- Intuitively one understands that if one is solving the Schroedinger's equation for energies $E$ such that $\{ x \vert U(x)\leq E \}$ 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 = \vert G(s_i)\vert ^2 + \vert p \vert ^2 \sum _ {i} \vert \frac{\partial G}{\partial s_i} \vert ^2 + \frac{e^2}{2}(\sum _i \vert s_i \vert ^2 - n^2 \vert p \vert^2 - r ) + 2\vert \sigma \vert ^2 (\sum _i \vert s_i \vert ^2 + n^2 \vert p \vert^2) $

where $e$ is a real constant, $s_i$, $p$ and $\sigma$ are complex and $r$ is a real field and $G$ is a degree $n$ transverse homogeneous function in $s_i$.

Now apparently the following claims are true,

If $r = 0$ then for any value of $\sigma$, the range of $s_i$, $p$ where $U(s_i,p) \leq E$ is true is compact and hence the spectrum is discrete.

If $\sigma \neq 0$ then for any fixed non-zero value of $r$, the region of $s_i$ and p where $U(s_i,p;r) \leq E$ is true is compact for small enough values of $E$ and hence the spectrum is discrete for low-lying values of $E$ below some $E_{critical}$.

(..the above is apparently motivated by the fact that at $s_i=p=0$, $U$ becomes constant and equal to $\frac{e^2 r^2}{2}$ and hence independent of $\sigma$..so apparently if one goes above some critical value of $E$ the spectrum is continuous thanks to field configurations with large $\vert \sigma \vert$ but at $s_i = p =0$... )

- 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 $s_i$, $p$, $\sigma$ to be maps from $S^1$ to $\mathbb{C}$ or $\mathbb{R}$. If the theory is on a circle then semiclassically apparently the estimate for $E_{critical}$ is $\frac{e^2 r^2}{2} 2\pi R$ 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