Albeverio, Sergio; Cacciapuoti, Claudio; Finco, Domenico Coupling in the singular limit of thin quantum waveguides. (English) Zbl 1137.81330 J. Math. Phys. 48, No. 3, 032103, 21 p. (2007). Summary: We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e., a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling. Cited in 1 ReviewCited in 20 Documents MSC: 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 47N50 Applications of operator theory in the physical sciences PDF BibTeX XML Cite \textit{S. Albeverio} et al., J. Math. Phys. 48, No. 3, 032103, 21 p. (2007; Zbl 1137.81330) Full Text: DOI arXiv References: [1] DOI: 10.1006/jfan.1995.1068 · Zbl 0822.35002 [2] Albeverio S., Solvable Models in Quantum Mechanics, 2. ed. (2005) [3] DOI: 10.1017/CBO9780511758904 [4] DOI: 10.1088/0305-4470/38/22/010 · Zbl 1071.47003 [5] Bollé D., J. Oper. Theory 13 pp 3– (1985) [6] Bonciocat, A. I. , Ph.D., thesis, Universität Bonn, 2006. [7] DOI: 10.1063/1.2213789 · Zbl 1112.81360 [8] DOI: 10.1142/S0129055X95000062 · Zbl 0837.35037 [9] DOI: 10.1016/j.geomphys.2004.08.003 · Zbl 1095.58007 [10] DOI: 10.1063/1.528538 · Zbl 0693.46066 [11] DOI: 10.1016/0034-4877(89)90023-2 · Zbl 0749.47038 [12] Feynman R. P., Eng. Sci. 23 pp 22– (1960) [13] DOI: 10.1016/j.physleta.2006.03.070 · Zbl 1160.81386 [14] DOI: 10.1142/S0129055X01000843 · Zbl 1029.81067 [15] DOI: 10.1088/0305-4470/32/4/006 · Zbl 0928.34066 [16] DOI: 10.1088/0959-7174/12/4/201 · Zbl 1063.35525 [17] DOI: 10.1088/0959-7174/14/1/014 · Zbl 1063.81058 [18] DOI: 10.1088/0305-4470/38/22/013 · Zbl 1070.81062 [19] DOI: 10.1006/jmaa.2000.7415 · Zbl 0982.35076 [20] DOI: 10.1016/S0034-4877(06)80048-0 · Zbl 1143.47017 [21] DOI: 10.1088/0305-4470/38/22/015 · Zbl 1072.81019 [22] DOI: 10.1007/s00023-006-0272-x · Zbl 1187.81124 [23] Reed M., Methods of Modern Mathematical Physics (1978) · Zbl 0401.47001 [24] DOI: 10.1007/s002050100164 · Zbl 0997.49003 [25] DOI: 10.1063/1.1699299 [26] Saitō Y., Anal. Int. Math. J. Anal. Appl. 21 pp 171– (2001) [27] Simon B., Quantum Mechanics for Hamiltonians Defined as Quadratic Forms (1971) · Zbl 0232.47053 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.