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Generalized \(Q\)-functions and Dirichlet-to-Neumann maps for elliptic differential operators. (English) Zbl 1179.47041

The notion of a \(Q\)-function associated with a pair \(\{S, A\}\) consisting of a symmetric operator \(S\) and a selfadjoint extension \(A\) of \(S\) in a Hilbert or Pontryagin space is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized \(Q\)-function. Krein type formulas for the difference of the resolvents of uniformly elliptic second-order differential expressions on bounded and unbounded domains are established, together with trace formulas in an \(H^2\) framework.

MSC:

47F05 General theory of partial differential operators
35J99 Elliptic equations and elliptic systems
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[1] Alpay, D.; Bruinsma, P.; Dijksma, A.; de Snoo, H. S.V., A Hilbert space associated with a Nevanlinna function, (Proceedings International Symposium MTNS 89, vol. III. Proceedings International Symposium MTNS 89, vol. III, Progr. Syst. Control Theory (1990), Birkhäuser: Birkhäuser Basel), 115-122
[2] Alpay, D.; Gohberg, I., A trace formula for canonical differential expressions, J. Funct. Anal., 197, 2, 489-525 (2003) · Zbl 1043.34024
[3] Alpay, D.; Gohberg, I., Pairs of selfadjoint operators and their invariants, Algebra i Analiz. Algebra i Analiz, St. Petersburg Math. J., 16, 1, 59-104 (2005), translation in: · Zbl 1084.47019
[4] Amrein, W. O.; Pearson, D. B., \(M\)-operators: A generalisation of Weyl-Titchmarsh theory, J. Comput. Appl. Math., 171, 1-26 (2004) · Zbl 1051.35047
[5] Bade, W. G.; Freeman, R. S., Closed extensions of the Laplace operator determined by a general class of boundary conditions, Pacific J. Math., 12, 395-410 (1962) · Zbl 0198.17403
[6] Beals, R., Non-local boundary value problems for elliptic operators, Amer. J. Math., 87, 315-362 (1965) · Zbl 0134.31302
[7] Behrndt, J.; Hassi, S.; de Snoo, H. S.V., Boundary relations, unitary colligations, and functional models, Complex Anal. Oper. Theory, 3, 57-98 (2009) · Zbl 1186.47007
[8] J. Behrndt, M. Kurula, A. van der Schaft, H. Zwart, Dirac structures and their composition on Hilbert spaces, preprint, 2008; J. Behrndt, M. Kurula, A. van der Schaft, H. Zwart, Dirac structures and their composition on Hilbert spaces, preprint, 2008 · Zbl 1209.47016
[9] Behrndt, J.; Langer, M., Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal., 243, 536-565 (2007) · Zbl 1132.47038
[10] Behrndt, J.; Malamud, M. M.; Neidhardt, H., Trace formulae for dissipative and coupled scattering systems, (Oper. Theory Adv. Appl., vol. 188 (2008), Birkhäuser: Birkhäuser Basel), 49-85 · Zbl 1175.47008
[11] S Birman, M., Perturbations of the continuous spectrum of a singular elliptic differential operator by varying the boundary and the boundary conditions, Vestnik Leningrad. Univ., 17, 22-55 (1962)
[12] Brodskiĭ, M. S., Unitary operator colligations and their characteristic functions, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Russian Math. Surveys, 33, 4, 159-191 (1978), (in Russian); English transl.: · Zbl 0415.47007
[13] Brown, M.; Grubb, G.; Wood, I., \(M\)-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems, Math. Nachr., 282, 3, 314-347 (2009) · Zbl 1167.47057
