## Resolvents of self-adjoint extensions with mixed boundary conditions.(English)Zbl 1143.47017

Let $$H$$ be a symmetric operator with equal deficiency numbers, $$(\mathcal H,\Gamma_1,\Gamma_2)$$ be its space of boundary values (boundary triplet). An abstract version of the Krein resolvent formula gives an expression for the resolvent of any selfadjoint extension $$\widetilde H$$ of $$H$$ in terms of a selfadjoint linear relation in $$\mathcal H$$ corresponding to $$\widetilde H$$.
The author considers the description of $$\widetilde H$$ by the boundary conditions of the form $$A\Gamma_1u=B\Gamma_2u$$, where $$A,B$$ are bounded linear operators, $$AB^*=BA^*$$ (earlier, mostly some special “canonical” forms of such abstract boundary conditions were used). For such a description, an explicit form of the resolvent formula is obtained, enabling a spectral analysis of selfadjoint extensions. As examples, the author considers point interactions in one dimension, Schrödinger operators on spaces consisting of pieces with different dimensions, and the Laplacian on a half-plane.

### MSC:

 47B25 Linear symmetric and selfadjoint operators (unbounded) 35P05 General topics in linear spectral theory for PDEs 47E05 General theory of ordinary differential operators 47N50 Applications of operator theory in the physical sciences 47F05 General theory of partial differential operators 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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### References:

 [1] Pavlov, B.S., The theory of extensions and explicitly-solvable models‘, Russian math. surveys, 42, 127-168, (1987) · Zbl 0665.47004 [2] Albeverio, S.; Kurasov, P., Singular perturbations of differential operators. solvable Schrödinger type operators, (2000), Cambridge Univ. Press Cambridge · Zbl 0945.47015 [3] Albeverio, S.; Gesztesy, F.; Hoegh-Krohn, R.; Holden, H., Solvable models in quantum mechanics, (), With an appendix by P. Exner · Zbl 0679.46057 [4] Gitman, D.; Tyutin, I.; Voronov, B., Self-adjoint extensions as a quantization problem, (2006), Birkhäuser Basel [5] Akhiezer, N.I.; Glazman, I.M., () [6] 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 [7] Gesztesy, F.; Makarov, K.A.; Tsekanovskii, E., An addendum to Krein’s formula, J. math. anal. appl., 222, 594-606, (1998) · Zbl 0922.47006 [8] Krein, M.G.; Langer, H.K.; Krein, M.G.; Langer, H.K., Defect subspaces and generalized resolvents of an Hermitian operator in the space π^{k}, Funct. anal. appl., Funct. anal. appl., 5, 217-228, (1971) · Zbl 0236.47035 [9] Langer, H.; Textorius, B., On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space, Pac. J. math., 72, 135-165, (1977) · Zbl 0335.47014 [10] Brasche, J.F.; Koshmanenko, V.; Neidhardt, H., New aspects of Krein’s extension theory, Ukr. math. J., 46, 34-53, (1994) · Zbl 0835.47002 [11] Posilicano, A., A kren-like formula for singular perturbations of self-adjoint operators and applications, J. funct. anal., 183, 109-147, (2001) · Zbl 0981.47022 [12] Asorey, M.; Ibort, A.; Marmo, S., Global theory of quantum boundary conditions and topology change, Int. J. mod. phys. A, 20, 1001-1025, (2005) · Zbl 1134.58302 [13] Cheon, T.; Fülöp, T.; Tsutsui, I., Symmetry, duality, and anholonomy of point interaction in one dimension, Ann. physics, 294, 1-23, (2001) · Zbl 1016.81026 [14] Gorbachuk, V.I.; Gorbachuk, M.L., Boundary value problems for operator differential equations, (1991), Kluwer Dordrecht · Zbl 0751.47025 [15] Kostrykin, V.; Schrader, R., Kirchhoff’s rule for quantum wires, J. phys. A: math. gen., 32, 595-630, (1999) · Zbl 0928.34066 [16] Brüning, J.; Geyler, V., Scattering on compact manifolds with infinitely thin horns, J. math. phys., 44, 371-405, (2003) · Zbl 1061.58025 [17] Kuchment, P., Quantum graphs I. some basic structures, Waves random media, 14, S107-S128, (2004) · Zbl 1063.81058 [18] Exner, P., The von Neumann way to treat systems of mixed dimensionality, Rep. math. phys., 55, 79-92, (2005) · Zbl 1140.81381 [19] Brüning, J.; Geyler, V.; Lobanov, I., Spectral properties of Schrödinger operators on decorated graphs, Math. notes, 77, 858-861, (2005) · Zbl 1090.47020 [20] Albeverio, S.; Pankrashkin, K., A remark on Krein’s resolvent formula and boundary conditions, J. phys. A: math. gen., 38, 4859-4865, (2005) · Zbl 1071.47003 [21] Arens, R., Operational calculus of linear relations, Pac. J. math., 11, 9-23, (1961) · Zbl 0102.10201 [22] Derkach, V.A.; Hassi, S.; Malamud, M.M.; de Snoo, H.S.V., Generalized resolvents of symmetric operators and admissibility, Methods funct. anal. topology, 6, 24-55, (2000) · Zbl 0973.47020 [23] Rofe-Beketov, F.S., Self-adjoint extensions of differential operators in a space of vector-valued functions, Sov. math. dokl., 184, 1034-1037, (1969) · Zbl 0181.15401 [24] Kochubei, A.N., Extensions of symmetric operators and symmetric binary relations, Math. notes, 17, 25-28, (1975) · Zbl 0322.47006 [25] Brasche, J.F.; Malamud, M.M.; Neidhardt, H., Weyl functions and spectral properties of self-adjoint extensions, Integr equ. oper theory, 43, 264-289, (2002) · Zbl 1008.47028 [26] Albeverio, S.; Brasche, J.; Malamud, M.M.; Neidhardt, H., Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions, J. funct. anal., 228, 144-188, (2005) · Zbl 1083.47020 [27] Brasche, J., Spectral theory for self-adjoint extensions, (), 51-96 · Zbl 1063.47002 [28] Maslov, V.P.; Maslov, V.P., Théorie des perturbations et méthodes asymptotiques, Izd. mosk. GoS. univ., Moscow, (1972), Dunod Paris, (French transl.). · Zbl 0247.47010 [29] Arnold, V.I., On a characteristic class entering in the quantization conditions, Funct. anal. appl., 1, 1-14, (1967) · Zbl 0175.20303 [30] Geyler, V.A.; Margulis, V.A., Anderson localization in the nondiscrete maryland model, Theor math. phys., 70, 133-140, (1987) [31] 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 [32] Albeverio, S.; Dgbrowski, L.; Kurasov, P., Symmetries of Schrödinger operators with point interactions, Lett. math. phys., 45, 33-47, (1998) · Zbl 0909.34074 [33] Šeba, P., The generalized point interaction in one dimension, Czech J. phys., 36, 667-673, (1986) [34] Borisov, D.; Exner, P., Exponential splitting of bound states in a waveguide with a pair of distant windows, J. phys. A: math. gen., 37, 3411-3428, (2004) · Zbl 1050.81008 [35] Dittrich, J.; Kříž, J., Bound states in straight quantum waveguides with combined boundary conditions, J. math. phys., 43, 3892-3915, (2002) · Zbl 1060.81019 [36] Dowker, J.S., The hybrid spectral problem and Robin boundary conditions, J. phys. A: math. gen., 38, 4735-4754, (2005) · Zbl 1064.81035 [37] Levitin, M.; Parnovski, L.; Polterovich, I., Isospectral domains with mixed boundary conditions, J. phys. A: math. gen., 39, 2073-2082, (2006) · Zbl 1089.58021 [38] Albeverio, S.; Kuzhel, S., One-dimensional Schrödinger operators with P-symmetric zero-range potentials, J. phys. A: math. gen., 38, 4975-4988, (2005) · Zbl 1070.81048 [39] Albeverio, S.; Nizhnik, L., A Schrödinger operator with δ’-interaction on a Cantor set and Krein-Feller operators, Math. nachr., 279, 467-476, (2006) · Zbl 1105.47039 [40] Pankrashkin, K., Reducible boundary conditions in coupled channels, J. phys. A: math. gen., 38, 8979-8992, (2005) · Zbl 1086.81044 [41] Brüning, J.; Geyler, V.; Pankrashkin, K., On-diagonal singularities of the Green functions for Schrödinger operators, J. math. phys., 46, 113-508, (2005) [42] Frank, R.L., On the scattering theory of the Laplacian with a periodic boundary condition, I. existence of wave operators, Doc. math., 8, 547-565, (2003) · Zbl 1142.35374
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