaces. Inverse Problems, 23:2367-2373, 2007.
[19] T. Aktosun. Inverse Schr(?)dinger scattering on the line with partial knowledge of the potential. SIAM J. Appl. Math., 56(1):219-231, 1996.
[20] T. Aktosun and P. E. Sacks. Inversion of reflectivity data for nondecaying potentials. SIAM J. Appl Math., 60(4):1340-1356, 2000.
[21] T. Aktosun and R. Weder. Inverse scattering with partial information on the potential. J. Math. Anal Appl, 270:247-266, 2002.
[22] K. Chadan and P. C. Sabatier. Inverse Problems in Quantum Scattering Theroy. Springer-Verlag, New York, Second Edition, 1989.
[23] R. G. Newton. Inverse Schrodinger Scattering in Three Dimensions. Springer-Verlag, New York, 1989.
[24] A. G. Ramm. Inverse Problems. Springer, New York, 2005.
[25] T. Aktosun, and C. V. D Mee. Solution of the inverse scattering problem for the three-dimensional Schrodinger equation using a Fredholm integral equation. SIAM J. Math. Anal, 22(3):717-731, 1991.
[26] A. G. Ramm. Finding potential from the fixed-energy scattering data via D-N map. J.Inverse Ill-Posed Probl, 4(N2):145-152, 1996.
[27] Z. Q. Sun and G. Uhlmann. Inverse scattering for singular potentials in two dimensions. Trans. Amer. Math. Soc, 338(1):363-374, 1993.
[28] L. Paivarinta, V. S. Serov and E. Somersalo. Reconstruction of singularities of a scattering potential in two dimensions. Adv. Appl. Math., 15:97-113, 1994.
[29] V. S. Serov and L. Paivarinta. New estimates of the Green-Faddeev function and recovering of singularities in the two-dimensional Schroodinger operator with fixed energy. Inverse Prolems. 21:1291-1301, 2005.
[30] V. S. Serov. Some inverse scattering problems for two-dimensional Schrodinger operator. Proceedings of 5th International Conference on Inverse Problems in Engineering: Theory and Practic, Cambridge, UK, 11-15th July 2005.
[31] J. M. Reyes. Inverse backscattering for the Schrodinger equation in 2D. Inverse Problmes, 23:625-643, 2007.
[32] V. Serov. Inverse fixed angle scattering and backscattering problems in two dimensions. Inverse Problems, 24(6):065002(14pp), 2008.
[33] R. G. Novikov. The inverse scattering problem on a fixed energy level for the two-dimensional Schrodinger operator. J. Funct. Anal, 103:409-463, 1992.
[34] V. Isakov and Z. Sun. The inverse scattering at fixed energies in two dimensions. Indiana Univ. Math. J., 44(3):883-896, 1995.
[35] I. Kay and H. E. Morses. A simple verification of the Gelfand-Levitan equation for the three-dimensional scattering problem. Comm. Pure Appl. Math., 14:435-445, 1961.
[36] L. Faddeev. Three-dimensional inverse problem in the quantum theory of scattering. J. Math. Phys., 4:72-104, 1963.
[37] A. G. Ramm. Completness of the products of solutions to PDE and uniqueness theorems in inverse scattering. Inverse Problems, 3:L77-L82, 1987.
[38] A. G. Ramm. Uniqueness theorems for multidimensional inverse problems with unbouded coefficients. J. Math. Anal. Appl.. 136:586-574, 1988.
[39] A. G. Ramm. Recovery of the potential from fixed-energy scattering data. Inverse Problems, 4:877-886, 1988.
[40] A. G. Ramm. Completeness of the products of solutions of PDE and inverse problems. Inverse Problems, 6:643-664, 1990.
[41] A. G. Ramm. Symmetry properties of scattering amplitudes and applications to inverse problems. J. Math. Anal. Appl., 156:333-340, 1991.
[42] A. G. Ramm. Stability of the inversion of 3D fixed-frequency scattering data. J. Math. Anal. AppL 169:329-349, 1992.
[43] A. G. Ramm. Stability of the solution to 3D fixed-energy inverse scattering problem. J. Math. Anal. Appl., 170:1-15, 1992.
[44] A. G. Ramm. Stability estimates in inverse scattering. Acta Appl. Math., 28:1-42, 1992.
[45] A. G. Ramm. Stability of the solutions to 3D inverse scattering problems with fixed-energy data. Milan J. Math., 70:97-161, 2002.
[46] A. G. Ramm. Fixed-energy inverse scattering. Nonlinear Analysis, 69:971-978, 2008.
[47] A. G. Ramm. Inverse scattering problem with data at fixed energy and fixed incident direction. Nonlinear Analysis, 69:1478-1484, 2008.
[48] M. Horavóth and M. Kiss. The stability of inverse scattering with fixed energy. Inverse problems, 25(1):015001(14pp), 2009.
