# Infinitesimal variation of spectrum of Schrödinger operator with changing domain

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.

• 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. – Christian Remling Jun 13 '15 at 0:07
• Thanks Christian... could you elaborate what you mean by "extra" boundary condition? I want to move the boundary. – Austen Jun 15 '15 at 9:52
• You can consider the sum of the operators on $L^2(a,a')\oplus L^2(a',b)$ and compare it with the original operator. – Christian Remling Jun 15 '15 at 16:48
• I guess you have to impose some basic smoothness assumption on $V$ to make the question less hopeless. Assume $V=\alpha\delta_{x_0}$, then the spectrum can become very different as soon as you reach $x_0$, playing a bit with $\alpha\in \mathbb C$. – Delio Mugnolo Oct 8 '15 at 6:16

## 1 Answer

Take $a=-1$, $b=1$. Expand the endpoints by infinitesimal $\epsilon$. Let $y=\frac{x}{1+\epsilon}$. Then you get an equation on [-1,1] with potential of the form $V + \epsilon yV'$, so this can be considered a perturbation problem on a fixed domain.