Number of bound states and index theorems in quantum mechanics?

[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?

 

[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>.

 

[tom.stoer]

and what would be the conditions for an index theorem?

 

[dextercioby]

I think the notion of "index theorem" you mentioned would be in the sense of http://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem which is way over my head. My assertion above is just my gut-feeling.

 

[tom.stoer]

yes, something like that

 

[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

 

[A. Neumaier]

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?

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

 

[tom.stoer]

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

Great, many thanks, this is exactly what I was looking for!