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Dirichlet-to-Neumann maps, abstract Weyl-Titchmarsh \(M\)-functions, and a generalized index of unbounded meromorphic operator-valued functions. (English) Zbl 1350.47009

Summary: We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrödinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl-Titchmarsh \(M\)-functions and Donoghue-type \(M\)-functions corresponding to closed extensions of symmetric operators belong to it.
The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schrödinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl-Titchmarsh \(M\)-functions, respectively.

MSC:

47A53 (Semi-) Fredholm operators; index theories
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A10 Spectrum, resolvent
47B07 Linear operators defined by compactness properties
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[1] Alpay, D.; Behrndt, J., Generalized \(Q\)-functions and Dirichlet-to-Neumann maps for elliptic differential operators, J. Funct. Anal., 257, 1666-1694 (2009) · Zbl 1179.47041
[2] Amrein, W. O.; Pearson, D. B., \(M\) operators: a generalization of Weyl-Titchmarsh theory, J. Comput. Appl. Math., 171, 1-26 (2004) · Zbl 1051.35047
[3] Arendt, W.; ter Elst, A. F.M., The Dirichlet-to-Neumann operator on rough domains, J. Differential Equations, 251, 2100-2124 (2011) · Zbl 1241.47036
[4] Arendt, W.; ter Elst, A. F.M.; Kennedy, J. B.; Sauter, M., The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266, 1757-1786 (2014) · Zbl 1314.47062
[5] Arendt, W.; Mazzeo, R., Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11, 2201-2212 (2012) · Zbl 1267.35139
[6] Arens, R., Operational calculus of linear relations, Pacific J. Math., 11, 9-23 (1961) · Zbl 0102.10201
[7] Behrndt, J.; Gesztesy, F.; Holden, H.; Nichols, R., On the index of meromorphic operator-valued functions and some applications, in: J. Dittrich, H. Kovarik, A. Laptev (Eds.), Functional Analysis and Operator Theory for Quantum Physics, EMS Publishing House, EMS, ETH-Zürich, Switzerland, in press
[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.; Langer, M., Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples, (Hassi, S.; de Snoo, H. S.V.; Safraniec, F. H., Operator Methods for Boundary Value Problems. Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Series, vol. 404 (2012), Cambridge University Press: Cambridge University Press Cambridge), 121-160 · Zbl 1331.47067
[11] Behrndt, J.; Micheler, T., Elliptic differential operators on Lipschitz domains and abstract boundary value problems, J. Funct. Anal., 267, 3657-3709 (2014) · Zbl 1300.35026
[12] Behrndt, J.; Rohleder, J., Spectral analysis of selfadjoint elliptic differential operators, Dirichlet-to-Neumann maps, and abstract Weyl functions, Adv. Math., 285, 1301-1338 (2015) · Zbl 1344.47018
[13] Brasche, J. F.; Malamud, M. M.; Neidhardt, H., Weyl function and spectral properties of self-adjoint extensions, Integral Equations Operator Theory, 43, 264-289 (2002) · Zbl 1008.47028
[14] Brown, B. M.; Grubb, G.; Wood, I. G., \(M\)-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems, Math. Nachr., 282, 314-347 (2009) · Zbl 1167.47057
[15] Brown, B. M.; Marletta, M.; Naboko, S.; Wood, I., Boundary triples 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
[16] Brown, B. M.; Marletta, M.; Naboko, S.; Wood, I., An abstract inverse problem for boundary triples with an application to the Friedrichs model · Zbl 1422.47007
[17] Bruk, V. M., A certain class of boundary value problems with a spectral parameter in the boundary condition, Math. USSR-Sb., 29, 186-192 (1976) · Zbl 0376.47007
[18] 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
[19] Derkach, V. A., On Weyl function and generalized resolvents of a Hermitian operator in a Krein space, Integral Equations Operator Theory, 23, 387-415 (1995) · Zbl 0837.47030