[49] R. G. Ramm. Fixed energy inverse scattering for noncompactly supported potentials. Mathl. Comput. Modelling, 18(1):57-64, 1993.
[50] R. G. Novikov. The inverse scattering problem at fixed energy for the three-dimensioanl Schr(?)dinger equation with an exponentially decreasing potential. Commun. Math. Phys., 161:569-595, 1994.
[51] M. Lassaut, S. Y. Larsen, S. A. Sofianos and J. C. Wallet. Construction of potentials using mixed scattering data. Inverse problems, 24(5):055014(15pp), 2008.
[52] F. Nicoleau. A constructive procedure to recover asympotics of short-range or long-range potentials. J. Differential Equations, 205:354-364, 2004.
[53] G. Nakamura, Z. Q. Sun and G. Uhlmann. Global identifiablity for an inverse problem for the Schrodinger equation in a magnetic field. Math. Ann., 303:377-388, 1995.
[54] G. Eskin and J. Ralson. Inverse scattering problem for the Schrodinger equation with magnetic potential at a fixed energy. Commun. Math. Phys., 173:199-224, 1995.
[55] A. Panchenko, An inverse problem for the magnetic Schrodinger equation and quasi-exponential solutions of nonsraooth partial differential equations, Inverse Problems, 18(2002), 1421-1434.
[56] G. Eskin. Inverse problems for the Schrodinger operators with electromagnetic potentials in domains with obstacles. Inverse Problems. 19:985-996, 2003.
[57] G. Eskin. Inverse boundary value problems and the Aharonov-Bohm effect. Inverse Problems, 19:49-62, 2003.
[58] G. Eskin. Inverse boundary value problems in domains with several obstacles. Inverse Problems, 20:1497-1516, 2004.
[59] G. Eskin. Global uniqueness in the inverse scattering problem for the Schrodinger operator with external Yang-Mills potentials. Commun. Math. Phys.. 222:503-531, 2001.
[60] G. Eskin. Inverse problems for Schrodinger equations with Yang-Mills potentials in domains with obstacles and the Aharonov-Bohm effect. Journal of Physics: Conference Series, 12:23-32, 2005.
[61] D. Givoli. Non reflecting boundary conditons. J. Comput. Phys., 94:1-29, 1991.
[62] M. J. Grote and J. E. Keller. Exact non-reflecting boundary condition for the time dependent wave equation. SIAM J. Appl. Math., 55:280-297, 1995.
[63] M. J. Grote and J. E. Keller. Nonreflecting boundary conditions for Maxwell equations. J. Comput. Phys., 139:327-342, 1998.
[64] S. V. Tsynkov. Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math., 27:465-532, 1998.
[65] J. P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114:185-200, 1994.
[66] J. P. Berenger. Three-dimentional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 127:363-379, 1996.
[67] D. S. Katz, E. T. Thiele and A. Taflove. Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FDTD meshes. IEEE Microwave Guided Wave Lett., 4:268-270, 1994.
[68] W. Andrew, C. Balanis, and P. Tirkas. A comparison of the Berenger perfectly matched layer and the Lindman high-order ABC's for the FDTD method. IEEE Microwave Guided Wave Lett, 5:192-194, 1995.
[69] B. Chen, D. G. Fang and B. H. Zhou. Modified Berenger PML absorbing boundary condition for FDTD meshes. IEEE Microwave Guided Wave Lett, 5:399-401, 1995.
[70] C. Eswarappa and W. J. R. Hoefer. Implementation of Berenger absorbing boundary conditions in TLM by interfacing FDTD perfectly matched layers. Electron. Lett, 31:1264-1266, 1995.
[71] J. De Moerloose and M. A. Stuchly. Behavior of Berenger's ABC for evanescent waves. IEEE Microwave Guided Wave Lett., 5:344-346, 1995.
[72] J. Fang and Z. Wu. Generalized perfectly matched layer-An extension of Berenger's perfectly matched layer boundary condition. IEEE Microwave Guided Wave Lett., 5:451-453, 1995.
[73] Z. Wu and J. Fang. Numerical implementation and performance of perfectly matched layer boundary condition for waveguide structures. IEEE Trans. Microwave Theory Tech., 43:2676-2683, 1995.
[74] C. M. Rappaport. Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space. IEEE Microwave Guided Wave Lett, 5:90-92, 1995.
[75] R. Mittra and (?). Pekel. A new look at the perfectly matched layer(PML) concept for the refiectionless absorption of electromagnetcis waves. IEEE Microwave Guided Wave Lett., 5:84-86, 1995.
[76] M. Cai and H. Zhou. Proof of perfectly mathed layer conditions in three dimensions. Electro. Lett, 31:1675-1676, 1995.
[77] F. L. Teixeira and W. C. Chew. Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates. IEEE Microwave Guided Wave Lett, 7:399-401, 1997.
