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With $z=x+iy$, we use the Fourier transformation in $(x,y)$ to see that $H$ is unitarily equivalent to $$ \frac12\begin{pmatrix}2m&\xi-i\eta\\ \xi+i\eta&-2m\end{pmatrix}, \text{whose eigenvalues are } …

The standard $n$-dimensional harmonic oscillator is the operator $ \mathcal H=\frac{1}{2}\sum_{1\le j\le n}(D_j^2+x_j^2), \text{ $D_j=-i\partial_{x_j}$}, $ and its spectral decomposition is $$ \mathca …

The class $\Sigma^1$ of symbols on $\mathbb R^{2n}$ is made with $C^\infty$ functions $a$ of $X=(x,\xi)\in \mathbb R^n\times\mathbb R^n$ such that $$ \vert\partial_X^\alpha a\vert\le C_\alpha(1+\vert …

Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small …

Here are some very classical references: M. Cwikel. Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. of Math. (2), 106(1):93–100, 1977. E. Lieb. …

Let us consider $\mathbb S^d$ the unit Euclidean sphere of $\mathbb R^{d+1}$ and let $\Delta_{\mathbb S^d}$ be the Laplace operator on $\mathbb S^d$. We have $$ -\Delta_{\mathbb S^d}=\sum_{k\in \mathb …

Let me point out a more specific result for hyperbolic polynomials, known as Bronshtein's theorem (see e.g. the preprint https://arxiv.org/abs/1309.2150 by A. Parusinski & A. Rainer). Let $p(X,y)$ be …

Take $f=g=1$, $\Delta$ the usual Laplace operator on $\mathbb S^1$, then $$ C_0=\partial_\theta e^{\partial_\theta^2}, $$ can be identified to the diagonal infinite matrix $ (ik e^{-k^2})_{k\in \mathb …

Consider the (stationary) Schrödinger equation $$ -\Delta u+ Vu=0, $$ or the differential inequality $\vert\Delta u \vert\le \vert V u\vert$, where $V$ is some "potential" function. The following uniq …

Yes: look at Theorem 8.1.4 in the first volume of Hörmander's ALPDO (Springer Grund. 256). For the classical (conic) wave-front-set, given any closed conic set $S$, you can construct a distribution $u …

Here is a statement, due to Lagrange. Take a square system $\dot x=f(x)$ with $f$ of class $C^2$ such that $f(0)=0$ and the spectrum of $df(0)$ is contained in {$z\in \mathbb C , \Re{z}\le -\delta$} …

Let me first recall what is the so-called paving conjecture: for any $\epsilon >0$, there exists $r\in \mathbb N$ such that for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a partitio …

Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that $$\forall u\in\mathscr S(\mathbb R^n),\quad \langle\mathbf 1_C^{Weyl}u,u\rangle\ …