[20] 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
[21] Derkach, V. A.; Hassi, S.; Malamud, M. M.; de Snoo, H. S.V., Boundary relations and generalized resolvents of symmetric operators, Russ. J. Math. Phys., 16, 17-60 (2009) · Zbl 1182.47026
[22] 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
[23] 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
[24] Derkach, V. A.; Malamud, M. M.; Tsekanovskii, E. R., Sectorial extensions of a positive operator, and the characteristic function, Sov. Math., Dokl., 37, 106-110 (1988) · Zbl 0698.47004
[25] Donoghue, W. F., On the perturbation of spectra, Comm. Pure Appl. Math., 18, 559-579 (1965) · Zbl 0143.16403
[26] Gesztesy, F.; Holden, H.; Nichols, R., Integral Equations Operator Theory, 85, 301-302 (2016), Erratum · Zbl 1465.47010
[27] Gesztesy, F.; Kalton, N. J.; Makarov, K. A.; Tsekanovskii, E., Some applications of operator-valued Herglotz functions, (Alpay, D.; Vinnikov, V., Operator Theory, System Theory and Related Topics. The Moshe Livšic Anniversary Volume. Operator Theory, System Theory and Related Topics. The Moshe Livšic Anniversary Volume, Oper. Theory Adv. Appl., vol. 123 (2001), Birkhäuser: Birkhäuser Basel), 271-321 · Zbl 0991.30020
[28] Gesztesy, F.; Makarov, K. A.; Tsekanovskii, E., An addendum to Krein’s formula, J. Math. Anal. Appl., 222, 594-606 (1998) · Zbl 0922.47006
[29] Gesztesy, F.; Mitrea, M., Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, (Mitrea, D.; Mitrea, M., Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz’ya’s 70th Birthday. Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz’ya’s 70th Birthday, Proceedings of Symposia in Pure Mathematics, vol. 79 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 105-173 · Zbl 1178.35147
[30] Gesztesy, F.; Mitrea, M., Self-adjoint extensions of the Laplacian and Krein-type resolvent formulas in nonsmooth domains, J. Anal. Math., 113, 53-172 (2011) · Zbl 1231.47044
[31] Gesztesy, F.; Naboko, S. N.; Weikard, R.; Zinchenko, M., Donoghue-type \(m\)-functions for Schrödinger operators with operator-valued potentials, J. Anal. Math. (2016), in press
[32] Gesztesy, F.; Tsekanovskii, E., On matrix-valued Herglotz functions, Math. Nachr., 218, 61-138 (2000) · Zbl 0961.30027
[33] Gohberg, I.; Goldberg, S.; Kaashoek, M. A., Classes of Linear Operators, Vol. I, Operator Theory: Advances and Applications, vol. 49 (1990), Birkhäuser: Birkhäuser Basel · Zbl 0745.47002
[34] Gohberg, I.; Kreĭn, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18 (1969), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0181.13504
[35] Gohberg, I.; Leiterer, J., Holomorphic Operator Functions of One Variable and Applications, Operator Theory: Advances and Applications, vol. 192 (2009), Birkhäuser: Birkhäuser Basel · Zbl 1182.47014
[36] Gohberg, I. C.; Sigal, E. I., An operator generalizations of the logarithmic residue theorem and the theorem of Rouché, Math. USSR-Sb., 13, 603-625 (1971) · Zbl 0254.47046
[37] Gorbachuk, V. I.; Gorbachuk, M. L., Boundary Value Problems for Operator Differential Equations (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0751.47025
[38] Hassi, S.; Malamud, M. M.; Mogilevskii, V., Unitary equivalence of proper extensions of a symmetric operator and the Weyl function, Integral Equations Operator Theory, 77, 449-487 (2013) · Zbl 1285.47018
[39] Jerison, D.; Kenig, C., The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130, 161-219 (1995) · Zbl 0832.35034
[40] Kac, I. S.; Krein, M. G., \(R\)-functions - analytic functions mapping the upper halfplane into itself, Amer. Math. Soc. Transl. Ser. 2, 103, 1-18 (1974), Supplement to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems, Mir, Moscow 1968; Engl. transl. in · Zbl 0291.34016
[41] Kato, T., Perturbation Theory for Linear Operators (1980), Springer: Springer Berlin, corr. printing of the 2nd ed.