[78] Z. Wu and J. Fang. Performance of the perfectly matched layer in modeling wave propagation in microwave and digital circuit interconnects, in 11th Annual Review of Progress in Applied Computational Electromagnetics, vol. 1, ACES, Monterey, CA, 504-511, Mar. 1995.
[79] M. Gribons, S.-K. Lee and A. C. Cangellaris. Modifications of Berengers perfectly matched layer for the absorption of electromagnetics waves in layered media, in 11th Annual Review of Progress in Applied Computational Electromagnetics, vol. 1, ACES, Monterey, CA, 498-503, Mar. 1995.
[80] W. C. Chew, W. H. Weedon and A. Sezginer. A 3-D perfectly matched medium by coordinate stretching and its absorption of static fields, in 11th Annual Review of Progress in Applied Computational Electromagnetics, vol. 1, ACES, Monterey, CA, 485-489, Mar. 1995.
[81] E. Kashdan and E. Turkel. Numerical solution of the time-dependent Maxwell's equations in spherical coordinates, in 19th Annual Review of Progress in Applied Computational Electromagnetics, 184-188, 2003.
[82] W. Chew and W. Weedon. A 3d perfectly matched medium from modified Maxwell's equations with stretched coordinates. Microwave Opt. Technol. Lett., 13(7):599-604, 1994.
[83] F. Collino. Perfectly matched layer for the paraxial equation. J. Comp. Phys., 131:164-170, 1997.
[84] F. Collino and P. Monk. The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput, 19(6):2061-2090, 1998.
[85] M. Lassas and E. Somersalo. On the existence and convergence of the solution of PML equations. Computing, 60:229-242, 1998.
[86] T. Hohage, F. Schmidt and L. Zschiedrich. Solving time-harmonic scattering problems based on the pole condition Ⅱ: Convergence of the PML method. SIAM J. Math. Anal, 35:547-560, 2003.
[87] M. Lassas and E. Somersalo. Analysis of the PML equations in general convex geometry. Proc. Royal Soc. Edinburgh Sect. A, 131(5):1183-1207, 2001.
[88] Z. Chen and X. Liu. An adaptive perfectly matched layer technique for time-harmonic scattering problems. SIAM J. Numer. Anal., 43:645-671, 2005.
[89] S. V. Tsynkov and E. Turkel. A Cartesian perfectly matched layer for the Helmholtz equation, in Absorbing Boundaries and Layers. Domain Decomposition Methods, Applications to Large Scale Computations, Nova Science. New York, 2001.
[90] Z. Chen and H. Wu. An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal., 41:799-826, 2003.
[91] P. Monk. Finite Element Methods for Maxwell's Equations. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, UK, 2003.
[92] G. Bao and H. Wu. Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell's equations. SIAM J. Numer. Anal., 43:2121-2143, 2005.
[93] J. H. Bramble and J. E. Pasciak. Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math. Comp., 76:597-614, 2007.
[94] E. Turkel and A. Yefet. Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math., 27:533-557, 1998.
[95] F. L. Teixeira and W. C. Chew. Advances in the theory of perfectly matched layers. Fast and Efficient Algorithms in Computational Electromagnetics (W.C.Chew et al, ed.), Artech House, Boston, 283-346, 2001.
[96] F. Q. Hu. On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. J. Comput. Phys., 129:201-219, 1996.
[97] F. Collino and P. Monk. Optimizing the perfectly matched layer. Comput. Methods Appl. Mech. Engrg., 164:157-171, 1998.
[98] D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. John Wiley&Sons, New York, 1983.
[99] A. Kirsch. Characterization of the shape of the scattering obstacle by the spectral data of the far-field operator. Inverse Problems, 14:1489-1512, 1998.
[100] A. Kirsch. Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory. Inverse Problems, 15:413-429, 1999.
[101] P. Hahner. An inverse problem in electrostatics. Inverse Problems, 15:961-975, 1999.
[102] M. Brühl and M. Hanke. Numerical implementation of two non-iterative methods for locating inclusions by impedence tomography. Inverse Problems, 16:1029-1042, 2000.
[103] A. Kirsch. New characterizations of solutions in inverse scattering theory. Appl. Anal., 76:319-350, 2000.
[104] M. Cheney. The linear sampling method and the music algorithm. Inverse Problems, 17:591-595, 2001.
[105] A. Kirsch. The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogenous media. Inverse Problems, 18:1025-1040, 2002.
[106] N. Grinberg. Obstacle visualization via the factorization method for the mixed boundary value problem. Inverse Problems, 18:1687-1704, 2002.
[107] N. I. Grinberg and A. Kirsch. The factorization method for obstacles with a-priori seperated sound-soft and sound-hard parts. Math. Comput. Simulations, 66:267-279, 2004.
[108] R. Kress. A sampling method for an inverse boundary value problem for harmonic vector fields. Ill-posed and Inverse problems ed S. Kabanikhin and V. G. Romanov(Utrecht:VSP), 243-262, 2002.
