Abstract
We consider Schrödinger operators H = −Δ + V in L 2(Ω) where the domain Ω ⊂ ℝ d+1+ and the potential V = V (x, y) are periodic with respect to the variable x ∈ ℝd. We assume that Ω is unbounded with respect to the variable y ∈ ℝ and that V decays with respect to this variable. V may contain a singular term supported on the boundary.
We develop a scattering theory for H and present an approach to prove absence of singular continuous spectrum. Moreover, we show that certain repulsivity conditions on the potential and the boundary of Ω exclude the existence of surface states. In this case, the spectrum of H is purely absolutely continuous and the scattering is complete.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
M.Sh. Birman, Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions, (Russian) Vestnik Leningrad. Univ. 17, no. 1 (1962), 22–55.
M.Sh. Birman, M.Z. Solomyak, On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential, J. Anal. Math. 83 (2001), 337–391.
E.B. Davies, B. Simon, Scattering Theory for Systems with Different Spatial Asymptotics on the Left and Right, Commun. Math. Phys. 63 (1978), 277–301.
M. Eastham, H. Kalf, Schrödinger-type operators with continuous spectra, Research Notes in Mathematics, 65. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982.
N. Filonov, F. Klopp, Absolute continuity of the spectrum of a Schrödinger operator with a potential which is periodic in some directions and decays in others, Documenta Math. 9 (2004), 107–121; Erratum: ibd., 135–136.
N. Filonov, F. Klopp, Absolutely continuous spectrum for the isotropic Maxwell operator with coefficients that are periodic in some directions and decay in others, Comm. Math. Phys., to appear.
N. Filonov, A.V. Sobolev, Absence of the singular continuous component in the spectrum of analytic direct integrals, Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 318 (2004), 298–307.
R.L. Frank, On the scattering theory of the Laplacian with a periodic boundary condition. I. Existence of wave operators, Documenta Math. 8 (2003), 547–565.
R.L. Frank, On the Laplacian in the halfspace with a periodic boundary condition, preprint, mp-arc 04-407.
R.L. Frank, R.G. Shterenberg, On the scattering theory of the Laplacian with a periodic boundary condition. II. Additional channels of scattering, Documenta Math. 9 (2004), 57–77.
R. Hempel, I. Herbst, Bands and gaps for periodic magnetic Hamiltonians, Partial differential operators and mathematical physics (Holzhau, 1994), 175–184, Oper. Theory Adv. Appl., 78, Birkhäuser, Basel, 1995.
I.V. Kamotskii, S.A. Nazarov, Exponentially decreasing solutions of the problem of diffraction by a rigid periodic boundary, Math. Notes 73, no. 1–2 (2003), 129–131.
P. Kuchment, On some spectral problems of mathematical physics, Partial differential equations and inverse problems, 241–276, Contemp. Math., 362, Amer. Math. Soc., Providence, RI, 2004.
S.T. Kuroda, Scattering theory for differential operators. III. Exterior problems, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974), 227–241, Lecture Notes in Math., 448, Springer, Berlin, 1975.
A.W. Saenz, Quantum-mechanical scattering by impenetrable periodic surfaces, J. Math. Phys. 22, no. 12 (1981), 2872–2884.
B. Simon, Phase space analysis of simple scattering systems: extensions of some work of Enss. Duke Math. J. 46, no. 1 (1979), 119–168.
L. Thomas, Time dependent approach to scattering from impurities in a crystal. Comm. Math. Phys. 33 (1973), 335–343.
D.R. Yafaev, Mathematical Scattering Theory, Amer. Math. Soc., 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Frank, R.L., Shterenberg, R.G. (2007). On the Spectrum of Partially Periodic Operators. In: Janas, J., Kurasov, P., Laptev, A., Naboko, S., Stolz, G. (eds) Operator Theory, Analysis and Mathematical Physics. Operator Theory: Advances and Applications, vol 174. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8135-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8135-6_4
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8134-9
Online ISBN: 978-3-7643-8135-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)