×

Spectral enclosures for non-self-adjoint extensions of symmetric operators. (English) Zbl 1472.47018

In the description of many quantum mechanical systems, operators appear as a consequence of heuristic arguments which suggest in a first step a formal expression for the Hamiltonian or Schrödinger operator describing the model. These operators \(S\) are typically unbounded and symmetric on a domain \(\operatorname{dom}{S}\) which is a dense subspace of a Hilbert space \(\mathcal{H}\). In a second crucial step for the description of the quantum mechanical system, one has to choose a closed (in many cases self-adjoint) extension of \(S\) in order to start the analysis of the model. Typically, fixing an extension means to specify the relevant boundary conditions for the system. The paper under review focuses on the description of closed non-selfadjoint extensions of \(S\) which appear as restrictions of the adjoint operator \(S^*\) and on the analysis of some of their spectral properties. The article also presents in its final part several applications of their results to elliptic operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with \(\delta\)-interactions, and to quantum graphs with non-self-adjoint vertex couplings.
In the first part of the article, the authors use an abstract and systematic approach to the description of extensions in terms of so-called boundary triples \((\Gamma_0,\Gamma_1,\mathcal{G})\), where \(\Gamma_{0,1}: \operatorname{dom}S^*\to \mathcal{G}\) satisfy a Green identity on the auxiliary Hilbert space \(\mathcal{G}\). Boundary triples (or their generalizations called quasi boundary triples) provide a useful technique to describe extensions encoding abstractly the boundary data of the problem. To formulate their main results in this part (e.g., Theorem 3.1), the authors use the Weyl function, which is an operator-valued function on the auxiliary Hilbert space defined in terms of the boundary triples. In addition, they introduce also a boundary operator \(B\) (in general non-symmetric) which serves to label the different extensions.
The article has an informative and well-written introduction to the topic describing their methods and results, but also introducing the reader to the literature and alternative approaches in this very active field. The bibliography list contains more than 130 references.

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47B28 Nonselfadjoint operators
35P05 General topics in linear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
35J10 Schrödinger operator, Schrödinger equation
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abels, H.; Grubb, G.; Wood, I., Extension theory and Kreĭn-type resolvent formulas for nonsmooth boundary value problems, J. Funct. Anal., 266, 4037-4100, (2014) · Zbl 1312.47019
[2] Abramov, A. A.; Aslanyan, A.; Davies, E. B., Bounds on complex eigenvalues and resonances, J. Phys. A, 34, 57-72, (2001) · Zbl 1123.81415
[3] Adler, M.; Bombieri, M.; Nagel, K.-J., Perturbation of analytic semigroups and applications to partial differential equations, J. Evol. Equ., 17, 1183-1208, (2017) · Zbl 06832391
[4] Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15, 119-147, (1962) · Zbl 0109.32701
[5] Albeverio, S.; Brasche, J. F.; 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
[6] Albeverio, S.; Fei, S.-M.; Kurasov, P., Point interactions: \(\mathcal{P} \mathcal{T}\)-hermiticity and reality of the spectrum, Lett. Math. Phys., 59, 227-242, (2002) · Zbl 1053.81026
[7] Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Holden, H., Solvable models in quantum mechanics. with an appendix by pavel exner, (2005), AMS Chelsea Publishing · Zbl 1078.81003
[8] Albeverio, S.; Kostenko, A.; Malamud, M. M., Spectral theory of semibounded Sturm-Liouville operators with local interactions on a discrete set, J. Math. Phys., 51, (2010), 24 pp · Zbl 1314.81059