[109] R. Kress. A factorization method for an inverse Neumann problem for harmonic vector fields. Georgian Math. J., 10:549-560, 2003.
[110] R. Kress and L. Kiihn. Linear sampling methods for inverse boundary value problems in potential theory. Appl. Numer. Math., 43:161-173, 2002.
[111] N. I. Grinberg. The operator factorization method in inverse obstacle scattering. Inter. equ. oper. therory, 54:333-348, 2006.
[112] T. Arens and A. Kirsch. The factorization method in inverse scattering from periodic structures. Inverse Problems, 19:1195-1211, 2003.
[113] T. Arens and N. Grinberg. A complete factorization method for scattering by periodic surfaces. Computing, 75:111-132, 2004.
[114] A. Kirsch. The factorization method for Maxwell's equation. Inverse Problems, 20:s117-s134, 2004.
[115] A. Kirsch. The factorization method for a class of inverse elliptic problems. Math. Nachr., 278(3):258-277, 2005.
[116] B. Gebauer. Factorization method and inclusions of mixed type an inverse elliptic boundary value problem. Inverse problems and imaging, 2(3):355-372, 2008.
[117] R. Potthast. A survey on sampling and probe methods for inverse problems. Inverse Problems, 22:R1-R47, 2006.
[118] D. Colton and R. Kress. Using fundamental solutions in inverse scattering. Inverse Problems, 22:R49-R66, 2006.
[119] A. Kirsch and N. Grinberg. The Factorization Method for inverse problems. Oxford lecture series in mathematics and its applications·36, Oxford University Press, 2008.
[120] D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edn, Number 93 in Applied Mathematical Sciences, Springer. New York, 1998.
[121] D. Colton and R. Kress. Eigenvalues of the far field operator and inverse scattering theory. SIAM J. Math. Anal., 26:601-615, 1995.
[122] C. Z. A perfectly matched layer approach to the nonlinear Schrodinger wave equations. J. Comput. Phys., 227:537-556, 2007.
[123] A. Ahland, D. Schulz and E. Voges. Accurate mesh truncation for Schrodinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra. Phys. Rev. B., 60(8):5109-5112, 1999.
[124] X. Li. Inverse Scattering Problems for System of Differential Operators. University of California, Los Angeles, 2005.
[125] S. Agmon. Spectral properties of Schrodinger operators and scattering theory. Annali di Pisa., Serie Ⅳ(2):151-218, 1975.
[126] G. N. Watson. A Treatise on the Theorey of Bessel Function. Cambridge, 1992.
[127] 曾谨言． 量子力学(卷1)．科学出版社，北京，第四版， 2007.
[128] J. Weidmann. Zur Spektral theorie von Sturm-Liouville Operatoren. Math. Z., 98:263-302, 1967.
[129] S. Kotani and N. Ushiroya. One-dimensional Schrodinger operators with random decaying potentials. Commun. Math. Phys., 115:247-266, 1988.
[130] B. Simon. Schrodinger operators in the twenty-first century. Mathematical Physics(eds. A. Fokas, A. Grigoryan. T. Kibble and B. Zegarlinski), Imperial College Press, London, 283-288, 2000.
[131] D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Second Edition, 1983.
[132] A. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Springer, Berlin, 1996.
[133] V. S. Serov. Some inverse scattering problems for two-dimensional Schrodinger operators. Proceedings of the 5th international conference on inverse problems in engineering: theory and practice, Cambrige, UK, 11-15th July 2005.
[2] M. Read and B. Simon. Methods of Modern Mathematical Physics Ⅲ: Scattering Theory. Academic, New York, 1979.
[3] R. Milson. Liouville transformation and exactly solvable Schrodinger equations. Int. J. Theor. Phys., 37(6): 1735-1752, 1998.
[4] C. Lin. C. W. Wang and X. L. Zhang. Seried solution for the Schrodinger equation with a long-range spherically symmetric potential. Proceedings: Mathematical, Physical and Engineering Sciences, 458:1285-1290, 2002.
[5] K. Chadan and R. Kobayashi. New classes of potentials for which the radial Schrodinger equation can be solved at zero energy. J. Phys. A: Math. Gen., 39:3381-3396, 2006.
[6] K. Chadan and R. Kobayashi. New classes of potentials for which the radial Schrodinger equation can be solved at zero energy: Ⅱ. J. Phys. A: Math. Gen., 39:13691-13699, 2006.
[7] T. Fevens and H. Jiang. Absorbing boundary conditions for the Schrodinger equations. SIAM J. Sci. Comput, 21(1):255-282, 1999.
[8] M. Ehrhardt. Discrete transparent boundary conditions for Schroodinger-type equations for non-compactly supported initial data. Appl. Numer. Math., 58(5):660-673, 2008.
[9] M. Ehrhardt and A. Arnold. Discrete transparent boundary conditions for the Schro(?)dinger equations. Riv. Mat. Univ. Parma., 6:57-108, 2001.
