Infinitesimal variation of spectrum of Schrödinger operator with changing domain

Question

Suppose we have a Schrödinger operator

$$-\frac{d^2}{dx^2}+V(x)$$

defined on $[a,b]$ with Dirichlet boundary conditions. I am interested in whether there are any results for the variation of the spectrum $\{\lambda_i\}$ with changing $a$ and $b$, i.e. the quantities

$$\frac{\partial \lambda_i}{\partial a}, \frac{\partial \lambda_i}{\partial b}$$

I assume that $V(x)$ is defined on $\mathbb{R}$, so that changing the endpoints will also lead to the inclusion of new parts of the potential.

This post imported from StackExchange MathOverflow at 2015-06-14 09:06 (UTC), posted by SE-user Austen

Comments

An obvious remark is that $\partial\lambda_j/\partial a>0$. Introducing an extra boundary condition is a rank one perturbation of the resolvent, so there's a lot of theory you can apply here.

This post imported from StackExchange MathOverflow at 2015-06-14 09:06 (UTC), posted by SE-user Christian Remling