Weyl-Titchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control

Complex Analysis and Operator Theory

Volumn 3, Issue 1, 2009, Pages 289-322

  Ryzhov, Vladimir ^a  

^a Bridges Transitions Inc   (Canada)

Author keywords

Abstract boundary value problem; Green formula; linear systems with boundary control; operator colligations; Schr dinger operator; Weyl Titchmarsh function

Indexed keywords

EID: 61549085915     PISSN: 16618254     EISSN: 16618262     Source Type: Journal    
DOI: 10.1007/s11785-007-0034-x     Document Type: Article

References (45)

1. M.S. Agranovich. Elliptic Boundary Problems. Encyclopaedia of Mathematical Sciences, Vol. 79, Partial Differential Equations IX. (Springer-Verlag, 1997).
2. S. Albeverio and P. Kurasov. Singular Perturbations of Differential Operators. (Cambridge University Press, 2000).
3. A. Alonso and B. Simon. The Birman-Krein-Vishik theory of self adjoint extensions of semibounded operators. J. Oper. Theory 4 (1980) 251-270.
4. W. O. Amrein and D. B. Pearson. M-operators: a generalisation of Weyl-Titchmarsh theory. J. Comput. Appl. Math. 171, 1-2 (2004) 1-26.
5. D. Z. Arov. Passive linear stationary dynamic systems. (Russian) Sibirsk. Mat. Zh. 20, 2 (1979), 211-228.;
6. English Transl.: Siberian Mathematical Journal. 20 (2) 149-162.
7. F. V. Atkinson. Discrete and Continuous Boundary Problems. (Academic Press, New York, London, 1964).
8. J. Behrndt, M. Langer. Boundary value problems for elliptic partial differential operators on bounded domains. Preprint. (2006) p. 35.
9. M. I. Belishev. Boundary control in reconstruction of manifolds and metrics (the BC-method). Inverse Problems 13 (1997) R1-R45.
10. M.I. Belishev, V. A. Ryzhov and V. B. Filippov. A spectral variant of the BCmethod: theory and numerical experiment. Phys. Dokl. 39 (1994) 466-470.
11. C. W. Bennewitz, W. N. Everitt. Some remarks on the Titchmarsh-Weyl coefficient. In Proc. of Plejel Conf (Uppsala).
12. S. Bergman, M. Schiffer. Kernel Functions and Elliptic Differential Equations in Mathematical Physics, (Academic Press, New York, 1953).
13. M. S. Birman. On the theory of self-adjoint extensions of positive definite operators. Mat. Sb. 38:80 (1956) 431-450. (Russian).
14. M. S. Birman. Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions. Vestnik Leningrad. Univ. 17, 1 (1962) 22-55. (Russian).
15. J. F. Brasche, P. Exner, Yu. A. Kuperin, and P. Šeba. Schrödinger operators with singular interactions, J. Math. Anal. Appl. 184 (1994) 112-139.
16. M.S. Brodskiǐ Triangular and Jordan Representations of Linear Operators, (American Mathematical Society, Providence, Rhode Island, 1971).
17. F. E. Browder. On the spectral theory of elliptic diferential operators. I, Math. Annalen 142 (1961) 22-130.
18. B. M. Brown and M. Marletta. Spectral inclusion and spectral exacteness for PDEs on exterior domains. IMA J. Num. Anal. 24 (2004) 21-43.
19. D. Colton and R. Kress. Integral Equation Methods in Scattering Theory. (John Wiley & Sons, New York, 1983).
20. R. F. Curtain. Reciprocals of regular linear systems: a survey. Electronic Proceedings of the 15th International Symposium on the Mathematical Theory of Networks and Systems. (University of Notre Dame, South Bend, Indiana, August 12-16, 2002).
21. R. F. Curtain, H. J. Zwart. An Introduction to Infinite-Dimensional Linear Systems Theory. (Springer, New York, Berlin, Heidelberg, 1995).
22. V. A. Derkach and M. M. Malamud. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95 (1991) 1-95.
23. F. Gesztesy, M. Mitrea, and M. Zinchenko. Variations on a Theme of Jost and Pais. Preprint. (2007), p. 39.
24. V. I. Gorbachuk and M. L. Gorbachuk. Boundary Value Problems for Operator Differential Equations. Translated and revised from the 1984 Russian original. Mathematics and its Applications (Soviet Series), 48. (Kluwer Academic Publishers Group, Dordrecht, 1991).
25. G. Grubb. A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Sup. Pisa 22 (1968) 425-513. (From Stanford University dissertation 1966).
26. G. Grubb. Known and unknown results on elliptic boundary problems. Bull. Amer. Math. Soc. 43 (2006) 227-230.
27. D. B. Hinton and J. K. Shaw. On Titchmarsh-Weyl m(A)-functions for Hamiltonian systems. J. Diff. Equat. 40 (1981) 316-342.
28. V. Isakov. Inverse Problems for Partial Differential Equations. (Springer, New York, Berlin, Heidelberg, 2006).
29. M. G. Krein. Concerning the resolvents of an Hermitian operator with the deficiencyindex (m, m). Comptes Rendue (Doklady) Acad. Sci. URSS (NS.), 52 (1946) 651-654.
30. J. L. Lions. Contrôle optimal de systèmes gouvernés par des équation aux derivees partielles. (Dunod Gauthier-Villars, Paris, 1968).
31. M. S. Livšic. Operators, Oscillations, Waves (Open Systems). (American Mathematical Society, Providence, Rhode Island, 1973).
32. S.I. Lyashko. Generalized Optimal Control of Linear Systems with Distributed Parameters. (Kluwer Academic Publishers, New York, 2002).
33. W. McLean. Strongly Elliptic Systems and Boundary Integral Equations. (Cambridge University Press, 2000).
34. C. Miranda. Partial Differential Equations of Elliptic Type. (Springer-Verlag, Berlin, 1970).
35. S. N. Naboko. Functional model of perturbation theory and its applications to scattering theory. (Russian) Boundary value problems of mathematical physics 10. Trudy Mat. Inst. Steklov. 147 (1980) 86-114.
36. J. R. Partington. Linear Operators and Linear Systems. An Analytical Approach to Control Theory. (Cambridge University Press, 2004).
37. B. S. Pavlov. The theory of extensions, and explicitly solvable models. Uspekhi Mat. Nauk 42, 6 (1987) 99-131. (Russian)
38. A. Posilicano. A Kreǐn-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183 (2001) 109-147.
39. V. A. Ryzhov. Functional model of a class of nonselfadjoint extensions of symmetric operators. Operator Theory: Advances and Applications 174 (2007) 117-158.
40. V. A. Ryzhov. A General Boundary Value Problem and its Weyl Function. To appear in Opuscula Math. 2007.
41. O. J. Staffans. Well-Posed Linear Systems. (Cambridge University Press, 2005).
42. J. Sylvester and G. Uhlmann. The Dirichlet to Neumann map and applications. Proceedings of the Conference "Inverse problems in partial differential equations (Arcata,1989)" 101-139 (SIAM, Philadelphia, 1990).
43. E. S. Titchmarsh. Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Second Edition. (Clarendon Press, Oxford, 1962).
44. M. Vishik. On general boundary conditions for elliptic differential equations. AMS Translations 24 (1963) 107-172.
45. H. Weyl. The method of orthogonal projection in potential theory. Duke Math. J. 7 (1940) 411-444.