[10] A. Schadle. Non-reflecting boundary conditions for the two-dimensional Schrodinger equation. Wave Motion., 35:181-188, 2002.
[11] X. Antoine, A. Arnold, C. Besse. M. Ehrhardt and A. Schadle. A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrodinger equations. Commun. Comput. Phys., 4(4):729-796, 2008.
[12] T. Monovasilis, Z. Kalogiratou and T. E. Simos. Numerical solution of the two-dimensional time independent Schr(?)dinger equation by symplectic schemes. Appl. Num. Anal. Comp. Math., 1(1):195-204, 2004.
[13] Z. Kalogiratou, T. Monovasilis and T. E. Simos. Numerical solution of the two-dimensional time independent Schrodinger equation with Numerov-type methods. J. Math. Chem., 37(3):271-279, 2005.
[14] A. S. Kadyrov, I. Bray. A. T. Stelbovics and B. Saha. Direct solution of the three-dimensional Lippmann-Schwinger equation. J. Phys. B: At. Mol. Opt. Phys., 38:509-515, 2005.
[15] V. Marchenko. Sturm-Liouville Operators and Applications. Birkhauser. Boston, 1986.
[16] B. Levitan. Inverse Strum-Liouville Problems. VNU Press, Utrecht, 1987.
[17] T. Aktosun and R. Weder. Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrodinger equation. Inverse Problems, 22:89-114, 2006.
[18] L. Amour and T. Raoux. Inverse spectral results for Schr(?)dinger operators on the unit interval with potential in L~p spaces. Inverse Problems, 23:2367-2373, 2007.
[19] T. Aktosun. Inverse Schr(?)dinger scattering on the line with partial knowledge of the potential. SIAM J. Appl. Math., 56(1):219-231, 1996.
[20] T. Aktosun and P. E. Sacks. Inversion of reflectivity data for nondecaying potentials. SIAM J. Appl Math., 60(4):1340-1356, 2000.
[21] T. Aktosun and R. Weder. Inverse scattering with partial information on the potential. J. Math. Anal Appl, 270:247-266, 2002.
[22] K. Chadan and P. C. Sabatier. Inverse Problems in Quantum Scattering Theroy. Springer-Verlag, New York, Second Edition, 1989.
[23] R. G. Newton. Inverse Schrodinger Scattering in Three Dimensions. Springer-Verlag, New York, 1989.
[24] A. G. Ramm. Inverse Problems. Springer, New York, 2005.
[25] T. Aktosun, and C. V. D Mee. Solution of the inverse scattering problem for the three-dimensional Schrodinger equation using a Fredholm integral equation. SIAM J. Math. Anal, 22(3):717-731, 1991.
[26] A. G. Ramm. Finding potential from the fixed-energy scattering data via D-N map. J.Inverse Ill-Posed Probl, 4(N2):145-152, 1996.
[27] Z. Q. Sun and G. Uhlmann. Inverse scattering for singular potentials in two dimensions. Trans. Amer. Math. Soc, 338(1):363-374, 1993.
[28] L. Paivarinta, V. S. Serov and E. Somersalo. Reconstruction of singularities of a scattering potential in two dimensions. Adv. Appl. Math., 15:97-113, 1994.
[29] V. S. Serov and L. Paivarinta. New estimates of the Green-Faddeev function and recovering of singularities in the two-dimensional Schroodinger operator with fixed energy. Inverse Prolems. 21:1291-1301, 2005.
[30] V. S. Serov. Some inverse scattering problems for two-dimensional Schrodinger operator. Proceedings of 5th International Conference on Inverse Problems in Engineering: Theory and Practic, Cambridge, UK, 11-15th July 2005.
[31] J. M. Reyes. Inverse backscattering for the Schrodinger equation in 2D. Inverse Problmes, 23:625-643, 2007.
[32] V. Serov. Inverse fixed angle scattering and backscattering problems in two dimensions. Inverse Problems, 24(6):065002(14pp), 2008.
[33] R. G. Novikov. The inverse scattering problem on a fixed energy level for the two-dimensional Schrodinger operator. J. Funct. Anal, 103:409-463, 1992.
[34] V. Isakov and Z. Sun. The inverse scattering at fixed energies in two dimensions. Indiana Univ. Math. J., 44(3):883-896, 1995.
[35] I. Kay and H. E. Morses. A simple verification of the Gelfand-Levitan equation for the three-dimensional scattering problem. Comm. Pure Appl. Math., 14:435-445, 1961.
[36] L. Faddeev. Three-dimensional inverse problem in the quantum theory of scattering. J. Math. Phys., 4:72-104, 1963.
[37] A. G. Ramm. Completness of the products of solutions to PDE and uniqueness theorems in inverse scattering. Inverse Problems, 3:L77-L82, 1987.
[38] A. G. Ramm. Uniqueness theorems for multidimensional inverse problems with unbouded coefficients. J. Math. Anal. Appl.. 136:586-574, 1988.
