Semiclassical analysis for highly degenerate potentials

Abstract:

This paper characterizes the semi-classical limit of the fundamental energy, \begin{equation*} E(h):= \sigma _1[-h^2\Delta +a(x);\Omega ], \end{equation*} 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 \begin{equation*} \lim _{h\downarrow 0} \frac {E(h)}{h^2}= \min \left \{\sigma _1[-\Delta ;\Omega _{0,i}], \; i=1,2 \right \} \end{equation*} and that $\psi _h$ approximates in $H_0^1(\Omega )$ the ground state of $-\Delta$ in $\Omega _{0,i}$ if \begin{equation*} \sigma _1[-\Delta ;\Omega _{0,i}]< \sigma _1[-\Delta ;\Omega _{0,j}],\qquad j \in \{1,2\}\setminus \{i\}. \end{equation*}