Schrödinger particle on a half line (Q1074845)

The half line Hamiltonian \(H_{\alpha}=-d^ 2/dx^ 2\), \[ D(H_{\alpha})=\{f\in L^ 2(0,\infty);\quad f,f'\in AC(0,\infty)\quad and\quad f(0_+)+\alpha f'(0_+)=0\} \] is investigated. It is shown that the operator \(H_{\alpha}\) represents a limit of Schrödinger operators with local potentials with support shriking to the endpoint 0. \[ H_{\alpha}=.R.\lim_{\epsilon \to o_+}H_{\epsilon}\quad where\quad \] \[ H_{\epsilon}=-(d^ 2/dx^ 2)+(1/\epsilon^ 2)V(x/\epsilon);\quad V\in L(0,\infty),\quad \epsilon >0 \] is defined by the Dirichlet boundary condition at 0 and N.R.lim means the norm resolent limit.