Beta Critical for the Schrodinger Operator with Delta Potential

Abstract

For the one dimensional Schrödinger operator in the case of Dirichlet boundary condition, we show that ${\beta }_{cr}$ is positive and zero for the case of Neumann and Robin boundary condition considering the potential energy of the form $V(x)=-\beta \delta (x-a)$ where, $\beta \ge 0,a>0.$ We prove that the ${\beta }_{cr}$ goes to infinity when the delta potential moves towards the boundary in dimension one with Dirichlet boundary condition. We also show that the ${\beta }_{cr}>0$ and $\beta \in (0,\frac{1}{2})$ considering Dirichlet problem with delta potential on the circle in dimension two.