[42] Kochubei, A. N., Extensions of symmetric operators and symmetric binary relations, Math. Notes, 17, 25-28 (1975) · Zbl 0322.47006
[43] Krein, M. G.; Ovčarenko, I. E., \(Q\)-functions and sc-resolvents of nondensely defined Hermitian contractions, Sib. Math. J., 18, 728-746 (1977) · Zbl 0409.47013
[44] Krein, M. G.; Ovčarenko, I. E., Inverse problems for \(Q\)-functions and resolvent matrices of positive Hermitian operators, Sov. Math., Dokl., 19, 1131-1134 (1978) · Zbl 0443.47026
[45] 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
[46] Malamud, M. M., Certain classes of extensions of a lacunary Hermitian operator, Ukrainian Math. J., 44, 190-204 (1992) · Zbl 0804.47011
[47] Malamud, M. M., On a formula of the generalized resolvents of a nondensely defined hermitian operator, Ukrainian Math. J., 44, 1522-1547 (1992)
[48] Malamud, M. M., Spectral theory of elliptic operators in exterior domains, Russ. J. Math. Phys., 17, 96-125 (2010) · Zbl 1202.35142
[49] Malamud, M. M.; Mogilevskii, V. I., Krein type formula for canonical resolvents of dual pairs of linear relations, Methods Funct. Anal. Topology, 8, 72-100 (2002) · Zbl 1074.47501
[50] Malamud, M. M.; Mogilevskii, V. I., Generalized resolvents of symmetric operators, Math. Notes, 73, 429-435 (2003) · Zbl 1076.47002
[51] Malamud, M. M.; Neidhardt, H., On the unitary equivalence of absolutely continuous parts of self-adjoint extensions, J. Funct. Anal., 260, 613-638 (2011) · Zbl 1241.47011
[52] Malamud, M. M.; Neidhardt, H., Sturm-Liouville boundary value problems with operator potentials and unitary equivalence, J. Differential Equations, 252, 5875-5922 (2012) · Zbl 1257.47053
[53] Mogilevskii, V., Boundary triples and Weyl-Titchmarsh functions of differential operators with arbitrary deficiency indices, Methods Funct. Anal. Topology, 15, 280-300 (2009) · Zbl 1199.47189
[54] Pankrashkin, K., Resolvents of self-adjoint extensions with mixed boundary conditions, Rep. Math. Phys., 58, 207-221 (2006) · Zbl 1143.47017
[55] Pankrashkin, K., An example of unitary equivalence between self-adjoint extensions and their parameters, J. Funct. Anal., 265, 2910-2936 (2013) · Zbl 1300.47029
[56] Posilicano, A., Boundary triples and Weyl functions for singular perturbations of self-adjoint operators, Methods Funct. Anal. Topology, 10, 57-63 (2004) · Zbl 1066.47024
[57] Posilicano, A., Self-adjoint extensions of restrictions, Oper. Matrices, 2, 483-506 (2008) · Zbl 1175.47025
[58] Posilicano, A.; Raimondi, L., Krein’s resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators, J. Phys. A, 42, Article 015204 pp. (2009) · Zbl 1161.81016
[59] Post, O., Boundary pairs associated with quadratic forms · Zbl 1380.47001
[60] Ryzhov, V., A general boundary value problem and its Weyl function, Opuscula Math., 27, 305-331 (2007) · Zbl 1155.47025
[61] Saakjan, Sh. N., Theory of resolvents of a symmetric operator with infinite defect numbers, Akad. Nauk Armjan. SSR Dokl., 41, 193-198 (1965), (in Russian) · Zbl 0163.37804
[62] Weidmann, J., Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, vol. 68 (1980), Springer: Springer New York
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