- Intuitively one understands that if one is solving the Schroedinger's equation for energies E such that {x|U(x)≤E} is compact (..is there a weaker criteria?..) then the spectrum will turn out to be discrete and the wave-functions will decay exponentially for large values of x. What is the most rigorous statement and proof of this?
I want to focus on one potential,
U=|G(si)|2+|p|2∑i|∂G∂si|2+e22(∑i|si|2−n2|p|2−r)2
+2|σ|2(∑i|si|2+n2|p|2)
where e is a real constant, si, p and σ are complex and r is a real field and G is a degree n transverse homogeneous function in si.
Now apparently the following claims are true,
-
If r=0 then for any value of σ, the range of si, p where U(si,p)≤E is true is compact and hence the spectrum is discrete.
-
If σ≠0 then for any fixed non-zero value of r, the region of si and p where U(si,p;r)≤E is true is compact for small enough values of E and hence the spectrum is discrete for low-lying values of E below some Ecritical.
(..the above is apparently motivated by the fact that at si=p=0, U becomes constant and equal to e2r22 and hence independent of σ..so apparently if one goes above some critical value of E the spectrum is continuous thanks to field configurations with large |σ| but at si=p=0... )
- My reading of the literature is that the above two claims are true independent of the topology of the space on which the fields are valued though in a case of interest one wants the theory to be on a circle and hence I guess one wants to think of si, p, σ to be maps from S1 to C or R. If the theory is on a circle then semiclassically apparently the estimate for Ecritical is e2r222πR where R is the radius of the circle.
I would be glad if someone can help justify the above three claims.
This paper is the reference for this question
This post imported from StackExchange MathOverflow at 2014-09-01 11:18 (UCT), posted by SE-user Anirbit