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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.

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  • $\begingroup$ 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. $\endgroup$
    – Christian Remling
    Commented Jun 13, 2015 at 0:07
  • $\begingroup$ Thanks Christian... could you elaborate what you mean by "extra" boundary condition? I want to move the boundary. $\endgroup$
    – Austen
    Commented Jun 15, 2015 at 9:52
  • $\begingroup$ 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. $\endgroup$
    – Christian Remling
    Commented Jun 15, 2015 at 16:48
  • $\begingroup$ 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 $\alpha\in \mathbb C$. $\endgroup$
    – Delio Mugnolo
    Commented Oct 8, 2015 at 6:16

1 Answer 1

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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.

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