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Let me define by $C$ the operator of complex conjugation, then $T= K+HC$; I presume by the conjugate operator $T^\ast$ you mean the Hermitian adjoint of $T$; as discussed here, $C^\ast=C$, so $T^\ast= …

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 …

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 …

These are indeed two different definitions, see the discussion in One-parameter Semigroups of Positive Operators. That reference also gives two different names for the two definitions, peripheral spec …

If $(d/dy)g(y)=g’(y)$, so no operator commutator, then $$e^{d/dy}g(y)=g(y+1),$$ so $$f(y)=y^{-n}/g(y+1).$$ If instead you choose the operator identity $(d/dy)g(y)=g’(y)+g(y)d/dy$, then $$f(y)=(1/g(y)) …

This may give you more information than you need, but this perturbed harmonic oscillator can be solved in terms of parabolic cylinder functions. The eigenvalues are given by Equation 3.19 of The energ …

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 …

My intuition for the bound state/discrete spectrum relationship is that the discrete spectrum $\lambda_1,\lambda_2,\ldots$ of a Hermitian operator allows the construction of a set of eigenfunctions $f …

This answer was for the original conjecture in the OP (before the edits). I'm leaving it for the record. To test the conjectured formula, let me take a simple case of diagonal $2\times 2$ matrices, $ …

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 …

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,\;\; \ …

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 …

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 …