On the Density of the Mass Distribution of a String at the Origin
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- 作者：Israel S. Kac ; Vyacheslav N. Pivovarchik
- 关键词：Primary 34A55 ; Secondary 34B09 ; 34L20 ; Spectral function ; Dirichlet boundary condition ; Neumann boundary condition ; regular string ; singular string ; Krein’s string
- 刊名：Integral Equations and Operator Theory
- 出版年：2015
- 出版时间：April 2015
- 年：2015
- 卷：81
- 期：4
- 页码：581-599
- 全文大小：319 KB
- 参考文献：1. Barcilon, V. (1983) Explicit solution of the inverse problem for a vibrating string. J. Math. Anal. Appl. 93: pp. 222-234 CrossRef
2. Shen, C.-L. (2005) The Liouville transformation, the density function, and the complete transformability problem of the potential equation. Inverse Probl. 21: pp. 615-634 CrossRef
3. Shen, C.-L. (2005) On the Barcilon formula for the string equation with a piecewise continuous density function. Inverse Probl. 21: pp. 635-655 CrossRef
4. Dehghan, M., Jodayree, A. (2013) A Reconstructing the potential function of the indefinite Sturm-Liouville problem using infinite product forms. Electron. J. Differ. Equ. 2013: pp. 116
5. Kac, I.S., Krein, M.G. (1974) R-functions analytic functions mapping the upper half plane into itself. Am. Math. Soc. Transl. Ser. 2 103: pp. 1-18
6. Kac, I.S., Krein, M.G. (1974) On spectral functions of the string Amer. Math. Soc. Transl. Ser. 2 103: pp. 19-102
7. Kac, I.S.: Generalization of an asymptotic formula of V.A. Marchenko for spectral functions of a second-order boundary value problem. Izv. Akad. Nauk SSSR, Ser. Mat. 37(2), 422-36 (1973) (in Russian) [English transl. in Math. USSR Izvestija 7(2), 424-8]
8. Kasahara, Y. (1975) Spectral theory of generalized second order differential operators and its applications to Markov processes. Japan J. Math. 1: pp. 67-84
9. Krein, M.G.: On some cases of effective determination of the density of an inhomogeneous cord from its spectral function. Doklady Akad. Nauk SSSR 93, 617-20 (1953) (in Russian)
10. Levin, B.Ja.: Distribution of Zeros of Entire Functions, Trans. Math. Monographs, vol. 5. American Mathematical Society, Providence (1980)
11. Dym, H., McKean, H.P.: Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Dover, NY (2008)
12. Krein, M.G.: On a generalization of Stieltjes investigations. Doklady Acad. Nauk SSSR 87, 881-84 (1952) (in Russian)
13. Gohberg, I.C., Krein, M.G.: Theory and Applications of Volterra Operators in Hilbert Space, AMS Transl. Math. Monographs, vol. 24 (1970)
14. Krein, M.G.: Chebyshev–Markov inequalities in spectral theory of string. Matem. Issledovaniya (Institute of Mathematics Akad. Sci. Moldavian SSR) 5(1), 78-01 (1970) (in Russian)
15. Krein, M.G.: Analogue of Chebyshev–Markov inequality in one-dimensional boundary value problem Doklady AN SSSR 89(1), 5- (1953) (in Russian) - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Analysis - 出版者：Birkh盲user Basel
- ISSN：1420-8989
文摘
In V. Barcilon (J Math Anal Appl 93:222-34, 1983) two boundary value problems were considered generated by the differential equation of a string $$y^{\prime\prime} + \lambda p(x)y = 0, \,\, 0 \leq x \leq L with continuous real function p(x) (density of the string) and the boundary conditions y(0) =? y(L)?=?0 for the first problem and \({y^{\prime}(0) = y(L) = 0}\) for the second one. In the above paper the following formula was stated $$p(0) = \frac{1}{L^2\mu_1} \mathop{\prod} \limits_{n=1}^{\infty}\frac{\lambda_n^2}{\mu_n \mu_{n+1}} \quad \quad \quad (**)$$ where \({\{\lambda_k\}_{k=1}^{\infty}}\) is the spectrum of the first boundary value problem and \({\{\mu_k\}_{k=1}^{\infty}}\) of the second one. A rigorous proof of (**) was given in C.-L. Shen (Inverse Probl 21:635-55, 2005) under the more restrictive conditions of piecewise continuity of \({p^{\prime}(x)}\) . In this paper (**) was deduced using $$p(0)=\lim\limits_{\lambda\to +\infty} \left(\frac{\phi(L,-\lambda)} {\lambda^{\frac{1}{2}} \psi(L,-\lambda)} \right)^2 \quad \quad \quad \quad (\ast\ast\ast)$$ where \({\phi(x,\lambda)}\) is the solution of (*) which satisfies the boundary conditions \({\phi(0) - 1 = \phi^{\prime}(0) = 0 \,\,{\rm and}\,\, \psi(x,\lambda)}\) is the solution of (*) which satisfies \({\psi(0) = \psi^{\prime}(0) - 1 = 0}\) . In our paper we prove that (***) is true for the so-called M.G. Krein strings which may have any nondecreasing mass distribution function M(x) with finite nonzero \({M^{\prime}(0)}\) . Also we show that (**) is true for a wide class of strings including those for which M(x) is a singular function, i.e. \({M^{\prime}(x) = p(x)\mathop{=} \limits^{a.e.}0}\) .