Transmission Eigenvalues and the Riemann Zeta Function in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type Ⅰ
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摘要
We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of Type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles,in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups.For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.
We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of Type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles,in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups.For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.
We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of Type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles,in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups.For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.
引文
[1]Blasten,E.,Paivarinta,L.,Sylvester,J.:Corners always scatter.Commun.Math.Phys.,331,725-753(2014)
[2]Borthwick,D.:Spectral Theory of Infinite-Area Hyperbolic Surfaces,Birkhauser,Boston,2000
[3]Cakoni,F.,Colton,D.,Haddar,H.:Inverse Scattering Theory and Transmission Eigenvalues,CBMS Series,SIAM Publications,88,2016
[4]Cakoni,F.,Gintides,D.,Haddar,H.:The existence of an infinite discrete set of transmission eigenvalues.SIAM J.Math.Anal.,42,237-255(2010)
[5]Colton,D.,Kress,R.:Inverse Acoustic and Electromagnetic Scattering Theory,3rd Edition,Springer,New York,2013
[6]Colton,D.,Leung,Y.J.:Complex eigenvalues and the inverse spectral problem for transmission eigenvalues.Inverse Problems,29,104008(2013)
[7]Colton,D.,Leung,Y.J.,Meng,S.:Distribution of complex transmission eigenvalues for spherically stratified media.Inverse Problems,31,035006(2015)
[8]Colton,D.,Leung,Y.J.:The existence of complex transmission eigenvalues for spherically stratified media.Appl.Anal.,96,39-47(2017)
[9]Dyatlov,S.,Zworski,M.:Mathematical Theory of Scattering Resonances,book in progress
[10]Faddeev,L.D.,Pavlov,B.S.:Scattering theory and automorphic functions.Proc.Steklov Inst.Math.,27,161-193(1972)
[11]Greenleaf,A.,Kurylev,Y.,Lassas,M.,et al.:Invisibility and inverse problems.Bull.Amer.Math.Soc.,46,55-97(2009)
[12]Guillop′e,L.,Zworski,M.:Scattering asymptotics for Riemann surfaces.Ann.of Math.,145,597-660(1997)
[13]Hardy,G.H.,Wright,E.M.:An Introduction to the Theory of Numbers,6th Edition additional authors A.Wiles,R.Heath-Brown and J.Silverman,Oxford University Press,New York,2008
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[15]Hitrik,M.,Krupchyk,K.,Ola,P.,et al.:The interior transmission problem and bounds of transmission eigenvalues.Math.Res.Lett.,18,279-293(2011)
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[19]Iwaniec,H.:Spectral Methods of Automorphic Forms,Graduate Studies in Math.,Vol.53,American Mathematical Society,Providence,RI,2002
[20]Katok,S.:Fuchsian Groups,Chicago Lectures in Mathematics,University of Chicago Press,Chicago,IL,1992
[21]Kloosterman,H.D.:On the representation of numbers in the form ax2+by2+cz2+dt2.Acta Math.,49,407-464(1926)
[22]Kirsch,A.,Lechleiter,A.:The inside-outside duality for scattering problems by inhomogeneous media.Inverse Problems,29,104011(2013)
[23]Lakshtanov,E.,Vainberg,B.:Remarks on interior transmission eigenvalues,Weyl formula and branching billiards.J.Phys.A,45,125202(2012)
[24]Lax,P.D.,Phillips,R.S.:Scattering Theory,2nd Edition,Pure and Applied Mathematics,26.Academic Press,Inc.,Boston,MA,1989
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[27]Melrose,R.B.:Geometric Scattering Theory,Cambridge University Press,Cambridge,1995
[28]Phillips,R.S.,Sarnak,P.:On cusp forms for cofinite subgroups of P SL2(R).Invent.Math.,80,339-364(1980)
[29]Robbiano,L.:Spectral analysis of the interior transmission eigenvalue problem.Inverse Problems,29,paper 104001(2013)
[30]Rudin,W.:Real and Complex Analysis,3rd Edition,McGraw-Hill Book Co.,New York,1987
[31]Selberg,A.:Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series.J.Indian Math.Soc.,20,47-87(1956)
[32]Shimura,G.:Introduction to the Arithmetic Theory of Automorphic Functions,Princeton Univ.Press,Princeton,NJ,1971
[33]Vodev,G.:Parabolic transmission eigenvalue-free regions in the degenerate isotropic case.Asympt.Anal.,106,147-168(2018)
