, Volume 73, Issue 1-2, pp 46-57

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

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Abstract

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 \(t > 0\) .