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On compact Riemannian surfaces (say without boundary), the Schrodinger operator I am interested in is of the form $-\Delta+2\kappa$, where $\kappa$ is the Gauss curvature. For minimal surfaces in $\mathbb{R}^3$, this is the Jacobi operator coming from the second variation of area. I am wondering if there are any recommended references/surveys on estimates on the first eigenvalue and other spectral properties of this operator. Thanks for your time!

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Surprisingly, not all is known about this Schrödinger type operator. The best I know is the following paper and reference therein.

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