[39] A. G. Ramm. Recovery of the potential from fixed-energy scattering data. Inverse Problems, 4:877-886, 1988.
[40] A. G. Ramm. Completeness of the products of solutions of PDE and inverse problems. Inverse Problems, 6:643-664, 1990.
[41] A. G. Ramm. Symmetry properties of scattering amplitudes and applications to inverse problems. J. Math. Anal. Appl., 156:333-340, 1991.
[42] A. G. Ramm. Stability of the inversion of 3D fixed-frequency scattering data. J. Math. Anal. AppL 169:329-349, 1992.
[43] A. G. Ramm. Stability of the solution to 3D fixed-energy inverse scattering problem. J. Math. Anal. Appl., 170:1-15, 1992.
[44] A. G. Ramm. Stability estimates in inverse scattering. Acta Appl. Math., 28:1-42, 1992.
[45] A. G. Ramm. Stability of the solutions to 3D inverse scattering problems with fixed-energy data. Milan J. Math., 70:97-161, 2002.
[46] A. G. Ramm. Fixed-energy inverse scattering. Nonlinear Analysis, 69:971-978, 2008.
[47] A. G. Ramm. Inverse scattering problem with data at fixed energy and fixed incident direction. Nonlinear Analysis, 69:1478-1484, 2008.
[48] M. Horavóth and M. Kiss. The stability of inverse scattering with fixed energy. Inverse problems, 25(1):015001(14pp), 2009.
[49] R. G. Ramm. Fixed energy inverse scattering for noncompactly supported potentials. Mathl. Comput. Modelling, 18(1):57-64, 1993.
[50] R. G. Novikov. The inverse scattering problem at fixed energy for the three-dimensioanl Schr(?)dinger equation with an exponentially decreasing potential. Commun. Math. Phys., 161:569-595, 1994.
[51] M. Lassaut, S. Y. Larsen, S. A. Sofianos and J. C. Wallet. Construction of potentials using mixed scattering data. Inverse problems, 24(5):055014(15pp), 2008.
[52] F. Nicoleau. A constructive procedure to recover asympotics of short-range or long-range potentials. J. Differential Equations, 205:354-364, 2004.
[53] G. Nakamura, Z. Q. Sun and G. Uhlmann. Global identifiablity for an inverse problem for the Schrodinger equation in a magnetic field. Math. Ann., 303:377-388, 1995.
[54] G. Eskin and J. Ralson. Inverse scattering problem for the Schrodinger equation with magnetic potential at a fixed energy. Commun. Math. Phys., 173:199-224, 1995.
[55] A. Panchenko, An inverse problem for the magnetic Schrodinger equation and quasi-exponential solutions of nonsraooth partial differential equations, Inverse Problems, 18(2002), 1421-1434.
[56] G. Eskin. Inverse problems for the Schrodinger operators with electromagnetic potentials in domains with obstacles. Inverse Problems. 19:985-996, 2003.
[57] G. Eskin. Inverse boundary value problems and the Aharonov-Bohm effect. Inverse Problems, 19:49-62, 2003.
[58] G. Eskin. Inverse boundary value problems in domains with several obstacles. Inverse Problems, 20:1497-1516, 2004.
[59] G. Eskin. Global uniqueness in the inverse scattering problem for the Schrodinger operator with external Yang-Mills potentials. Commun. Math. Phys.. 222:503-531, 2001.
[60] G. Eskin. Inverse problems for Schrodinger equations with Yang-Mills potentials in domains with obstacles and the Aharonov-Bohm effect. Journal of Physics: Conference Series, 12:23-32, 2005.
[61] D. Givoli. Non reflecting boundary conditons. J. Comput. Phys., 94:1-29, 1991.
[62] M. J. Grote and J. E. Keller. Exact non-reflecting boundary condition for the time dependent wave equation. SIAM J. Appl. Math., 55:280-297, 1995.
[63] M. J. Grote and J. E. Keller. Nonreflecting boundary conditions for Maxwell equations. J. Comput. Phys., 139:327-342, 1998.
[64] S. V. Tsynkov. Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math., 27:465-532, 1998.
[65] J. P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114:185-200, 1994.
[66] J. P. Berenger. Three-dimentional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 127:363-379, 1996.
[67] D. S. Katz, E. T. Thiele and A. Taflove. Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FDTD meshes. IEEE Microwave Guided Wave Lett., 4:268-270, 1994.
[68] W. Andrew, C. Balanis, and P. Tirkas. A comparison of the Berenger perfectly matched layer and the Lindman high-order ABC's for the FDTD method. IEEE Microwave Guided Wave Lett, 5:192-194, 1995.
[69] B. Chen, D. G. Fang and B. H. Zhou. Modified Berenger PML absorbing boundary condition for FDTD meshes. IEEE Microwave Guided Wave Lett, 5:399-401, 1995.
