Michael Eckhoff
Precise Asymptotics of Small Eigenvalues of Reversible Diffusions 
in the Metastable Regime
(442K, postscript)

ABSTRACT.  We investigate the connection between metastability and the 
spectrum near zero corresponding to the elliptic, second-order, 
differential operator 
$L_\e\equiv -\e\D+\nabla F\cdot\nabla$, $\e>0$, 
with $F:\R^d\to\R$. For generic $F$ to each local minimum of $F$ 
there corresponds an eigenvalue of $L_\e$ which converges to 
zero exponentially fast as $\e\downarrow 0$. Modulo errors 
of exponentially small order in $\e$ this eigenvalue is 
given as the inverse of the expected metastable relaxation 
time. The corresponding eigenstate, which may be 
viewed as a metastable state, is highly concentrated in the 
basin of attraction of this trap.