[34]Vodev,G.:High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues.Anal.PDE,11,213-236(2018)
[35]Zworski,M.:Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces.Invent.Math.,136,353-409(1999)
[2]Borthwick,D.:Spectral Theory of Infinite-Area Hyperbolic Surfaces,Birkhauser,Boston,2000
[3]Cakoni,F.,Colton,D.,Haddar,H.:Inverse Scattering Theory and Transmission Eigenvalues,CBMS Series,SIAM Publications,88,2016
[4]Cakoni,F.,Gintides,D.,Haddar,H.:The existence of an infinite discrete set of transmission eigenvalues.SIAM J.Math.Anal.,42,237-255(2010)
[5]Colton,D.,Kress,R.:Inverse Acoustic and Electromagnetic Scattering Theory,3rd Edition,Springer,New York,2013
[6]Colton,D.,Leung,Y.J.:Complex eigenvalues and the inverse spectral problem for transmission eigenvalues.Inverse Problems,29,104008(2013)
[7]Colton,D.,Leung,Y.J.,Meng,S.:Distribution of complex transmission eigenvalues for spherically stratified media.Inverse Problems,31,035006(2015)
[8]Colton,D.,Leung,Y.J.:The existence of complex transmission eigenvalues for spherically stratified media.Appl.Anal.,96,39-47(2017)
[9]Dyatlov,S.,Zworski,M.:Mathematical Theory of Scattering Resonances,book in progress
[10]Faddeev,L.D.,Pavlov,B.S.:Scattering theory and automorphic functions.Proc.Steklov Inst.Math.,27,161-193(1972)
[11]Greenleaf,A.,Kurylev,Y.,Lassas,M.,et al.:Invisibility and inverse problems.Bull.Amer.Math.Soc.,46,55-97(2009)
[12]Guillop′e,L.,Zworski,M.:Scattering asymptotics for Riemann surfaces.Ann.of Math.,145,597-660(1997)
[13]Hardy,G.H.,Wright,E.M.:An Introduction to the Theory of Numbers,6th Edition additional authors A.Wiles,R.Heath-Brown and J.Silverman,Oxford University Press,New York,2008
[14]Hejhal,D.A.:The Selberg Trace Formula for P SL(2,R),Vol.2.Lecture Notes in Mathematics,Vol.1001,Springer-Verlag,Berlin,1983
[15]Hitrik,M.,Krupchyk,K.,Ola,P.,et al.:The interior transmission problem and bounds of transmission eigenvalues.Math.Res.Lett.,18,279-293(2011)
[16]Huxley,M.N.:Scattering Matrices for Congruence Subgroups,Modular Forms(R.Rankin ed.),Ellis Horwood Ser.Math.Appl.,Halsted Press,New York,157-196,1984
[17]Iwaniec,H.,Kowalski,E.:Analytic Number Theory,AMS Colloquium Publications,Vol.53,Providence RI,2004
[18]Iwaniec,H.:Lectures on the Riemann Zeta Function,University Lecture Series,Vol.62,American Mathematical Society,Providence,RI,2014
[19]Iwaniec,H.:Spectral Methods of Automorphic Forms,Graduate Studies in Math.,Vol.53,American Mathematical Society,Providence,RI,2002
[20]Katok,S.:Fuchsian Groups,Chicago Lectures in Mathematics,University of Chicago Press,Chicago,IL,1992
[21]Kloosterman,H.D.:On the representation of numbers in the form ax2+by2+cz2+dt2.Acta Math.,49,407-464(1926)
[22]Kirsch,A.,Lechleiter,A.:The inside-outside duality for scattering problems by inhomogeneous media.Inverse Problems,29,104011(2013)
[23]Lakshtanov,E.,Vainberg,B.:Remarks on interior transmission eigenvalues,Weyl formula and branching billiards.J.Phys.A,45,125202(2012)
[24]Lax,P.D.,Phillips,R.S.:Scattering Theory,2nd Edition,Pure and Applied Mathematics,26.Academic Press,Inc.,Boston,MA,1989
[25]Lax,P.D.,Phillips,R.S.:Scattering Theory for Automorphic Functions.Ann.of Math.Stud,87,Princeton Univ.Press,Princeton,NJ,1976
[26]Lax,P.D.,Phillips,R.S.:Scattering theory for automorphic functions.Bull.Amer.Math.Soc.,2,261-295(1980)
[27]Melrose,R.B.:Geometric Scattering Theory,Cambridge University Press,Cambridge,1995
[28]Phillips,R.S.,Sarnak,P.:On cusp forms for cofinite subgroups of P SL2(R).Invent.Math.,80,339-364(1980)
[29]Robbiano,L.:Spectral analysis of the interior transmission eigenvalue problem.Inverse Problems,29,paper 104001(2013)
[30]Rudin,W.:Real and Complex Analysis,3rd Edition,McGraw-Hill Book Co.,New York,1987
[31]Selberg,A.:Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series.J.Indian Math.Soc.,20,47-87(1956)
[32]Shimura,G.:Introduction to the Arithmetic Theory of Automorphic Functions,Princeton Univ.Press,Princeton,NJ,1971
[33]Vodev,G.:Parabolic transmission eigenvalue-free regions in the degenerate isotropic case.Asympt.Anal.,106,147-168(2018)
[34]Vodev,G.:High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues.Anal.PDE,11,213-236(2018)
[35]Zworski,M.:Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces.Invent.Math.,136,353-409(1999)