Number of bound states and index theorems in quantum mechanics?

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Discussion Overview

The discussion revolves around the existence of an index theorem for an n-dimensional Hamiltonian of the form H = -∇² + V(x) that could potentially count the bound states in the discrete spectrum. Participants explore the implications of such a theorem and the conditions necessary for its validity.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes the idea of an index theorem that counts bound states for the Hamiltonian H = -∇² + V(x).
  • Another participant expresses skepticism, suggesting that the arbitrary nature of the potential in multiple dimensions likely negates the possibility of such an index theorem.
  • A question is raised regarding the specific conditions that would be necessary for an index theorem to apply in this context.
  • One participant references the Atiyah-Singer index theorem, indicating that the concept may be complex and beyond their current understanding.
  • Another participant suggests that instead of seeking an index theorem, it might be more appropriate to look for an analytical index that provides a formula for calculating the number of bound states without solving the Schrödinger equation.
  • One participant mentions that the topic has been studied extensively in the past and points to specific references, including Thirring's work and recent papers, as useful resources for further exploration.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of an index theorem for counting bound states. There are competing views regarding the feasibility and conditions necessary for such a theorem.

Contextual Notes

Participants acknowledge the complexity of the potential in n dimensions and the potential limitations of applying index theorems in this context. The discussion reflects uncertainty regarding the applicability of established mathematical results to the problem at hand.

tom.stoer
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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?
 
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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>.
 
and what would be the conditions for an index theorem?
 
yes, something like that
 
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:
tom.stoer said:
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
 

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