Spectral and propagation results for magnetic Schrödinger operators; a \(C^*\)-algebraic framework. (English) Zbl 1173.46048
Authors’ abstract: We study generalized magnetic Schrödinger operators of the form \(H_h(A, V)= h(\Pi^A)+ V\), where \(h\) is an elliptic symbol, \(\Pi^A= -i\nabla- A\), with \(A\) a vector potential defining a variable magnetic field \(B\), and \(V\) is a scalar potential. We are mainly interested in anisotropic functions \(B\) and \(V\). The first step is to show that these operators are affiliated to suitable \(C^*\)-algebras of (magnetic) pseudodifferential operators. A study of the quotient of these \(C^*\)-algebras by the ideal of compact operators leads to formulae for the essential spectrum of \(H_h(A, V)\), expressed as a union of spectra of some asymptotic operators, supported by the quasi-orbits of a suitable dynamical system. The quotient of the same \(C^*\)-algebras by other ideals give localization results on the functional calculus of the operators \(H_h(A, V)\), which can be interpreted as non-propagation properties of their unitary groups.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 35Q40 | PDEs in connection with quantum mechanics |
| 46L55 | Noncommutative dynamical systems |
| 46N50 | Applications of functional analysis in quantum physics |
| 47F05 | General theory of partial differential operators |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Keywords:
magnetic field; pseudodifferential operators; dynamical system; twisted crossed-product; essential spectrum; Schrödinger operatorsReferences:
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