Gap Control by Singular Schrödinger Operators in a Periodically Structured Metamaterial
DOI:
https://doi.org/10.15407/mag14.03.270Анотація
Ми розглядаємо сiм'ю $\{\mathcal{H}^\varepsilon\}_{\varepsilon >0}$ of $\varepsilon\mathbb{Z}^n$-перiодичних операторiв Шредiнгера з $\delta'$-взаємодiями, якi локалiзованi на сiм'ї замкнених компактних поверхонь; мiнiмальна комiрка перiодичностi мiстить $m\in\mathbb{N}$ таких поверхонь. Показано, що при $\varepsilon\to 0$ i при певному порядку сили взаємодiї $\mathcal{H}^\varepsilon$ має на кiнцевих iнтервалах не бiльше m спектральних лакун. Крiм того, гранична поведiнка перших m лакун повнiстю контролюється за допомогою належного вибору цих поверхонь i сили взаємодiї.
Mathematics Subject Classification: 35P05, 35P20, 35J10, 35B27
Ключові слова:
перiодичний оператор Шредiнгера, $\delta'$-взаємодiя, спектральна лакуна, асимптотика власних значеньПосилання
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd edition, Amer. Math. Soc. Chelsea Publishing, Providence, R.I., 2005.
D. Barseghyan and A. Khrabustovskyi, Gaps in the spectrum of a periodic quantum graph with periodically distributed δ 0 -type interactions, J. Phys. A: Math. Theor. 48 (2015), 255201. https://doi.org/10.1088/1751-8113/48/25/255201
J. Behrndt, P. Exner, and V. Lotoreichik, Schrödinger operators with δ- and δ 0 interactions on Lipschitz surfaces and chromatic numbers of associated partitions, Rev. Math. Phys. 26 (2014), 1450015. https://doi.org/10.1142/S0129055X14500159
J. Behrndt, M. Langer, and V. Lotoreichik, Schrödinger operators with δ and δ 0 potentials supported on hypersurfaces, Ann. Henri Poincaré 14 (2013), 385–423. https://doi.org/10.1007/s00023-012-0189-5
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Amer. Math. Soc., Providence, R.I., 2013.
B.M. Brown, V. Hoang, M. Plum, and I.G. Wood, Floquet–Bloch theory for elliptic problems with discontinuous coefficients. Spectral theory and analysis, Oper. Theory Adv. Appl., 214, Birkhäuser, Basel, 2011, 1–20. https://doi.org/10.1007/978-3-7643-9994-8_1
P. Exner and A. Khrabustovskyi, On the spectrum of narrow Neumann waveguide with periodically distributed δ 0 traps, J. Phys. A: Math. Theor. 48 (2015), 315301. https://doi.org/10.1088/1751-8113/48/31/315301
P. Exner and O. Post, Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005), 77–115. https://doi.org/10.1016/j.geomphys.2004.08.003
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966. https://doi.org/10.1007/978-3-642-53393-8
A. Khrabustovskyi, Opening up and control of spectral gaps of the Laplacian in periodic domains, J. Math. Phys. 55 (2014), 121502. https://doi.org/10.1063/1.4902935
P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser, Basel, 1993.
M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York-London, 1978.
B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28 (1978), 377–385. https://doi.org/10.1016/0022-1236(78)90094-0
