Non-random perturbations of the Anderson Hamiltonian in the 1-D case

Abstract

Recently (S. Molchanov and B. Vainberg, Non-random perturbations of the Anderson Hamiltonian, J. Spectral Theory 50 (2) (2011), pp. 179–195), two of the authors applied the Lieb method to the study of the negative spectrum for particular operators of the form H = H ₀ − W. Here, H ₀ is the generator of the positive stochastic (or sub-stochastic) semigroup, W(x) ≥ 0 and W(x) → 0 as x → ∞ on some phase space X. They used the general results in several ‘exotic’ situations, among them the Anderson Hamiltonian H ₀. In the 1-D case, the subject of this article, we will prove similar, but more precise results.

Keywords:

• Anderson Hamiltonian
• Schrödinger operator
• negative eigenvalues

AMS Subject Classifications:

• Primary 34L20
• Secondary 34L15