[14] Brown, M.; Marletta, M.; Naboko, S.; Wood, I., Boundary triplets and \(M\)-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, J. Lond. Math. Soc. (2), 77, 700-718 (2008) · Zbl 1148.35053
[15] P. Bruinsma, A. Dijksma, H.S.V. de Snoo, Models for generalized Carathéodory and Nevanlinna functions, in: Koninklijke Nederlandse Akademie van Wetenschappen, Verhandelingen, Afd. Natuurkunde, Eerste Reeks, deel 40, Amsterdam, 1993, pp. 161-178; P. Bruinsma, A. Dijksma, H.S.V. de Snoo, Models for generalized Carathéodory and Nevanlinna functions, in: Koninklijke Nederlandse Akademie van Wetenschappen, Verhandelingen, Afd. Natuurkunde, Eerste Reeks, deel 40, Amsterdam, 1993, pp. 161-178 · Zbl 0828.93019
[16] Brüning, J.; Geyler, V.; Pankrashkin, K., Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Rev. Math. Phys., 20, 1-70 (2008) · Zbl 1163.81007
[17] Derkach, V. A.; Hassi, S.; Malamud, M. M.; de Snoo, H. S.V., Boundary relations and their Weyl families, Trans. Amer. Math. Soc., 358, 5351-5400 (2006) · Zbl 1123.47004
[18] Derkach, V. A.; Malamud, M. M., Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal., 95, 1-95 (1991) · Zbl 0748.47004
[19] Derkach, V. A.; Malamud, M. M., The extension theory of Hermitian operators and the moment problem, J. Math. Sci., 73, 141-242 (1995) · Zbl 0848.47004
[20] Exner, P.; Kondej, S., Schrödinger operators with singular interactions: A model of tunneling resonances, J. Phys. A, 37, 34, 8255-8277 (2004) · Zbl 1072.81056
[21] Exner, P.; Kondej, S., Strong-coupling asymptotic expansion for Schrödinger operators with a singular interaction supported by a curve in \(R^3\), Rev. Math. Phys., 16, 5, 559-582 (2004) · Zbl 1053.81030
[22] Gesztesy, F.; Latushkin, Y.; Mitrea, M.; Zinchenko, M., Nonselfadjoint operators, infinite determinants, and some applications, Russ. J. Math. Phys., 12, 443-471 (2005) · Zbl 1201.47028
[23] Gesztesy, F.; Mitrea, M.; Zinchenko, M., Multi-dimensional versions of a determinant formula due to Jost and Pais, Rep. Math. Phys., 59, 365-377 (2007) · Zbl 1207.47043
[24] Gesztesy, F.; Mitrea, M.; Zinchenko, M., Variations on a theme of Jost and Pais, J. Funct. Anal., 253, 399-448 (2007) · Zbl 1133.47010
[25] Gesztesy, F.; Mitrea, M., Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, (Proceedings of Symposia in Pure Mathematics, vol. 79 (2008), American Mathematical Society: American Mathematical Society Providence, RI), 105-173 · Zbl 1178.35147
[26] Gesztesy, F.; Mitrea, M., Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, (Oper. Theory Adv. Appl., vol. 191 (2009), Birkhäuser: Birkhäuser Basel), 81-113 · Zbl 1180.35186
[27] Gesztesy, F.; Tsekanovskii, E., On matrix-valued Herglotz functions, Math. Nachr., 218, 61-138 (2000) · Zbl 0961.30027
[28] Gohberg, I.; Kreĭn, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18 (1969), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0181.13504
[29] Gorbachuk, V. I.; Gorbachuk, M. L., Boundary Value Problems for Operator Differential Equations, Math. Appl. (Soviet Ser.), vol. 48 (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0751.47025
[30] Grisvard, P., Elliptic Problems in Nonsmooth Domains, Monogr. Studies Math., vol. 24 (1985), Pitman: Pitman Boston, MA · Zbl 0695.35060
[31] Grubb, G., A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Sc. Norm. Super. Pisa, 22, 425-513 (1968) · Zbl 0182.14501
[32] Grubb, G., On the coerciveness and semiboundedness of general boundary problems, Israel J. Math., 10, 32-95 (1971) · Zbl 0231.35027
