Mathematical Notes

, Volume 73, Issue 1, pp 46–57

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

  • L. I. Danilov

DOI: 10.1023/A:1022169916738

Cite this article as:
Danilov, L.I. Mathematical Notes (2003) 73: 46. doi:10.1023/A:1022169916738


We prove the absolute continuity of the spectrum of the Schrödinger operator in \(L^2 ({\mathbb{R}}^n )\), \(n \geqslant 3\), with periodic (with a common period lattice \(\Lambda\)) scalar \(V\) and vector \(A \in C^1 ({\mathbb{R}}^n ,{\mathbb{R}}^n )\) potentials for which either \(\user1{A} \in H_{\user2{loc}}^\user1{q} \user2{(}\mathbb{R}^\user1{n} \user2{;}\mathbb{R}^\user1{n} \user2{)}\), \(2q > n - 2\), or the Fourier series of the vector potential \(A\) converges absolutely, \(V \in L_w^{p(n)} (K)\), where \(K\) is an elementary cell of the lattice \(\Lambda\), \(p(n) = n/2\) for \(n = 3, 4, 5, 6\), and \(p(n) = n - 3\) for \(n \geqslant 7\), and the value of \(\user2{lim}_{\user1{t} \to \user2{ + }\infty } \left\| {\theta _\user1{t} V} \right\|_{\user1{L}_\user1{w}^{\user1{p}\user2{(}\user1{n}\user2{)}} \user2{(}\user1{K}\user2{)}} \) is sufficiently small, where \(\theta _\user1{t} \user2{(}\user1{x}\user2{) = 0 if }\left| {V\user2{(}\user1{x}\user2{)}} \right| \leqslant \user1{t}\) and \(\theta _t (x) = 1\) otherwise, \(x \in K\), and \(t > 0\).

absolute continuity Schrödinger operator spectrum vector and scalar potentials periodic potential quadratic form 

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • L. I. Danilov
    • 1
  1. 1.Physico-Technical Institute, Udmurtia Scientific Center, The Ural DivisionRussian Academy of SciencesIzhevsk