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3
votes
1answer
171 views

Random Hankel matrix and eigenvalues distribution

I would like to know if there are any theoretical results on the distribution of the eigenvalues of Hankel matrices. I seek a result like the Marchenko–Pastur distribution for random matrices.
9
votes
4answers
488 views

Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?

I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
2
votes
1answer
105 views

When does the “norm of quasi-eigenvectors” matter in calculations? For which physical results are these even used?

Which physical system in nonrelativistic quantum mechanics is actually described by a model, where the norm of the "position eigenstate" (i.e. the delta distribution as limit of vectors in the ...
2
votes
1answer
118 views

Are Quantum Physics and statistical theory always the same as semiclassical approximations?

Quantum Mechanics and Statistical physics is a bit hard , could we then study only the WKB approximation ? In the form: replace $ \sum_{n=0}^{\infty}exp(- \beta E_{n})=Z(\beta)\sim\iint ...
1
vote
1answer
206 views

Inverse of a sum of two easy matrices

Let $A$ be a symmetric positive semidefinite matrix and $I$ the identity matrix. Given the linear equation $$ y = A(A + \sigma^2I)^{-1} x $$ Write $A$ in terms of its eigenvectors $|u_i\rangle$, ...
1
vote
2answers
134 views

eigenvalue staircase and hamiltonians

Let two Hamiltonians $H_{1}$ and $H_{2}$ be defined in such a manner that their eigenvalue staircases satisfy $ N_{1} (E) = N_{2} (E)+A +O(E^{-1})$ What can we say about their potentials $ V_{1} ...
24
votes
4answers
1k views

In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?

A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential. Is there a one to one correspondence between the potential and its spectrum? If the ...