Abstract
We consider the Laplace operator in an infinite planar strip with a periodic delta interaction. The width of the strip is fixed and for simplicity is chosen equal to π. The delta interaction is introduced on a periodic system of curves. Each curve consists of a finite number of segments, each having smoothness C1. The curves are supposed to be strictly internal and do not intersect the boundaries of the strip. The period of their location is 2επ, where ε is a sufficiently small number. The function describing the delta interaction is also periodic on the system of curves and is assumed to be bounded and measurable. The main result is the following fact. If ε ≤ ε0, where ε0 is a certain explicitly calculated number and the norm of the function describing the delta interaction is smaller than some explicit constant, then the lower part of the spectrum of the operator has no internal gaps. The lower part is understood as the band of the spectrum until some point, which is explicitly calculated in terms of the parameter ε as a rather simple function. This result can be considered as a first step to the proof of the strong Bethe-Sommerfeld conjecture on the complete absence of gaps in the spectrum of an operator for a sufficiently small period of location of delta interactions.
Similar content being viewed by others
References
D. I. Borisov, “On gaps in the lower part of the spectrum of a periodic magnetic operator in a strip,” Sovrem. Mat. Fundam. Napravl. 63(3), 373–391 (2017). doi https://doi.org/10.22363/2413-3639-2017-63-3-373-391
M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators,” Proc. Steklov Inst. Math. 171, 1–121 (1987).
C. B. E. Beeken, Periodic Schrödinger Operators in Dimension Two: Constant Magnetic Fields and Boundary Value Problems, PhD Thesis (Univ. Sussex, Brighton, 2002).
D. Borisov and G. Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions,” J. Phys. A: Math. Theor. 42(36), article 365205 (2009). doi https://doi.org/10.1088/1751-8113/42/36/365205
D. Borisov, R. Bunoiu, and G. Cardone, “On a waveguide with frequently alternating boundary conditions: Homogenized Neumann condition,” Ann. H. Poincaré 11(8), 1591–1627 (2010). doi https://doi.org/10.1007/s00023-010-0065-0
D. Borisov, R. Bunoiu, and G. Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: Homogenization and asymptotics,” Zeit. Angew. Math. Phys. 64(3), 439–472 (2013). doi https://doi.org/10.1007/s00033-012-0264-2
B. E. J. Dahlberg and E. Trubowitz, “A remark on two dimensional periodic potentials,” Comment. Math. Helv. 57(1), 130–134 (1982). doi https://doi.org/10.1007/BF02565850
Y. Karpeshina, “Spectral properties of the periodic magnetic Schrödinger operator in the high-energy region. Two-dimensional case,” Comm. Math. Phys. 251(3), 473–514 (2004). doi https://doi.org/10.1007/s00220-004-1129-0
A. Mohamed, “Asymptotic of the density of states for the Schrodinger operator with periodic electromagnetic potential,” J. Math. Phys. 38(8), 4023–4051 (1997). doi https://doi.org/10.1063/1.532105
L. Parnovski, “Bethe-Sommerfeld conjecture,” Ann. H. Poincaré 9(3), 457–508 (2008). doi https://doi.org/10.1007/s00023-008-0364-x
L. Parnovski and A. V. Sobolev, “Bethe-Sommerfeld conjecture for periodic operators with strong perturbations,” Invent. Math. 181(3), 467–540 (2010). doi https://doi.org/10.1007/s00222-010-0251-1
Funding
This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00046).
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2018, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Vol. 24, No. 2, pp. 46–53.
Rights and permissions
About this article
Cite this article
Borisov, D.I. Gaps in the Spectrum of the Laplacian in a Strip with Periodic Delta Interaction. Proc. Steklov Inst. Math. 305 (Suppl 1), S16–S23 (2019). https://doi.org/10.1134/S0081543819040047
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543819040047