Rank-one perturbations and Anderson-type Hamiltonians. (English) Zbl 1419.81013

Liaw, Constanze

Banach J. Math. Anal. 13, No. 3, 507-523 (2019).

Summary: Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators \[ H_\omega=H+V_\omega \] on a separable Hilbert space \(\mathcal{H}\), where the perturbation is given by \[ V_\omega=\sum_n\omega_n(\cdot,\phi_n)\phi_n \] with a sequence \(\{\phi_n\}\subset \mathcal{H}\) and independent identically distributed random variables \(\omega_n\). We show that the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank-one perturbation. This result connects one of the least trackable perturbation problem (with almost surely noncompact perturbations) with one where the perturbation is “only” of rank-one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has nonzero Lebesgue measure.

MSC:

81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
30E20	Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47B80	Random linear operators
81Q15	Perturbation theories for operators and differential equations in quantum theory

Keywords:

rank-one perturbations; Anderson-type Hamiltonian; Krein-Lifshits spectral shift; discrete random Schrödinger operator

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