Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity. (English. Russian original) Zbl 0941.35015

St. Petersbg. Math. J. 11, No. 2, 203-232 (2000); translation from Algebra Anal. 11, No. 2, 1-40 (1999).
The authors of the paper discuss the absolute continuity of the spectrum for the periodic Schrödinger operator \(H\) in \(L_2(\mathbb{R}^d)\) \((d\geq 2)\), with magnetic and electric potential. Besides a survey of the results already published and the analysis of the difficulties arising, the paper contains detailed proofs of some statements. The operator \(H\) is defined by a metric tensor field \(g(x)\equiv \{g^{jl}(x)\}\) \((j,l=1,2,\dots,d)\), \(x\in \mathbb{R}^d\) \((d\geq 3)\), \(H\equiv H(g,A,v) =(D-A(x))^*g(x) (D-A(x))+V(x)\), where \(D\equiv -i\nabla\), \(A\) is a magnetic vector-valued potential and \(V\) is an electric scalar-valued potential. Here the functions \(g,A,V\) are assumed to be real and periodic with respect to the lattice \(\Gamma\) in \(\mathbb{R}^d\). According to the Floquet theory the spectrum of \(H\) has band structure. For the periodic operator \(H\equiv -\Delta+V(x)\) the absence of degenerate bands has been proved by L. Thomas (1972). He suggests a method of analytic continuation in the complex quasimomentum. Here the authors show that \(H\) is absolutely continuous provided that \(V\) is a \(\Gamma\)-periodic function in \(V\in L_r(\Omega)\) with \(r>1\) \((d=2)\), in \(L^0_{d/2,\infty}(\Omega)\) \((d=3,4)\) and in \(L^0_{d-2,\infty} (\Omega)\) \((d\geq 5)\). Another result is that under similar assumptions and also \(\text{div} aA=0\) \((g=a)\), \(\int_\Omega A dx=0\) and \(A\in L_{2r}(\Omega)\) with \(r>1\) \((d=2)\) it follows that \(A\in C^{2d+3}(d\geq 3)\).


35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
35P15 Estimates of eigenvalues in context of PDEs