# zbMATH — the first resource for mathematics

Spectral asymptotics for Schrödinger operators with periodic point interactions. (English) Zbl 1006.34081
Spectral asymptotics are considered for Schrödinger operators with periodic point interactions generated in $$L_2(\mathbb{R})$$ by the expression $H=-{d^2\over dx^2}+\sum\limits_{n\in \mathbb{Z}}\alpha_n \delta(x-n),$ where $$\delta$$ is the Dirac delta-function and $$\alpha_n$$ are real constants. It is shown that the first terms in the asymptotics determine the class of unitary equivalent operators uniquely.

##### MSC:
 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
Full Text:
##### References:
 [1] Akhiezer, N.I.; Glazman, I.M., Theory of linear operators in Hilbert space, Monographs and studies in mathematics, I, II, (1981), Pitman Boston · Zbl 0467.47001 [2] Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Kirsch, W., On point interactions in one dimension, J. operator theory, 12, 101-126, (1984) · Zbl 0561.35023 [3] Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Holden, H., Solvable models in quantum mechanics, (1988), Springer-Verlag Berlin/New York · Zbl 0679.46057 [4] Albeverio, S.; Kurasov, P., Singular perturbations of differential operators, (2000), Cambridge Univ. Press · Zbl 0945.47015 [5] Avron, J.; Exner, P.; Last, Y., Periodic Schrödinger operators with large gaps and wannier – stark ladders, Phys. rev. lett., 72, 896-899, (1994) · Zbl 0942.34503 [6] Boman, J.; Kurasov, P., Finite rank singular perturbations and distributions with discontinuous test functions, Proc. amer. math. soc., 126, 1673-1683, (1998) · Zbl 0894.34079 [7] Carreau, M., Four-parameter point-interaction in 1D quantum systems, J. phys. A, 26, 427-432, (1993) · Zbl 0778.58069 [8] Cheon, T.; Shigehara, T., some aspects of generalized contact interaction in one-dimensional quantum mechanics, mathematical results in quantum mechanics (Prague, 1998), Oper. theory adv. appl., 108, 203-208, (1999) · Zbl 0970.81015 [9] Chernoff, P.R.; Hughes, R.J., A new class of point interactions in one dimension, J. funct. anal., 111, 97-117, (1993) · Zbl 0790.47039 [10] Demkov, Yu.N.; Ostrovsky, V.N., Zero-range potentials and their applications in atomic physics, (1988), Plenum New York [11] P. Exner, and, H. Grosse, Some properties of the one-dimensional generalized point interactions (a torso), electronic preprint arXiv:math-ph/9910029, 1999. [12] P. Exner, H. Neidhardt, and, V. Zagrebnov, Potential approximations to δ′: An inverse Klauder phenomenon with norm-resolvent convergence, electronic preprint arXiv:math-ph/0103027. [13] Gesztesy, F.; Holden, H., A new class of solvable models in quantum mechanics describing point interactions on the line, J. phys. A, 20, 5157-5177, (1987) · Zbl 0627.34030 [14] Gesztesy, F.; Holden, H.; Kirsch, W., On energy gaps in a new type of analytically solvable model in quantum mechanics, J. math. anal. appl., 134, 9-29, (1988) · Zbl 0669.47002 [15] Gesztesy, F.; Kirsch, W., One-dimensional Schrödinger operators with interactions singular on a discrete set, J. reine angew. math., 362, 28-50, (1985) · Zbl 0565.34018 [16] Hughes, R., Generalized kronig – penney Hamiltonians, J. math. anal. appl., 222, 151-166, (1998) · Zbl 0914.47066 [17] Kiselev, A., Some examples in one-dimensional “geometric” scattering on manifolds, J. math. anal. appl., 212, 263-280, (1997) · Zbl 0914.58037 [18] Korotyaev, E., Inverse problem for periodic “weighted” operators, J. funct. anal., 170, 188-218, (2000) · Zbl 0970.47021 [19] Kreǐn, M., On the characteristic function A(λ) of a linear canonical system of differential equations of second order with periodic coefficients, Prikl. mat. mehk., 21, 320-329, (1957) [20] Kronig, R.de L.; Penney, W.G., Quantum mechanics of electrons in crystal lattices, Proc. roy. soc. London, 130, 499-513, (1931) · Zbl 0001.10601 [21] Kurasov, P.; Pavlov, B., An electron in a homogeneous crystal of point-like atoms with internal structure. II, Teoret. mat. fiz., 74, 82-93, (1988) · Zbl 1268.81144 [22] Kurasov, P., Zero-range potentials with internal structures and the inverse scattering problem, Lett. math. phys., 25, 287-297, (1992) · Zbl 0761.35075 [23] Kurasov, P., Distribution theory for discontinuous test functions and differential operators with generalized coefficients, J. math. anal. appl., 201, 297-323, (1996) · Zbl 0878.46030 [24] K. Makarov, On delta-like interactions with internal structure and semibounded from below three-body Hamiltonian, preprint FUB/HEP, 88-13:1-16, 1988. [25] Mikhailets, V.A.; Sobolev, A.V., Common eigenvalue problem and periodic Schrödinger operators, J. funct. anal., 165, 150-172, (1999) · Zbl 0933.47032 [26] Pavlov, B.S., The theory of extensions, and explicitly solvable models, Uspekhi mat. nauk, 42, 99-131, (1987) · Zbl 0648.47010 [27] Pavlov, B.S., Boundary conditions on thin manifolds and the semiboundedness of the three-body Schrödinger operator with point potential, Mat. sb. (N.S.), 136, 163-177, (1988) [28] Šeba, P., Some remarks on the δ′-interaction in one dimension, Rep. math. phys., 24, 111-120, (1986) · Zbl 0638.70016 [29] Šeba, P., The generalized point interaction in one dimension, Czechoslovak J. phys., 36, 667-673, (1986) [30] Šeba, P., A remark about the point interaction in one dimension, Ann. physik, 44, 323-328, (1987) · Zbl 0648.35026 [31] T. Shigehara, H. Mizoguchi, T. Mishima, and, T. Cheon, Realization of a four-Parameter family of generalized one-dimensional contact interactions by three nearby delta potentials with renormalized strengths, electronic preprint arXiv:quant-ph/9812006, 1998. [32] T. Shigehara, H. Mizoguchi, T. Mishima, and, T. Cheon, Band spectra of Kronig-Penney model with generalized contact interaction, electronic preprint arXiv:quant-ph/9912049, 1999.
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.