[70] C. Eswarappa and W. J. R. Hoefer. Implementation of Berenger absorbing boundary conditions in TLM by interfacing FDTD perfectly matched layers. Electron. Lett, 31:1264-1266, 1995.
[71] J. De Moerloose and M. A. Stuchly. Behavior of Berenger's ABC for evanescent waves. IEEE Microwave Guided Wave Lett., 5:344-346, 1995.
[72] J. Fang and Z. Wu. Generalized perfectly matched layer-An extension of Berenger's perfectly matched layer boundary condition. IEEE Microwave Guided Wave Lett., 5:451-453, 1995.
[73] Z. Wu and J. Fang. Numerical implementation and performance of perfectly matched layer boundary condition for waveguide structures. IEEE Trans. Microwave Theory Tech., 43:2676-2683, 1995.
[74] C. M. Rappaport. Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space. IEEE Microwave Guided Wave Lett, 5:90-92, 1995.
[75] R. Mittra and (?). Pekel. A new look at the perfectly matched layer(PML) concept for the refiectionless absorption of electromagnetcis waves. IEEE Microwave Guided Wave Lett., 5:84-86, 1995.
[76] M. Cai and H. Zhou. Proof of perfectly mathed layer conditions in three dimensions. Electro. Lett, 31:1675-1676, 1995.
[77] F. L. Teixeira and W. C. Chew. Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates. IEEE Microwave Guided Wave Lett, 7:399-401, 1997.
[78] Z. Wu and J. Fang. Performance of the perfectly matched layer in modeling wave propagation in microwave and digital circuit interconnects, in 11th Annual Review of Progress in Applied Computational Electromagnetics, vol. 1, ACES, Monterey, CA, 504-511, Mar. 1995.
[79] M. Gribons, S.-K. Lee and A. C. Cangellaris. Modifications of Berengers perfectly matched layer for the absorption of electromagnetics waves in layered media, in 11th Annual Review of Progress in Applied Computational Electromagnetics, vol. 1, ACES, Monterey, CA, 498-503, Mar. 1995.
[80] W. C. Chew, W. H. Weedon and A. Sezginer. A 3-D perfectly matched medium by coordinate stretching and its absorption of static fields, in 11th Annual Review of Progress in Applied Computational Electromagnetics, vol. 1, ACES, Monterey, CA, 485-489, Mar. 1995.
[81] E. Kashdan and E. Turkel. Numerical solution of the time-dependent Maxwell's equations in spherical coordinates, in 19th Annual Review of Progress in Applied Computational Electromagnetics, 184-188, 2003.
[82] W. Chew and W. Weedon. A 3d perfectly matched medium from modified Maxwell's equations with stretched coordinates. Microwave Opt. Technol. Lett., 13(7):599-604, 1994.
[83] F. Collino. Perfectly matched layer for the paraxial equation. J. Comp. Phys., 131:164-170, 1997.
[84] F. Collino and P. Monk. The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput, 19(6):2061-2090, 1998.
[85] M. Lassas and E. Somersalo. On the existence and convergence of the solution of PML equations. Computing, 60:229-242, 1998.
[86] T. Hohage, F. Schmidt and L. Zschiedrich. Solving time-harmonic scattering problems based on the pole condition Ⅱ: Convergence of the PML method. SIAM J. Math. Anal, 35:547-560, 2003.
[87] M. Lassas and E. Somersalo. Analysis of the PML equations in general convex geometry. Proc. Royal Soc. Edinburgh Sect. A, 131(5):1183-1207, 2001.
[88] Z. Chen and X. Liu. An adaptive perfectly matched layer technique for time-harmonic scattering problems. SIAM J. Numer. Anal., 43:645-671, 2005.
[89] S. V. Tsynkov and E. Turkel. A Cartesian perfectly matched layer for the Helmholtz equation, in Absorbing Boundaries and Layers. Domain Decomposition Methods, Applications to Large Scale Computations, Nova Science. New York, 2001.
[90] Z. Chen and H. Wu. An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal., 41:799-826, 2003.
[91] P. Monk. Finite Element Methods for Maxwell's Equations. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, UK, 2003.
[92] G. Bao and H. Wu. Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell's equations. SIAM J. Numer. Anal., 43:2121-2143, 2005.
[93] J. H. Bramble and J. E. Pasciak. Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math. Comp., 76:597-614, 2007.
[94] E. Turkel and A. Yefet. Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math., 27:533-557, 1998.
[95] F. L. Teixeira and W. C. Chew. Advances in the theory of perfectly matched layers. Fast and Efficient Algorithms in Computational Electromagnetics (W.C.Chew et al, ed.), Artech House, Boston, 283-346, 2001.
[96] F. Q. Hu. On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. J. Comput. Phys., 129:201-219, 1996.
[97] F. Collino and P. Monk. Optimizing the perfectly matched layer. Comput. Methods Appl. Mech. Engrg., 164:157-171, 1998.
[98] D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. John Wiley&Sons, New York, 1983.
