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As mentioned in the comments, knowing both eigenvalues and eigenfunctions gives you enough information to find the shape of the domain, so to make this problem more challenging one might ask what mini …

I can identify one individual who scrutinized the Bernstein-Robinson manuscript and established its validity before it was published in PJM: Paul Halmos, I Want to be a Mathematician: An Automathog …

An explicit construction of generalized ladder operators $A^\pm=\mp d/dx+W(x)$ exists if the Hamiltonian can be factorized as $$H=-\frac{d^2}{dx^2}+V(x)=A^+ A^- +E_0,$$ with $E_0$ the lowest eigenvalu …

Indeed, this is the result of Davies - Pseudo-Spectra, the Harmonic Oscillator and Complex Resonances (1982): The resolvent operator $(H-zI)^{-1}$ of $$H=-d^2/dx^2+cx^2,\;\;\operatorname{Re}c>0,\;\; \ …

Q: Does anybody know how to numerically overcome this pseudospectral effect? The key idea is "normal ordering". Rewrite the problem in such a way that annihilation operators $a$ appear to the right of …

The $1/r$ Coulomb potential needs to be regularized, typically this is done by studying the Yukawa potential $e^{-\alpha r}/r$ and taking the limit $\alpha\rightarrow 0$ at the end. A recent critical …

Well, the absolute value squared of a wave packet $\Psi_t(x)$ has the interpretation of a time-dependent probability distribution $P_t(x)=|\Psi_t(x)|^2$ for the stochastic variable $x$ (position on a …

Upon Fourier transformation $x\mapsto k$, this becomes a diagonal operator with matrix elements $\langle k|\ln D|k'\rangle=2\pi \delta(k-k')\ln k$. So to find the matrix elements in the $x$-representa …

In the physics context, with $A$ and $B$ creation operators of two identical particles, the fact that only $AB=+BA$ and $AB=-BA$ are nontrivially allowed implies that the particles must be either boso …

This line of argument goes back to 1874 papers by J. W. L. Glaisher and J. O’Kinealy. A discussion of this early work and a critical examination (conditions on $a_n$ for which the theorem holds) is gi …

Q: Is the square root of the Moore-Penrose inverse equal to the Moore-Penrose inverse of the square root? A: No, here is a simple counter example, for the positive semi-definite matrix $M={{1\,0}\choo …

This line of thought has been explored by several authors: Miller and Thaheem, Derivatives of matrix order (1997) Naber, Matrix Order Differintegration (2003) da Porciuncula, Derivatives and integral …

This is the adiabatic theorem. You need a gap condition, where the eigenvalue remains separated from other eigenvalues during the time evolution, to prevent the eigenstate of $A$ from evolving into a …

The Euler beta function governs the statistics of Preferential attachment processes. A preferential attachment process is any of a class of processes in which some quantity, typically some form …

This review from 2007 contains a very extensive reference list. Some bound state problems in quantum mechanics (Dirk Hundertmark). “Spectral Theory and Mathematical Physics: A Festschrift in Honor o …