×

On the Lax-Phillips scattering matrix of the abstract wave equation. (English) Zbl 1482.47020

Summary: We study the dependence of singularities of scattering matrices of the abstract wave equation on the choice of asymptotically equivalent outgoing/incoming subspaces. We apply the obtained results to the radial wave equation with nonlocal potential. In the latter case, the concept of associated inner function – introduced by Douglas, Shapiro, and Shields in [R. G. Douglas et al., Ann. Inst. Fourier 20, No. 1, 37–76 (1970; Zbl 0186.45302)] – plays an essential role.

MSC:

47A40 Scattering theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
35L90 Abstract hyperbolic equations

Citations:

Zbl 0186.45302
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] V. M. Adamyan, Nondegenerate unitary couplings of semiunitary operators (in Russian), Funktsional. Anal. i Prilozhen. 7 (1973), no. 4, 1-16; English translation in Funct. Anal. Appl. 7 (1973), no. 4, 255-267. · Zbl 0289.47008
[2] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, I, Monagr. Stud. Math. 9, Pitman, Boston, 1981. Theory of Linear Operators in Hilbert Space, II, Monagr. Stud. Math. 10. · Zbl 0467.47001
[3] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms, I, McGraw-Hill, New York, 1954. · Zbl 0055.36401
[4] M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-adjoint Operators in Hilbert Space, Math. Appl. (Soviet Ser.) 5, Reidel, Dordrecht, 1987. · Zbl 0744.47017
[5] R. G. Douglas, H. S. Shapiro, and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970), no. 1, 37-76. · Zbl 0186.45302 · doi:10.5802/aif.338
[6] A. V. Kuzhel and S. A. Kuzhel, Regular Extensions of Hermitian Operators, VSP, Utrecht, 1998. · Zbl 0930.47003
[7] S. A. Kuzhel, About dependence of the Lax-Phillips scattering matrix on the choice of the incoming and outgoing subspaces, Methods Funct. Anal. Topology 7 (2001), no. 1, 45-52. · Zbl 0980.47009
[8] S. A. Kuzhel, Nonlocal perturbations of the radial wave equation: Lax-Phillips approach, Methods Funct. Anal. Topology 8 (2002), no. 2, 59-68. · Zbl 1006.47013
[9] S. A. Kuzhel, On an inverse problem in the Lax-Phillips scattering scheme for a class of operator-differential equations (in Russian), Algebra i Analiz 13 (2002), no. 1, 60-83; English translation in St. Petersburg Math. J. 13 (2002), no. 1, 41-56. · Zbl 0992.35060
[10] S. A. Kuzhel, On conditions for the applicability of the Lax-Phillips scattering scheme to an investigation of an abstract wave equation (in Ukrainian), Ukraïn. Mat. Zh. 55 (2003), no. 5, 621-630; English translation in Ukrainian Math. J. 55 (2003), no. 5, 749-760. · Zbl 1100.47521
[11] P. D. Lax and R. F. Phillips, Scattering Theory for Automorphic Functions, Ann. of Math. Stud. 87, Princeton Univ. Press, New York, 1976. · Zbl 0362.10022
[12] P. D. Lax and R. F. Phillips, Scattering Theory, 2nd ed., with appendices by C. S. Morawetz and G. Schmidt, Pure Appl. Math. 26, Academic Press, Boston, 1989. · Zbl 0697.35004
[13] R. A. Martínez-Avendaño and P. Rosenthal, An Introduction to Operators on the Hardy-Hilbert Space, Grad. Texts in Math. 237, Springer, New York, 2007. · Zbl 1116.47001
[14] N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading, I, Math. Surveys Monogr. 92, Amer. Math. Soc., Providence, 2002. · Zbl 1007.47001
[15] A. Plessner, Zur Spektraltheorie maximaler Operatoren, Dokl. Akad. Nauk SSSR 22 (1939), no. 5, 227-230.
[16] A. Plessner, Über Funktionen eines maximalen Operatoren, Dokl. Akad. Nauk SSSR 23 (1939), no. 4, 327-330.
[17] A. Plessner, Über halbunitäre Operatoren, Dokl. Akad. Nauk SSSR 25 (1939), no. 9, 710-712.
[18] B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, 2nd ed., Universitext, Springer, New York, 2010. · Zbl 1234.47001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.