# 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