All Questions

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the ...

The Cwikel–Lieb–Rosenbljum inequality asserts that, for any potential $V:\mathbb{R}^n\to\mathbb{R}$, we have $$(\mbox{number of eigenvalue} \leq 0\mbox{ , counted with multiplicity, of }-\Delta+V\,)\...

Let $H=H_0+V$ be a Hamiltonian on $\mathbb{R}^3$ where $H_0=-\frac{\Delta}{2m}$ is the free Hamiltonian and $V$ is a potential. Let us assume first that $V$ decays sufficiently fast at infinity and ...

The Mathieu equation is an important eigenvalue problem in Mathematical Physics that is completely understood in its properties, although there is no "direct way" of expressing eigenvalues and ...

I'm a physicist who needs mathematical advice: Let $A= \sum_{i=1}^{\infty} a_i P_{\phi_i}$ be a self-adjoint operator with projectors $P_{\phi_i}$ on the orthonormal eigenbasis $(\phi_i).$ Let $$B= \...

Let $\mu$ be a Borel measure on $\mathbb{R}$. By the Lebesgue decomposition theorem, there exists measures $\mu_\text{pp}$, $\mu_\text{ac}$ and $\mu_\text{sing}$ such that $\mu = \mu_\text{pp}+\mu_\...