Scattering and characteristic functions of a dissipative operator generated by a system of equations

In this paper, we consider a system of first-order equations with the same eigenvalue parameter together with dissipative boundary conditions. Applying Lax-Phillips scattering theory and Sz.-Nagy-Foiaş model operator theory we prove a completeness theorem.

PDF

___

[1] Akhiezer NI. The Classical Moment Problem and Some Related Questions in Analysis. Hafner Publishing Company. New York: 1965.
[2] Allahverdiev BP. On dilation theory and spectral analysis of dissipative Schrödinger operators in Weyl’s limit-circle case. Mathematics of the USSR Izvestiya 1991; 36: 247-262.
[3] Allahverdiev BP, Canoglu A. Spectral analysis of dissipative Schrodinger operators. Proceedings of the Royal Society of Edinburgh Section A 1997; 127A: 1113-1121.
[4] Allahverdiev BP, Bairamov E, Ugurlu E. Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions. Journal of Mathematical Analysis and Applications 2013; 401: 388-396.
[5] Askey R. Orthogonal Polynomials and Special Functions. Society for Industrial and Applied Mathematics. Bristol: 1975.
[6] Atkinson FV. Discrete and Continuous Boundary Problems. Academic Press. New York: 1964.
[7] Chihara TS. An Introduction to Orthogonal Polynomials. Gordon and Breach Science Publication. New York: 1978.
[8] Gesztesy F, Deift P, Galvez C, Perry P, Schlag W. Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proceedings of Symposia in Pure Mathematics. Vol:76, Part 2. American Mathematical Society Providence. Rhode Island: 2007.
[9] Gorbachuk VI, Gorbachuk ML. Boundary Value Problems for Ordinary Differential Equations.Kluwer Academic Publishers, Dordrecht: 1991.
[10] Ismail MEH. Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press. Cambridge: 2005.
[11] Krein M. Continuous analogous of proposition on polynomials orthogonal on the unit circle. Doklady Akademii Nauk SSSR 1955; 105, 637-640.
[12] Lax PD, Phillips RS. Scattering Theory. New York: 1967.
[13] Nagy BSz, Foiaş C. Harmonic Analysis of Operators on Hilbert Space. Academia Kioda. Budapest: 1970.
[14] Naimark MA. Linear Differential Operators. Ungar, New York: 1967.
[15] Pavlov BS. Spectral analysis of a dissipative singular Schrödinger operator in terms of a functional model. In: Shubin MA. (eds) Partial Differential Equations VIII, Encyclopaedia of Mathematical Sciences. Volume 65. Springer, Berlin: 1996.
[16] Sakhnovich LA. Spetral theory of a class of canonical differential systems.Functional Analysis and Its Applications 2000; 34, 119-128.
[17] Simon B. Orthogonal Polynomials on the Unit Circle Part 1: Classical Theory. American Mathematical Society Providence. Rhode Island: 2004.
[18] Szego G. Orthogonal Polynomials. American Mathematical Society Providence. Rhode Island: 1939.
[19] Uğurlu E, Bairamov E. Spectral analysis of eigenparameter dependent boundary value transmission problems. Journal of Mathematical Analysis and Applications 2014; 413: 482-494.
[20] Uğurlu E. Extensions of a minimal third-order formally symmetric operator. Bulletin of the Malaysian Mathematical Sciences Society 2020; 43: 453-470.
[21] Uğurlu E. The spectral analysis of a system of first-order equations with dissipative boundary conditions. Mathematical Methods in the Applied Sciences 2021; 44: 11046-11058.