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Given any bounded domain $\Omega\subset \mathbb R^n$, $n\geq 2$, with a Lipschitz boundary, let $\{(\lambda_k,\phi_k)\}_{k\in \mathbb N}$ be the Dirichlet eigenvalues and eigenfunctions of $-\Delta$ ...

I have asked the same question on math.SE, without much success so I'm trying my luck here too. Let $M$ be an $n$-dimensional manifold, with $n$ even and consider the heat kernel of the Laplacian on $...

In M.A. Shubin's book Pseudodifferential Operators and Spectral Theory, we have the following statement. Let $X\subset\mathbb{R}^n$ be an open set, and fix a symbol $a\in S_{\varrho,\delta}^m(X\...

Let $S$ be a nonnegative self-adjoint operator on a complex Hilbert space $X$. (For example, $X$ consists of functions on $\mathbb R^d$; it could be $L^2(\mathbb R^d), \dot{H}^2(\mathbb R^d)$, etc.)...