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Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the metric $g$ and $q \in C^\infty(M)$. Also let $\nu$ denote the outward unit normal to $\partial M$ and $f \in C^\infty(\partial M)$.

I am looking for either a reference or a proof that the eigenvalue problem

$$\cases{Lu = \lambda u, \quad \text{on } M \\ \dfrac{\partial u}{\partial \nu} = fu, \quad \text{on } \partial M}$$

has a discrete real spectrum of the form

$$\lambda_1 < \lambda_2 \leq \cdots \leq \lambda_k \leq \cdots \nearrow \infty.$$

Moreover, is there a variational (minmax) characterization for these eigenvalues?

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  • $\begingroup$ If I understand things correctly, $\Delta_g$ is a non-negative Laplace operator. Are you assuming that $q\ge 0$? $\endgroup$
    – Bazin
    Apr 12 at 17:41
  • $\begingroup$ No, I’m not assuming this. $\endgroup$
    – Eduardo Longa
    Apr 12 at 18:54

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