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The method of stationary phase is very well-known and employed in many areas of physics and mathematics, and, of course, included in various versions as theorem in textbooks, especially on pseudors an …

It is well-known that if $\phi:\mathbb{R}^n \longrightarrow \mathbb{R}$ is a function with $\phi(0) = 0$, $\phi(x)>0$ if $x \neq 0$ and $D^2\phi(0) \geq 0$, then the integral $$\int_{\mathbb{R}^n} e^{ …

Let $f:(0, \infty) \longrightarrow (0, \infty)$ be a monotonously increasing function (in fact, a step function) and let $P$ be a polynomial of degree $N$. Suppose I know that for some $k$, the limit …

Considering Schrödinger operators $$ H(\hbar) = \hbar \Delta + V $$ where $V$ is some potential, perturbation theory tells that the eigenvalues of $H(\hbar)$ are holomorphic on some region containing …

Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous. Question: Is there still an asymptotic expansion of the heat kernel of the form $$ p_t(x, y) \sim (4 …

This is crosspost from math.stackexchange https://math.stackexchange.com/questions/311213/heat-kernel-asymptotics-on-manifold-with-boundary where the question did not yield any answer On a closed Riemannian … Edit: Clearly, the same asymptotics as above cannot hold on a manifold with boundary. …

Let $\|\cdot \|$ be some matrix norm on the space of $n \times n$ matrices. Denote $$ M(r) := \{ A \in \mathrm{Mat}_{n \times n}(\mathbb{Z}) \mid \| M \| \leq r \}.$$ Denote by $p(r)$ the fraction of …