Man’ko, S. S. Schrödinger operators on star graphs with singularly scaled potentials supported near the vertices. (English) Zbl 1278.81097 J. Math. Phys. 53, No. 12, 123521, 13 p. (2012). Summary: We study Schrödinger operators on star metric graphs with potentials of the form \(\alpha \varepsilon^{-2} Q(\varepsilon^{1}x)\). In dimension 1 such potentials, with additional assumptions on \(Q\), approximate in the sense of distributions as \(\varepsilon \to 0\) the first derivative of the Dirac delta-function. We establish the convergence of the Schrödinger operators in the uniform resolvent topology and show that the limit operator depends on {\(\alpha\)} and \(Q\) in a very nontrivial way.{©2012 American Institute of Physics} Cited in 5 Documents MSC: 81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 47A10 Spectrum, resolvent PDF BibTeX XML Cite \textit{S. S. Man'ko}, J. Math. Phys. 53, No. 12, 123521, 13 p. (2012; Zbl 1278.81097) Full Text: DOI arXiv References: [1] DOI: 10.1063/1.2710197 · Zbl 1137.81330 [2] Albeverio S., Solvable Models in Quantum Mechanics. With an appendix by Pavel Exner (2005) [3] Albeverio S., Singular Perturbations of Differential Operators and Solvable Schrödinger Type Operators (2000) [4] DOI: 10.1103/PhysRevB.23.4828 [5] DOI: 10.1103/PhysRevB.45.1074 [6] DOI: 10.1088/1751-8113/40/26/F02 · Zbl 1136.81442 [7] DOI: 10.1088/0022-3719/21/14/008 [8] DOI: 10.1016/S0375-9601(00)00690-3 · Zbl 1135.81336 [9] DOI: 10.1016/j.aop.2009.11.010 · Zbl 1192.81159 [10] DOI: 10.1088/0305-4470/36/27/311 · Zbl 1047.81567 [11] Coddington E., Theory of ordinary differential equations (1955) · Zbl 0064.33002 [12] DOI: 10.1088/1751-8113/41/41/415206 · Zbl 1192.81167 [13] DOI: 10.1088/0305-4470/29/1/011 · Zbl 0916.47056 [14] DOI: 10.1088/1751-8113/42/41/415305 · Zbl 1179.81080 [15] DOI: 10.1209/0295-5075/3/4/017 [16] Golovaty Yu., Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. 5 pp 16– (2009) [17] Golovaty Yu., Ukr. Mat. Visn. 6 pp 173– (2009) [18] DOI: 10.1088/1751-8113/43/15/155204 · Zbl 1187.81104 [19] DOI: 10.1088/0305-4470/32/4/006 · Zbl 0928.34066 [20] DOI: 10.1002/1521-3978(200008)48:8<703::AID-PROP703>3.0.CO;2-O · Zbl 0977.81163 [21] DOI: 10.1007/s00220-003-0831-7 · Zbl 1037.81044 [22] DOI: 10.1103/PhysRevLett.79.4794 [23] DOI: 10.1103/PhysRevB.42.9009 [24] DOI: 10.1088/0959-7174/12/4/201 · Zbl 1063.35525 [25] DOI: 10.1088/0959-7174/14/1/014 · Zbl 1063.81058 [26] DOI: 10.1023/A:1014481629504 · Zbl 0993.65122 [27] Man’ko S., Visn. L’viv. Univ., Ser. Mekh.-Mat. 71 pp 142– (2009) [28] Man’ko S. S., Nauk. Visn. Chernivets’kogo Univ., Mat. 1 pp 61– (2011) [29] DOI: 10.1088/1751-8113/43/44/445304 · Zbl 1202.81201 [30] DOI: 10.1112/S0024611500012272 · Zbl 1046.34092 [31] DOI: 10.1063/1.1749766 [32] DOI: 10.1063/1.1747293 [33] DOI: 10.1016/0003-4916(72)90285-0 [34] Šeba P., Rep. Math. Phys. 1 pp 111– (1986) · Zbl 0638.70016 [35] DOI: 10.1016/S0021-9045(03)00033-9 · Zbl 1027.46037 [36] DOI: 10.1088/0959-7174/14/1/017 · Zbl 1077.47513 [37] DOI: 10.1088/0305-4470/39/29/023 · Zbl 1095.81061 [38] DOI: 10.1088/1751-8113/40/20/013 · Zbl 1113.81050 [39] DOI: 10.1166/asl.2008.019 [40] DOI: 10.1088/1751-8113/43/10/105302 · Zbl 1205.81093 [41] DOI: 10.1016/j.physleta.2010.02.005 · Zbl 1236.81094 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.