The essential spectrum of two-dimensional Schrödinger operators with perturbed constant magnetic fields.

*(English)*Zbl 0564.35021The author considers the Schrödinger operator \(L:=((1/i)\nabla -a)^ 2\upharpoonright C_ 0^{\infty}({\mathbb{R}}^ 2)\) when the vector potential a is smooth and the vector field \(curl a\) tends to a positive number \(B_ 0\) at infinity. He shows that the essential spectrum of the closure of L consists of the odd multiples of \(B_ 0\) by establishing the following interesting result about commutators.

Theorem. Let P and Q be symmetric operators in a Hilbert space which are defined on a dense domain \(\Omega\) which is left invariant of P and Q. Suppose that \(P^ 2+Q^ 2\) is essentially self-adjoint and \(i(PQ- QP)u=(1+K)u\) (u\(\in \Omega)\) for some K which is relatively compact with respect to \(P^ 2+Q^ 2\). Then \(\sigma_ e(\overline{P^ 2+Q^ 2})\) is either empty or consists of the positive odd integers.

\(\{\) Reviewer’s remark. The spectrum of [may be totally different when curl a tends to zero at infinity. See K. Miller and B. Simon, Phys. Rev. Lett. 44, 1706-1707 (1980)\(\}\).

Theorem. Let P and Q be symmetric operators in a Hilbert space which are defined on a dense domain \(\Omega\) which is left invariant of P and Q. Suppose that \(P^ 2+Q^ 2\) is essentially self-adjoint and \(i(PQ- QP)u=(1+K)u\) (u\(\in \Omega)\) for some K which is relatively compact with respect to \(P^ 2+Q^ 2\). Then \(\sigma_ e(\overline{P^ 2+Q^ 2})\) is either empty or consists of the positive odd integers.

\(\{\) Reviewer’s remark. The spectrum of [may be totally different when curl a tends to zero at infinity. See K. Miller and B. Simon, Phys. Rev. Lett. 44, 1706-1707 (1980)\(\}\).

Reviewer: H.Kalf