A duality between Schrödinger operators on graphs and certain Jacobi matrices

Exner, P.

Annales de l'I.H.P. Physique théorique, Tome 67 (1997), p. 359-371 / Harvested from Numdam

Détail

Publié le : 1997-01-01

MR 1459512 | Zbl 0949.34073

Bibliographie

[Ad] V.M. Adamyan, Scattering matrices for microschemes, Oper.Theory: Adv. Appl., Vol. 59, 1992, pp. 1-10. | MR 1246807 | Zbl 0793.47006

[AGHH] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer, Heidelberg 1988. | MR 926273 | Zbl 0679.46057

[AL] Y. Avishai and J.M. Luck, Quantum percolation and ballistic conductance on a lattice of wires, Phys. Rev., Vol. B45, 1992, pp. 1074-1095.

[AEL] J.E. Avron, P. Exner and Y. Last, Periodic Schrödinger operators with large gaps and Wannier-Stark ladders, Phys.Rev.Lett., Vol. 72 (1994, pp. 896-899. | Zbl 0942.34503

[ARZ] J.E. Avron, A. Raveh and B. Zur, Adiabatic transport in multiply connected systems, Rev.Mod.Phys., Vol. 60, 1988, pp. 873-915. | MR 969997

[Az] M.Ya. Azbel, Energy spectrum of a conduction electron in a magnetic field, Sov. Phys. JETP, Vol. 19, 1964, pp. 634-647.

[Be] J. Bellissard, Gap labelling theorem for Schrödinger operators in Number Theory and Physics, M. WALDSCHMIDT et al., eds., Springer, Heidelberg 1992, pp. 538-630. | MR 1221111 | Zbl 0833.47056

[BFLT] J. Bellissard, A. Formoso, R. Lima and D. Testard, Quasi-periodic interactions with a metal-insulator transition, Phys. Rev., Vol. B26, 1982, pp. 3024-3030.

[BKN] J.F. Brasche, V. Koshmanenko and H. Neidhardt, New aspects of Krein's extension theory, Ukrainian J.Math., Vol. 46, 1994, pp. 37-54. | MR 1294812 | Zbl 0835.47002

[BT] W. Bulla and T. Trenckler, The free Dirac operator on compact and non-compact graphs, J. Math. Phys., Vol. 31, 1990, pp. 1157-1163. | MR 1050469 | Zbl 0699.47032

[CFKS] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrödinger Operators, Springer, Berlin 1987. | Zbl 0619.47005

[DSS] F. Delyon, B. Simon and B. Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré, Vol. A42, 1985, pp. 283-309. | Numdam | MR 797277 | Zbl 0579.60056

[E1] P. Exner, The absence of the absolutely continuous spectrum for δ' Wannier-Stark ladders, J. Math. Phys., Vol. 36, 1995, pp. 4561-4570 . | MR 1347099 | Zbl 0884.47049

[E2] P. Exner, Lattice Kronig-Penney models, Phys. Rev. Lett., Vol. 74, 1995, pp. 3503- 3506.

[E3] P. Exner, Contact interactions on graphs, J. Phys., Vol. A29, 1996, pp. 87-102 . | MR 1381435 | Zbl 0916.47056

[EG] P. Exner and R. Gawlista, Band spectra of rectangular graph superlattices, Phys. Rev., Vol. B53, 1996, pp. 7275-7286.

[EŠ] P. Exner and P. Šeba, Free quantum motion on a branching graph, Rep. Math. Phys., Vol. 28, 1989, pp. 7-26. | MR 1109248 | Zbl 0749.47038

[GH] F. Gesztesy and H. Holden, A new class of solvable models in quantum mechanics describing point interactions on the line, J. Phys., Vol. A20, 1987, pp. 5157-5177. | MR 914699 | Zbl 0627.34030

[GHK] F. Gesztesy, H. Holden and W. Kirsch, On energy gaps in a new type of analytically solvable model in quantum mechanics, J. Math. Anal. Appl., Vol. 134, 1988, pp. 9-29. | MR 958850 | Zbl 0669.47002

[GLRT] J. Gratus, C.J. Lambert, S.J. Robinson and R.W. Tucker, Quantum mechanics on graphs, J. Phys., Vol. A27, 1994, pp. 6881-6892. | MR 1306837 | Zbl 0858.47041

[GP] N.I. Gerasimenko and B.S. Pavlov, Scattering problem on noncompact graphs, Teor. Mat. Fiz., Vol. 74, 1988, pp. 345-359. | MR 953298 | Zbl 0659.47006

[Ha] P.G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Roy. Soc. (London), Vol. A68, 1955, pp. 874-878. | Zbl 0065.23708

[Ho] D.R. Hofstadter, Energy levels and wavefunctions for Bloch electrons in rational and irrational magnetic fields, Phys. Rev., Vol. B14, 1976, pp. 2239-2249.

[Kr] M.G. Krein, The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications I, II, Mat. Sbornik, Vol. 20, 1947, pp. 431-495, Vol. 21, 1947, pp. 365-404. | Zbl 0029.14103

[La] Y. Last, Zero measure spectrum for almost Mathieu operator, Commun. Math. Phys., Vol. 164, 1994, pp. 421-432. | MR 1289331 | Zbl 0814.11040

[Ma] L. Malozemov, The integrated density of states for the difference Laplacian on the modified Koch curve, Commun. Math. Phys. Vol. 156, 1993, pp. 387-397. | MR 1233851 | Zbl 0786.58039

[Mar] V.A. Marchenko, Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications, Vol. 22; Birkhäuser, Basel 1986. | MR 897106 | Zbl 0592.34011

[Ne] G. Nenciu, To the theory of self-adjoint extensions of symmetric operators with a spectral gap, Funkc. Anal. Appl., Vol. 19, 1985, pp. 81-82. | Zbl 0581.47005

[Ph] P. Phariseau, The energy spectrum of an amorhous substance, Physica, Vol. 26, 1960, pp. 1185-1191. | MR 127344 | Zbl 0095.44604

[PGF] R. Praivge, D. Grempel and S. Fishman, A solvable model of quantum motion in an incommensurate potential, Phys. Rev., Vol. B29, 1984, pp. 6500-6512. | MR 749700

[RS] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis. Self-adjointness, IV. Analysis of Operators, Academic Press, New York 1975, 1978. | MR 493420

[Re] W.T. Reid, Ordinary Differential Equations, J. Wiley, New York 1971. | MR 273082 | Zbl 0212.10901

[RuS] K. Ruedenberg and C.W. Scherr, Free-electron network model for conjugated systems, I. Theory, J. Chem. Phys., Vol. 21, 1953, pp. 1565-1581.

[Sh] M.A. Shubin, Discrete magnetic Laplacian, Commun. Math. Phys. 164, 1994, pp. 259-275. | MR 1289325 | Zbl 0811.46079

[Si] B. Simon, Almost periodic Schrödinger operators: a review, Adv. Appl. Math., Vol. 3, 1982, pp. 463-490. | MR 682631 | Zbl 0545.34023

[Su] T. Sunada, Generalized Harper operator on a graph, in Workshop on Zeta Function in Number Theory and Geometric Analysis in Honor of Jun-Ichi Igusa, Johns Hopkins University 1993.