@article{AIHPA_1997__66_4_359_0, author = {Exner, P.}, title = {A duality between {Schr\"odinger} operators on graphs and certain {Jacobi} matrices}, journal = {Annales de l'I.H.P. Physique th\'eorique}, pages = {359--371}, publisher = {Gauthier-Villars}, volume = {66}, number = {4}, year = {1997}, mrnumber = {1459512}, zbl = {0949.34073}, language = {en}, url = {http://www.numdam.org/item/AIHPA_1997__66_4_359_0/} }
TY - JOUR AU - Exner, P. TI - A duality between Schrödinger operators on graphs and certain Jacobi matrices JO - Annales de l'I.H.P. Physique théorique PY - 1997 SP - 359 EP - 371 VL - 66 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/item/AIHPA_1997__66_4_359_0/ LA - en ID - AIHPA_1997__66_4_359_0 ER -
Exner, P. A duality between Schrödinger operators on graphs and certain Jacobi matrices. Annales de l'I.H.P. Physique théorique, Tome 66 (1997) no. 4, pp. 359-371. http://www.numdam.org/item/AIHPA_1997__66_4_359_0/
[Ad] Scattering matrices for microschemes, Oper.Theory: Adv. Appl., Vol. 59, 1992, pp. 1-10. | MR | Zbl
,[AGHH] Solvable Models in Quantum Mechanics, Springer, Heidelberg 1988. | MR | Zbl
, , and ,[AL] Quantum percolation and ballistic conductance on a lattice of wires, Phys. Rev., Vol. B45, 1992, pp. 1074-1095.
and ,[AEL] Periodic Schrödinger operators with large gaps and Wannier-Stark ladders, Phys.Rev.Lett., Vol. 72 (1994, pp. 896-899. | Zbl
, and ,[ARZ] Adiabatic transport in multiply connected systems, Rev.Mod.Phys., Vol. 60, 1988, pp. 873-915. | MR
, and ,[Az] Energy spectrum of a conduction electron in a magnetic field, Sov. Phys. JETP, Vol. 19, 1964, pp. 634-647.
,[Be] Gap labelling theorem for Schrödinger operators in Number Theory and Physics, M. WALDSCHMIDT et al., eds., Springer, Heidelberg 1992, pp. 538-630. | MR | Zbl
,[BFLT] Quasi-periodic interactions with a metal-insulator transition, Phys. Rev., Vol. B26, 1982, pp. 3024-3030.
, , and ,[BKN] New aspects of Krein's extension theory, Ukrainian J.Math., Vol. 46, 1994, pp. 37-54. | MR | Zbl
, and ,[BT] The free Dirac operator on compact and non-compact graphs, J. Math. Phys., Vol. 31, 1990, pp. 1157-1163. | MR | Zbl
and ,[CFKS] Schrödinger Operators, Springer, Berlin 1987. | Zbl
, , and ,[DSS] From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré, Vol. A42, 1985, pp. 283-309. | Numdam | MR | Zbl
, and ,[E1] The absence of the absolutely continuous spectrum for δ' Wannier-Stark ladders, J. Math. Phys., Vol. 36, 1995, pp. 4561-4570 . | MR | Zbl
,[E2] Lattice Kronig-Penney models, Phys. Rev. Lett., Vol. 74, 1995, pp. 3503- 3506.
,[E3] Contact interactions on graphs, J. Phys., Vol. A29, 1996, pp. 87-102 . | MR | Zbl
,[EG] Band spectra of rectangular graph superlattices, Phys. Rev., Vol. B53, 1996, pp. 7275-7286.
and ,[EŠ] Free quantum motion on a branching graph, Rep. Math. Phys., Vol. 28, 1989, pp. 7-26. | MR | Zbl
and ,[GH] A new class of solvable models in quantum mechanics describing point interactions on the line, J. Phys., Vol. A20, 1987, pp. 5157-5177. | MR | Zbl
and ,[GHK] 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 | Zbl
, and ,[GLRT] Quantum mechanics on graphs, J. Phys., Vol. A27, 1994, pp. 6881-6892. | MR | Zbl
, , and ,[GP] Scattering problem on noncompact graphs, Teor. Mat. Fiz., Vol. 74, 1988, pp. 345-359. | MR | Zbl
and ,[Ha] Single band motion of conduction electrons in a uniform magnetic field, Proc. Roy. Soc. (London), Vol. A68, 1955, pp. 874-878. | Zbl
,[Ho] Energy levels and wavefunctions for Bloch electrons in rational and irrational magnetic fields, Phys. Rev., Vol. B14, 1976, pp. 2239-2249.
,[Kr] 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
,[La] Zero measure spectrum for almost Mathieu operator, Commun. Math. Phys., Vol. 164, 1994, pp. 421-432. | MR | Zbl
,[Ma] The integrated density of states for the difference Laplacian on the modified Koch curve, Commun. Math. Phys. Vol. 156, 1993, pp. 387-397. | MR | Zbl
,[Mar] Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications, Vol. 22; Birkhäuser, Basel 1986. | MR | Zbl
,[Ne] To the theory of self-adjoint extensions of symmetric operators with a spectral gap, Funkc. Anal. Appl., Vol. 19, 1985, pp. 81-82. | Zbl
,[Ph] The energy spectrum of an amorhous substance, Physica, Vol. 26, 1960, pp. 1185-1191. | MR | Zbl
,[PGF] A solvable model of quantum motion in an incommensurate potential, Phys. Rev., Vol. B29, 1984, pp. 6500-6512. | MR
, and ,[RS] Methods of Modern Mathematical Physics, II. Fourier Analysis. Self-adjointness, IV. Analysis of Operators, Academic Press, New York 1975, 1978. | MR
and ,[Re] Ordinary Differential Equations, J. Wiley, New York 1971. | MR | Zbl
,[RuS] Free-electron network model for conjugated systems, I. Theory, J. Chem. Phys., Vol. 21, 1953, pp. 1565-1581.
and ,[Sh] Discrete magnetic Laplacian, Commun. Math. Phys. 164, 1994, pp. 259-275. | MR | Zbl
,[Si] Almost periodic Schrödinger operators: a review, Adv. Appl. Math., Vol. 3, 1982, pp. 463-490. | MR | Zbl
,[Su] 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.
,