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Also asked on MSE (https://math.stackexchange.com/questions/4987654/in-what-sense-is-a-change-of-boundary-conditions-a-finite-rank-perturbation and https://math.stackexchange.com/questions/4875398/how-...

Let $\Omega$ be an open region or a non-compact complete manifold, $L$ be an elliptic operator with possibly non vanishing zero-order term, e.g. $-\Delta+q$. Suppose $W$ is an operator such that $W(...

How do you prove basic facts on the Schrodinger equation using the spectral theorem? More precisely, here is what I have in mind. The version of the Spectral Theorem I am familiar with is the ...