Number of bound states and index theorems in quantum mechanics?

1. Oct 9, 2012

tom.stoer

Just an idea: is there an index theorem for an n-dimensional Hamiltonian

$$H = -\triangle^{(n)} + V(x)$$

which "counts" the bound states

$$(H - E) \,u_E(x) = 0$$

i.e. eigenfunctions and eigenvalues in the discrete spectrum of H?

2. Oct 9, 2012

dextercioby

Due to the huge dosis of arbitrary in your setting (the potential in n variables needn't be seprarable), I'm quite sure the answer is <no>.

3. Oct 9, 2012

tom.stoer

and what would be the conditions for an index theorem?

4. Oct 9, 2012

5. Oct 9, 2012

tom.stoer

yes, something like that

6. Oct 9, 2012

tom.stoer

I probably should not ask for an index theorem but simply for an analytical index; the idea is to have a formula to calculate the number of bound states instead of solving the SE and counting them

Last edited: Oct 9, 2012
7. Oct 10, 2012

A. Neumaier

This was studies a lot about 40 years ago. Volume 3 of Thirring's ''Course in mathematical physics'' has some results. scholar.google for >>bounds on the number of "bound states"<<
or variations turns up useful references. For related recent results see, e.g.,
http://arxiv.org/pdf/1108.1002 or
http://www2.imperial.ac.uk/~alaptev/Papers/Lapt_Beijing.pdf

8. Oct 10, 2012