p-Adic Schrödinger-type operator with point interactions

A p-adic Schrödinger-type operator Dα + VY is studied. Dα (α > 0) is the operator of fractional differentiation and VY = n i,j=1 bij δxj , · δxi (bij ∈ C) is a singular potential containing the Dirac delta functions δx concentrated on a set of points Y = {x1, . . . , xn} of the field of p-adic numbers Qp. It is shown that such a problem is well posed for α >1/2 and the singular perturbation VY is form-bounded forα >1. In the latter case, the spectral analysis of η-self-adjoint operator realizations ofDα+VY in L2(Qp) is carried out. © 2007 Elsevier Inc. All rights reserved