[33] Hassi, S.; de Snoo, H. S.V.; Woracek, H., Some interpolation problems of Nevanlinna-Pick type. The Krein-Langer method, (Oper. Theory Adv. Appl., vol. 106 (1998), Birkhäuser: Birkhäuser Basel), 201-216 · Zbl 0912.30023
[34] Kato, T., Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., vol. 132 (1976), Springer-Verlag: Springer-Verlag Berlin
[35] Krein, M. G., The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications, I, Rec. Math. [Mat. Sbornik] N.S., 20, 62, 431-495 (1947), (in Russian) · Zbl 0029.14103
[36] Krein, M. G., The fundamental propositions of the theory of representations of Hermitian operators with deficiency index \((m, m)\), Ukrain. Mat. Zh., 1, 3-66 (1949), (in Russian) · Zbl 0258.47025
[37] Krein, M. G.; Langer, H., Über die \(Q\)-Funktion eines \(π\)-hermiteschen Operators im Raume \(\Pi_\kappa \), Acta Sci. Math. (Szeged), 34, 191-230 (1973) · Zbl 0276.47036
[38] Krein, M. G.; Langer, H., Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume \(\Pi_\kappa\) zusammenhängen, I, Einige Funktionenklassen und ihre Darstellungen, Math. Nachr., 77, 187-236 (1977) · Zbl 0412.30020
[39] Langer, H.; Textorius, B., On generalized resolvents and \(Q\)-functions of symmetric linear relations (subspaces) in Hilbert space, Pacific J. Math., 72, 135-165 (1977) · Zbl 0335.47014
[40] Lions, J.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, I (1972), Springer-Verlag: Springer-Verlag New York · Zbl 0223.35039
[41] Malamud, M. M.; Malamud, S. M., Spectral theory of operator measures in Hilbert space, St. Petersburg Math. J., 15, 3, 1-77 (2003) · Zbl 1076.47001
[42] Marletta, M., Eigenvalue problems on exterior domains and Dirichlet to Neumann maps, J. Comput. Appl. Math., 171, 367-391 (2004) · Zbl 1055.65093
[43] Mazya, V., Sobolev Spaces (1985), Springer-Verlag: Springer-Verlag Berlin
[44] Mikhailova, A. B.; Pavlov, B.; Prokhorov, L. V., Intermediate Hamiltonian via Glazman’s splitting and analytic perturbation for meromorphic matrix-functions, Math. Nachr., 280, 12, 1376-1416 (2007) · Zbl 1135.47060
[45] Mikhailova, A. B.; Pavlov, B.; Ryzhii, V. I., Dirichlet-to-Neumann techniques for the plasma-waves in a slot-diode, (Oper. Theory Adv. Appl., vol. 174 (2007), Birkhäuser: Birkhäuser Basel), 77-103 · Zbl 1123.82024
[46] Pavlov, B., A star-graph model via operator extension, Math. Proc. Cambridge Philos. Soc., 142, 365-384 (2007) · Zbl 1115.81027
[47] Posilicano, A., Self-adjoint extensions of restrictions, Oper. Matrices, 2, 483-506 (2008) · Zbl 1175.47025
[48] Posilicano, A.; Raimondi, L., Krein’s resolvent formula for self-adjoint extensions of symmetric second order elliptic differential operators, J. Phys. A, 42, 1 (2009), Article ID 015204, 11 pp · Zbl 1161.81016
[49] Post, O., First-order operators and boundary triples, Russ. J. Math. Phys., 14, 4, 482-492 (2007) · Zbl 1181.35040
[50] Ryzhov, V., Weyl-Titchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control, Complex Anal. Oper. Theory, 3, 289-322 (2009) · Zbl 1182.47011
[51] Višik, M. I., On general boundary problems for elliptic differential equations, Tr. Mosk. Mat. Obs., 1, 187-246 (1952) · Zbl 0047.09502
[52] Wloka, J., Partial Differential Equations (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0623.35006
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