Semiclassical analysis for highly degenerate potentials

This paper characterizes the semi-classical limit of the fundamental energy,

$E(h):= \sigma_1[-h^2\Delta+a(x);\Omega],$

and ground state $\psi_h$ of the Schrödinger operator $-h^2\Delta+a$ in a bounded domain $\Omega$, in the highly degenerate case when $a\geq 0$ and $a^{-1}(0)$ consists of two components, say $\Omega_{0,1}$ and $\Omega_{0,2}$. The main result establishes that

$\lim_{h\downarrow 0} \frac{E(h)}{h^2}= \min\left\{\sigma_1[-\Delta;\Omega_{0,i}], \; i=1,2\,\right\}$

and that $\psi_h$ approximates in $H_0^1(\Omega)$ the ground state of $-\Delta$ in $\Omega_{0,i}$ if

$\sigma_1[-\Delta;\Omega_{0,i}]< \sigma_1[-\Delta;\Omega_{0,j}],\qquad j \in \{1,2\}\setminus\{i\}.$