On point interactions in one dimension. (English) Zbl 0561.35023

Albeverio, Sergio; Gesztesy, Friedrich; Høegh-Krohn, Raphael; Kirsch, W.

J. Oper. Theory 12, 101-126 (1984).

The authors study the limit behavior of the Hamiltonians \[ H_{\epsilon,N}=-d^ 2/dx^ 2+\epsilon^{- 2}\sum^{N}_{j=1}\lambda_ j(\epsilon)V_ j((1/\epsilon)(x-x_ j)),\quad \epsilon >0, \] in the space \(L^ 2({\mathbb{R}}^ 1,dx)\), where \(\epsilon\) \(\downarrow 0\). In particular, the sufficient conditions are given for the norm resolvent convergence of \(H_{\epsilon,N}\) to the Hamiltonian describing N point interactions centered at \(x_ j\) of strengths \(\alpha_ j\equiv \lambda '\!_ j(0)\int V_ j(x)dx.\) The behavior of eigenvalues and resonances in the limit \(\epsilon\) \(\downarrow 0\) is discussed.

Reviewer: M.A.Perelmuter

MSC:

35J10	Schrödinger operator, Schrödinger equation
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

second order differential operator of the Schrödinger type; limit behavior; Hamiltonians; point interactions