[9] Albeverio, S.; Nizhnik, L., Schrödinger operators with nonlocal point interactions, J. Math. Anal. Appl., 332, 884-895, (2007) · Zbl 1122.47040
[10] Ando, T., Topics on operator inequalities, (1978), Division of Applied Mathematics, Research Institute of Applied Electricity, Hokkaido University Sapporo · Zbl 0388.47024
[11] 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
[12] Arendt, W.; ter Elst, A. F.M., The Dirichlet-to-Neumann operator on exterior domains, Potential Anal., 43, 313-340, (2015) · Zbl 1331.46022
[13] 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
[14] Arlinskii, Yu., Abstract boundary conditions for maximal sectorial extensions of sectorial operators, Math. Nachr., 209, 5-36, (2000) · Zbl 0949.47008
[15] Arlinskiĭ, Yu., Boundary triplets and maximal accretive extensions of sectorial operators, (Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Ser., vol. 404, (2012)), 35-72 · Zbl 1329.47036
[16] Arlinskiĭ, Yu.; Kovalev, Y.; Tsekanovskii, E., Accretive and sectorial extensions of nonnegative symmetric operators, Complex Anal. Oper. Theory, 6, 677-718, (2012) · Zbl 1294.47010
[17] Arlinskiĭ, Yu.; Popov, A. B., On m-accretive extensions of a sectorial operator, Mat. Sb., Sb. Math., 204, 1085-1121, (2013), [Russian]; English translation: · Zbl 1283.47015
[18] Arlinskiĭ, Yu.; Popov, A., On m-sectorial extensions of sectorial operators, Zh. Mat. Fiz. Anal. Geom., 13, 205-241, (2017) · Zbl 1489.47040
[19] Bade, W.; Freeman, R., Closed extensions of the Laplace operator determined by a general class of boundary conditions, Pacific J. Math., 12, 395-410, (1962) · Zbl 0198.17403
[20] Beals, R., Non-local boundary value problems for elliptic operators, Amer. J. Math., 87, 315-362, (1965) · Zbl 0134.31302
[21] Behrndt, J.; Grubb, G.; Langer, M.; Lotoreichik, V., Spectral asymptotics for resolvent differences of elliptic operators with δ and \(\delta^\prime\)-interactions on hypersurfaces, J. Spectr. Theory, 5, 697-729, (2015) · Zbl 1353.47090
[22] 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
[23] Behrndt, J.; Langer, M., On the adjoint of a symmetric operator, J. Lond. Math. Soc. (2), 82, 563-580, (2010) · Zbl 1209.47010
[24] Behrndt, J.; Langer, M., Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples, (Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Ser., vol. 404, (2012)), 121-160 · Zbl 1331.47067
[25] Behrndt, J.; Langer, M.; Lobanov, I.; Lotoreichik, V.; Popov, I. Yu., A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains, J. Math. Anal. Appl., 371, 750-758, (2010) · Zbl 1198.35159
[26] Behrndt, J.; Langer, M.; Lotoreichik, V., Schrödinger operators with δ and \(\delta^\prime\)-potentials supported on hypersurfaces, Ann. Henri Poincaré, 14, 385-423, (2013) · Zbl 1275.81027
[27] Behrndt, J.; Langer, M.; Lotoreichik, V., Spectral estimates for resolvent differences of self-adjoint elliptic operators, Integral Equations Operator Theory, 77, 1-37, (2013) · Zbl 1311.47059
[28] Behrndt, J.; Langer, M.; Lotoreichik, V., Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators, J. Lond. Math. Soc. (2), 88, 319-337, (2013) · Zbl 1296.35097
[29] Behrndt, J.; Langer, M.; Lotoreichik, V.; Rohleder, J., Quasi boundary triples and semibounded self-adjoint extensions, Proc. Roy. Soc. Edinburgh Sect. A, 147, 895-916, (2017) · Zbl 1386.35290
[30] 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
[31] Berkolaiko, G., An elementary introduction to quantum graphs, (Geometric and Computational Spectral Theory, Contemp. Math., vol. 700, (2017), Amer. Math. Soc. Providence, RI), 41-72 · Zbl 1388.34020
[32] Berkolaiko, G.; Kuchment, P., Introduction to quantum graphs, Mathematical Surveys and Monographs, vol. 186, (2013), American Mathematical Society Providence, RI · Zbl 1318.81005
[33] Birman, M. Sh.; Solomjak, M. Z., Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations in unbounded domains, Funktsional. Anal. i Prilozhen., Funct. Anal. Appl., 14, 267-274, (1981), [Russian]; English translation: · Zbl 0463.35061
