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The essential spectrum of two-dimensional Schrödinger operators with perturbed constant magnetic fields. (English) Zbl 0564.35021
The 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)$$\}$$.
Reviewer: H.Kalf

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 35P05 General topics in linear spectral theory for PDEs 47A10 Spectrum, resolvent
##### Keywords:
Schrödinger operator; essential spectrum; commutators; curl a
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