[99] A. Kirsch. Characterization of the shape of the scattering obstacle by the spectral data of the far-field operator. Inverse Problems, 14:1489-1512, 1998.
[100] A. Kirsch. Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory. Inverse Problems, 15:413-429, 1999.
[101] P. Hahner. An inverse problem in electrostatics. Inverse Problems, 15:961-975, 1999.
[102] M. Brühl and M. Hanke. Numerical implementation of two non-iterative methods for locating inclusions by impedence tomography. Inverse Problems, 16:1029-1042, 2000.
[103] A. Kirsch. New characterizations of solutions in inverse scattering theory. Appl. Anal., 76:319-350, 2000.
[104] M. Cheney. The linear sampling method and the music algorithm. Inverse Problems, 17:591-595, 2001.
[105] A. Kirsch. The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogenous media. Inverse Problems, 18:1025-1040, 2002.
[106] N. Grinberg. Obstacle visualization via the factorization method for the mixed boundary value problem. Inverse Problems, 18:1687-1704, 2002.
[107] N. I. Grinberg and A. Kirsch. The factorization method for obstacles with a-priori seperated sound-soft and sound-hard parts. Math. Comput. Simulations, 66:267-279, 2004.
[108] R. Kress. A sampling method for an inverse boundary value problem for harmonic vector fields. Ill-posed and Inverse problems ed S. Kabanikhin and V. G. Romanov(Utrecht:VSP), 243-262, 2002.
[109] R. Kress. A factorization method for an inverse Neumann problem for harmonic vector fields. Georgian Math. J., 10:549-560, 2003.
[110] R. Kress and L. Kiihn. Linear sampling methods for inverse boundary value problems in potential theory. Appl. Numer. Math., 43:161-173, 2002.
[111] N. I. Grinberg. The operator factorization method in inverse obstacle scattering. Inter. equ. oper. therory, 54:333-348, 2006.
[112] T. Arens and A. Kirsch. The factorization method in inverse scattering from periodic structures. Inverse Problems, 19:1195-1211, 2003.
[113] T. Arens and N. Grinberg. A complete factorization method for scattering by periodic surfaces. Computing, 75:111-132, 2004.
[114] A. Kirsch. The factorization method for Maxwell's equation. Inverse Problems, 20:s117-s134, 2004.
[115] A. Kirsch. The factorization method for a class of inverse elliptic problems. Math. Nachr., 278(3):258-277, 2005.
[116] B. Gebauer. Factorization method and inclusions of mixed type an inverse elliptic boundary value problem. Inverse problems and imaging, 2(3):355-372, 2008.
[117] R. Potthast. A survey on sampling and probe methods for inverse problems. Inverse Problems, 22:R1-R47, 2006.
[118] D. Colton and R. Kress. Using fundamental solutions in inverse scattering. Inverse Problems, 22:R49-R66, 2006.
[119] A. Kirsch and N. Grinberg. The Factorization Method for inverse problems. Oxford lecture series in mathematics and its applications·36, Oxford University Press, 2008.
[120] D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edn, Number 93 in Applied Mathematical Sciences, Springer. New York, 1998.
[121] D. Colton and R. Kress. Eigenvalues of the far field operator and inverse scattering theory. SIAM J. Math. Anal., 26:601-615, 1995.
[122] C. Z. A perfectly matched layer approach to the nonlinear Schrodinger wave equations. J. Comput. Phys., 227:537-556, 2007.
[123] A. Ahland, D. Schulz and E. Voges. Accurate mesh truncation for Schrodinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra. Phys. Rev. B., 60(8):5109-5112, 1999.
[124] X. Li. Inverse Scattering Problems for System of Differential Operators. University of California, Los Angeles, 2005.
[125] S. Agmon. Spectral properties of Schrodinger operators and scattering theory. Annali di Pisa., Serie Ⅳ(2):151-218, 1975.
[126] G. N. Watson. A Treatise on the Theorey of Bessel Function. Cambridge, 1992.
[127] 曾谨言． 量子力学(卷1)．科学出版社，北京，第四版， 2007.
[128] J. Weidmann. Zur Spektral theorie von Sturm-Liouville Operatoren. Math. Z., 98:263-302, 1967.
[129] S. Kotani and N. Ushiroya. One-dimensional Schrodinger operators with random decaying potentials. Commun. Math. Phys., 115:247-266, 1988.
[130] B. Simon. Schrodinger operators in the twenty-first century. Mathematical Physics(eds. A. Fokas, A. Grigoryan. T. Kibble and B. Zegarlinski), Imperial College Press, London, 283-288, 2000.
[131] D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Second Edition, 1983.
[132] A. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Springer, Berlin, 1996.
[133] V. S. Serov. Some inverse scattering problems for two-dimensional Schrodinger operators. Proceedings of the 5th international conference on inverse problems in engineering: theory and practice, Cambrige, UK, 11-15th July 2005.