[34] Borisov, D.; Krejčiřík, D., \(\mathcal{P} \mathcal{T}\)-symmetric waveguides, Integral Equations Operator Theory, 62, 489-515, (2008) · Zbl 1178.35141
[35] Borisov, D.; Krejčiřík, D., The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions, Asymptot. Anal., 76, 49-59, (2012) · Zbl 1241.47042
[36] Borisov, D.; Znojil, M., On eigenvalues of a \(\mathcal{P} \mathcal{T}\)-symmetric operator in a thin layer, Mat. Sb., Sb. Math., 208, 173-199, (2017), [Russian]; English translation: · Zbl 1371.35178
[37] Brasche, J. F.; Exner, P.; Kuperin, Yu. A.; Šeba, P., Schrödinger operators with singular interactions, J. Math. Anal. Appl., 184, 112-139, (1994) · Zbl 0820.47005
[38] Browder, F. E., Estimates and existence theorems for elliptic boundary value problems, Proc. Natl. Acad. Sci. USA, 45, 365-372, (1959) · Zbl 0093.29402
[39] Browder, F. E., On the spectral theory of elliptic differential operators. I, Math. Ann., 142, 22-130, (1961) · Zbl 0104.07502
[40] 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
[41] Brown, B. M.; Marletta, M.; Naboko, S.; Wood, I. G., 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
[42] Brown, B. M.; Marletta, M.; Naboko, S.; Wood, I., Inverse problems for boundary triples with applications, Studia Math., 237, 241-275, (2017) · Zbl 1422.47007
[43] Bruk, V. M., A certain class of boundary value problems with a spectral parameter in the boundary condition, Mat. Sb. (N.S.), 100, 142, 210-216, (1976) · Zbl 0334.47010
[44] 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
[45] Carlson, R., Adjoint and self-adjoint differential operators on graphs, Electron. J. Differential Equations, 1998, 6, (1998), 10 pp · Zbl 0888.34055
[46] Cherednichenko, K. D.; Kiselev, A. V.; Silva, L. O., Functional model for extensions of symmetric operators and applications to scattering theory, Netw. Heterog. Media, (2018), in press
[47] Conway, J. B., Functions of one complex variable I, Graduate Texts in Mathematics, vol. 11, (1978), Springer
[48] Cuenin, J.-C., Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials, J. Funct. Anal., 272, 2987-3018, (2017) · Zbl 1436.47012
[49] Cuenin, J.-C.; Tretter, C., Non-symmetric perturbations of self-adjoint operators, J. Math. Anal. Appl., 441, 235-258, (2016) · Zbl 1341.47017
[50] Davies, E. B., Non-self-adjoint differential operators, Bull. Lond. Math. Soc., 34, 513-532, (2002) · Zbl 1052.47042
[51] Demuth, M.; Hansmann, M.; Katriel, G., On the discrete spectrum of non-selfadjoint operators, J. Funct. Anal., 257, 2742-2759, (2009) · Zbl 1183.47016
[52] Derkach, V. A.; Hassi, S.; Malamud, M. M.; de Snoo, H., Boundary relations and their Weyl families, Trans. Amer. Math. Soc., 358, 5351-5400, (2006) · Zbl 1123.47004
[53] Derkach, V. A.; Hassi, S.; Malamud, M. M.; de Snoo, H., Boundary relations and generalized resolvents of symmetric operators, Russ. J. Math. Phys., 16, 17-60, (2009) · Zbl 1182.47026
[54] Derkach, V. A.; Hassi, S.; Malamud, M. M.; de Snoo, H., Boundary triplets and Weyl functions. recent developments, (Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Ser., vol. 404, (2012)), 161-220 · Zbl 1320.47024
[55] 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
[56] 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
[57] Edmunds, D. E.; Evans, W. D., Spectral theory and differential operators, Oxford Mathematical Monographs, (1987), The Clarendon Press, Oxford University Press New York · Zbl 0628.47017
[58] Edmunds, D. E.; Triebel, H., Function spaces, entropy numbers, differential operators, Cambridge Tracts in Mathematics, vol. 120, (1996), Cambridge University Press Cambridge · Zbl 0865.46020
[59] Ershova, Y.; Karpenko, I.; Kiselev, A., Isospectrality for graph Laplacians under the change of coupling at graph vertices, J. Spectr. Theory, 6, 43-66, (2016) · Zbl 1346.81048
