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The ladder operator in quantum mechanics are the operators $$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$ and $$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\rig …

We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$. I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\ma …

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator on …

I wonder if the following is true: Let $\alpha >0$ be a positive real number, do we have $$\Vert H^{\alpha} \psi''\Vert \le \Vert H^{\alpha+1} \psi\Vert,$$ where $H = -\frac{d^2}{dx^2} + x^2$ is the q …

Let $A$ and $B$ be positive-definite matrices such that $A \le B.$ By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$ I am now curious under …

We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is stric …

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am wonde …

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a …