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The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator $$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$ Fix two numbers $\alpha,\beta \ …

This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows: We fix the vector $v=(1,1)$ (yet, it seems the final result does not depen …

Let $A$ be a symmetric matrix in $\mathbb R^n$ such that $A$ is positive definite and hence satisfies $0< \lambda \le A \le \Lambda < \infty.$ Let $T$ be a densely defined and closed operator from so …

We shall consider the matrix-valued differential operator $$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$ This is a $1$-pe …

In the question here the author asks for the eigenvalues of an operator $$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$ Here I would like to ask if one can extend this idea t …

The generator an OU process is given by $$A = \operatorname{tr}(QD^2)+\langle Bx,D\rangle.$$ This one possesses an invariant measure given by $$d\mu(x) = b(x) \ dx \text{ with } b(x) = \frac{1}{(4\pi) …