[60] Ershova, Y.; Kiselev, A., Trace formulae for graph Laplacians with applications to recovering matching conditions, Methods Funct. Anal. Topology, 18, 343-359, (2012) · Zbl 1289.34088
[61] Ershova, Y.; Kiselev, A., Trace formulae for Schrödinger operators on metric graphs with applications to recovering matching conditions, Methods Funct. Anal. Topology, 20, 134-148, (2014) · Zbl 1313.34093
[62] Exner, P., Leaky quantum graphs: a review, (Analysis on Graphs and Its Applications. Selected Papers Based on the Isaac Newton Institute for Mathematical Sciences Programme, Cambridge, UK, 2007, Proc. Symp. Pure Math., vol. 77, (2008)), 523-564 · Zbl 1153.81487
[63] Exner, P.; Fraas, M., On geometric perturbations of critical Schrödinger operators with a surface interaction, J. Math. Phys., 50, (2009), 12 pp · Zbl 1300.81034
[64] Exner, P.; Ichinose, T., Geometrically induced spectrum in curved leaky wires, J. Phys. A, 34, 1439-1450, (2001) · Zbl 1002.81024
[65] Exner, P.; Jex, M., On the ground state of quantum graphs with attractive δ-coupling, Phys. Lett. A, 376, 713-717, (2012) · Zbl 1255.81160
[66] Exner, P.; Kondej, S., Bound states due to a strong δ-interaction supported by a curved surface, J. Phys. A, 36, 443-457, (2003) · Zbl 1050.81009
[67] Exner, P.; Kovařík, H., Quantum waveguides, (2015), Springer Heidelberg · Zbl 1314.81001
[68] Exner, P.; Rohleder, J., Generalized interactions supported on hypersurfaces, J. Math. Phys., 57, (2016), 23 pp · Zbl 1339.81038
[69] Exner, P.; Yoshitomi, K., Asymptotics of eigenvalues of the Schrödinger operator with a strong δ-interaction on a loop, J. Geom. Phys., 41, 344-358, (2002) · Zbl 1083.35034
[70] Fanelli, L.; Krejčiřík, D.; Vega, L., Spectral stability of Schrödinger operators with subordinated complex potentials, J. Spectr. Theory, (2018), in press · Zbl 1391.35296
[71] Frank, R. L., Eigenvalue bounds for Schrödinger operators with complex potentials, Bull. Lond. Math. Soc., 43, 745-750, (2011) · Zbl 1228.35158
[72] Frank, R. L., Eigenvalues of Schrödinger operators with complex surface potentials, (Functional Analysis and Operator Theory for Quantum Physics, (2017), Europ. Math. Soc. Zürich), 245-259 · Zbl 1372.81057
[73] Freeman, R., Closed extensions of the Laplace operator determined by a general class of boundary conditions, for unbounded regions, Pacific J. Math., 12, 121-135, (1962) · Zbl 0109.07102
[74] Freeman, R., Closed operators and their adjoints associated with elliptic differential operators, Pacific J. Math., 22, 71-97, (1967) · Zbl 0168.08405
[75] Galkowski, J.; Smith, H. F., Restriction bounds for the free resolvent and resonances in lossy scattering, Int. Math. Res. Not., 16, 7473-7509, (2015) · Zbl 1347.35188
[76] 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
[77] Gesztesy, F.; Mitrea, M., Generalized Robin boundary conditions, Robin-to-Dirichlet maps, Kreĭn-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, (Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Proc. Sympos. Pure Math., vol. 79, (2008), Amer. Math. Soc. Providence, RI), 105-173 · Zbl 1178.35147
[78] Gesztesy, F.; Mitrea, M., A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains, J. Anal. Math., 113, 53-172, (2011) · Zbl 1231.47044
[79] Gesztesy, F.; Mitrea, M.; Zinchenko, M., Variations on a theme of Jost and Pais, J. Funct. Anal., 253, 399-448, (2007) · Zbl 1133.47010
[80] Gesztesy, F.; Mitrea, M.; Zinchenko, M., On Dirichlet-to-Neumann maps and some applications to modified Fredholm determinants, (Methods of Spectral Analysis in Mathematical Physics, Oper. Theory Adv. Appl., vol. 186, (2009), Birkhäuser Verlag Basel), 191-215 · Zbl 1169.47014
[81] Gohberg, I. C.; Kreĭn, M. G., Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., vol. 18, (1969), Amer. Math. Soc. Providence, RI · Zbl 0181.13503
[82] Gorbachuk, V. I.; Gorbachuk, M. L., Boundary value problems for operator differential equations, (1991), Kluwer Academic Publ. Dordrecht · Zbl 0751.47025
