All Questions

Let $D$ be a first-order self-adjoint elliptic differential operator acting on sections of a vector bundle $S$ over a closed manifold $M$. Then it is well-known that the various eigenspaces $E_\lambda$...

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...

Let $a>0$ be a fixed number and consider the Hermite operator (or harmonic oscillator) defined by \begin{equation} Hf(x)=x^2f(x)-f''(x) \end{equation} for any smooth function $f$ compactly ...

Let $\Omega\subset \mathbb R^d$ be a smooth, bounded domain. Let $(e_n)_{n\geq 0}\subset L^2(\Omega)$ be the Hilbert basis generated by the Dirichlet-Laplacian eigenfunctions, i-e $-\Delta e_n=\...

Consider a generalised harmonic oscillator of the form $$ A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n, $$ where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The ...

Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$. My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...