[83] Grod, A.; Kuzhel, S., Schrödinger operators with non-symmetric zero-range potentials, Methods Funct. Anal. Topology, 20, 34-49, (2014) · Zbl 1313.81011
[84] Grubb, G., A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Sc. Norm. Super. Pisa (3), 22, 425-513, (1968) · Zbl 0182.14501
[85] Grubb, G., Remarks on trace estimates for exterior boundary problems, Comm. Partial Differential Equations, 9, 231-270, (1984) · Zbl 0553.35022
[86] Grubb, G., Krein resolvent formulas for elliptic boundary problems in nonsmooth domains, Rend. Semin. Mat. Univ. Politec. Torino, 66, 271-297, (2008) · Zbl 1206.35090
[87] Grubb, G., Distributions and operators, Graduate Texts in Mathematics, vol. 252, (2009), Springer New York · Zbl 1171.47001
[88] Grubb, G., Spectral asymptotics for Robin problems with a discontinuous coefficient, J. Spectr. Theory, 1, 155-177, (2011) · Zbl 1223.35146
[89] Hussein, A., Maximal quasi-accretive Laplacians on finite metric graphs, J. Evol. Equ., 14, 477-497, (2014) · Zbl 1305.34046
[90] Hussein, A.; Krejčiřík, D.; Siegl, P., Non-self-adjoint graphs, Trans. Amer. Math. Soc., 367, 2921-2957, (2015) · Zbl 1312.34068
[91] Ibort, A.; Lledó, F.; Pérez-Pardo, J. M., Self-adjoint extensions of the Laplace-Beltrami operator and unitaries at the boundary, J. Funct. Anal., 268, 634-670, (2015) · Zbl 1310.47032
[92] Kac, I. S.; Krein, M. G., On the spectral function of a string, (Atkinson, F. V., Discrete and Continuous Boundary Problems (Russian translation), Addition II, (1968), Mir Moscow), Amer. Math. Soc. Transl., 2, 103, 19-102, (1974), English translation: · Zbl 0291.34017
[93] Kac, I. S.; Krein, M. G., R-functions—analytic functions mapping the upper halfplane into itself, Amer. Math. Soc. Transl. (2), 103, 1-18, (1974) · Zbl 0291.34016
[94] Karreskog, G.; Kurasov, P.; Trygg Kupersmidt, I., Schrödinger operators on graphs: symmetrization and Eulerian cycles, Proc. Amer. Math. Soc., 144, 1197-1207, (2016) · Zbl 1355.34053
[95] Kato, T., Perturbation theory for linear operators, Classics in Mathematics, (1995), Springer-Verlag Berlin, Reprint of the 1980 edition · Zbl 0836.47009
[96] Kochubei, A. N., Extensions of symmetric operators and symmetric binary relations, Mat. Zametki, Math. Notes, 17, 25-28, (1975), [Russian]; English translation: · Zbl 0322.47006
[97] Kochubei, A. N., One-dimensional point interactions, Ukraïn. Mat. Zh., Ukrainian Math. J., 41, 1198-1201, (1989), [Russian]; English translation: · Zbl 0703.34086
[98] Kondej, S.; Krejčiřík, D., Asymptotic spectral analysis in colliding leaky quantum layers, J. Math. Anal. Appl., 446, 1328-1355, (2017) · Zbl 1355.82052
[99] Kondej, S.; Lotoreichik, V., Weakly coupled bound state of 2-D Schrödinger operator with potential-measure, J. Math. Anal. Appl., 420, 1416-1438, (2014) · Zbl 1298.35128
[100] Kovalev, Yu. G., 1D nonnegative Schrödinger operators with point interactions, Mat. Stud., 39, 150-163, (2013) · Zbl 1298.34168
[101] Kostenko, A.; Malamud, M., 1-D Schrödinger operators with local point interactions on a discrete set, J. Differential Equations, 249, 253-304, (2010) · Zbl 1195.47031
[102] Kostenko, A.; Malamud, M., 1-D Schrödinger operators with local point interactions: a review, (Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday, Proc. Symp. Pure Math., vol. 87, (2013), Amer. Math. Soc. Providence, RI), 235-262 · Zbl 1319.34003
[103] Kostrykin, V.; Schrader, R., Kirchhoff’s rule for quantum wires, J. Phys. A, 32, 595-630, (1999) · Zbl 0928.34066
[104] Kostrykin, V.; Schrader, R., Laplacians on metric graphs: eigenvalues, resolvents and semigroups, (Quantum Graphs and Their Applications, Contemp. Math., vol. 415, (2006), Amer. Math. Soc. Providence, RI), 201-225 · Zbl 1122.34066
[105] Kronig, R. de L.; Penney, W. G., Quantum mechanics of electrons in crystal lattices, Proc. R. Soc. Lond. A, 130, 499-513, (1931) · JFM 57.1587.02
[106] Kuchment, P., Quantum graphs. I. some basic structures, Waves Random Media, 14, S107-S128, (2004) · Zbl 1063.81058
[107] Kuzhel, S.; Znojil, M., Non-self-adjoint Schrödinger operators with nonlocal one-point interactions, Banach J. Math. Anal., 11, 923-944, (2017) · Zbl 1489.47065
[108] Laptev, A.; Safronov, O., Eigenvalue estimates for Schrödinger operators with complex potentials, Comm. Math. Phys., 292, 29-54, (2009) · Zbl 1185.35045
[109] Lenz, D.; Schubert, C.; Veselić, I., Unbounded quantum graphs with unbounded boundary conditions, Math. Nachr., 287, 962-979, (2014) · Zbl 1328.47023
[110] Lions, J.; Magenes, E., Non-homogeneous boundary value problems and applications I, (1972), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0223.35039
[111] Lotoreichik, V.; Ourmières-Bonafos, T., On the bound states of Schrödinger operators with δ-interactions on conical surfaces, Comm. Partial Differential Equations, 41, 999-1028, (2016) · Zbl 1346.35139
[112] Lotoreichik, V.; Rohleder, J., Schatten-von Neumann estimates for resolvent differences of Robin Laplacians on a half-space, (Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, Oper. Theory Adv. Appl., vol. 221, (2012), Birkhäuser/Springer Basel AG Basel), 453-468 · Zbl 1287.47038
[113] Lotoreichik, V.; Siegl, P., Spectra of definite type in waveguide models, Proc. Amer. Math. Soc., 145, 1231-1246, (2017) · Zbl 1369.81035
[114] Malamud, M. M., Operator holes and extensions of sectorial operators and dual pairs of contractions, Math. Nachr., 279, 625-655, (2006) · Zbl 1111.47017
[115] Malamud, M. M., Spectral theory of elliptic operators in exterior domains, Russ. J. Math. Phys., 17, 96-125, (2010) · Zbl 1202.35142
[116] Malamud, M. M.; Malamud, S. M., Spectral theory of operator measures in a Hilbert space, Algebra i Analiz, St. Petersburg Math. J., 15, 323-373, (2004), [Russian]; English translation: · Zbl 1076.47001
[117] 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
[118] Mantile, A.; Posilicano, A.; Sini, M., Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces, J. Differential Equations, 261, 1-55, (2016) · Zbl 1337.35032
[119] Maz’ya, V. G.; Shaposhnikova, T. O., Theory of Sobolev multipliers, (2009), Springer-Verlag Berlin Heidelberg
[120] Pankrashkin, K., Spectra of Schrödinger operators on equilateral quantum graphs, Lett. Math. Phys., 77, 139-154, (2006) · Zbl 1113.81056
[121] Pietsch, A., Eigenvalues and s-numbers, Cambridge Studies in Advanced Mathematics, vol. 13, (1987), Cambridge University Press Cambridge · Zbl 0615.47019
[122] Posilicano, A., Self-adjoint extensions of restrictions, Oper. Matrices, 2, 1-24, (2008) · Zbl 1175.47025
[123] Post, O., Equilateral quantum graphs and boundary triples, (Analysis on Graphs and its Applications, Proc. Sympos. Pure Math., vol. 77, (2008), Amer. Math. Soc. Providence, RI), 469-490 · Zbl 1153.81499
[124] Post, O., Boundary pairs associated with quadratic forms, Math. Nachr., 289, 1052-1099, (2016) · Zbl 1380.47001
[125] Schmüdgen, K., Unbounded self-adjoint operators on Hilbert space, (2012), Springer Dordrecht · Zbl 1257.47001
[126] Schubert, C.; Seifert, C.; Voigt, J.; Waurick, M., Boundary systems and (skew-)self-adjoint operators on infinite metric graphs, Math. Nachr., 288, 1776-1785, (2015) · Zbl 1375.47022
[127] Teschl, G., Mathematical methods in quantum mechanics. with applications to Schrödinger operators, (2009), American Mathematical Society Providence · Zbl 1166.81004
[128] Vishik, M. I., On general boundary problems for elliptic differential equations, Tr. Mosk. Mat. Obŝ., 1, 187-246, (1952), [Russian]
[129] Weidmann, J., Lineare operatoren in hilberträumen. teil 1. grundlagen, Mathematische Leitfäden, (2000), B. G. Teubner Stuttgart, [German] · Zbl 0972.47002
[130] Winkler, H., Spectral estimations for canonical systems, Math. Nachr., 220, 115-141, (2000) · Zbl 0968.34016
[131] Wyss, C., Riesz bases for p-subordinate perturbations of normal operators, J. Funct. Anal., 258, 208-240, (2010) · Zbl 1185.47014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.