  We investigate the close connection between metastability of the reversible
diffusion process X defined by the stochastic differential equation
dX_t=-\nabla F(X_t) dt+\sqrt2\epsilon dW_t,\qquad \epsilon >0, and the spectrum
near zero of its generator -L_{\epsilon}\equiv \epsilon \Delta -\nabla
F\cdot\nabla, where F:R^d\to R and W denotes Brownian motion on R^d. For
generic F to each local minimum of F there corresponds a metastable state. We
prove that the distribution of its rescaled relaxation time converges to the
exponential distribution as \epsilon \downarrow 0 with optimal and uniform
error estimates. Each metastable state can be viewed as an eigenstate of
L_{\epsilon} with eigenvalue which converges to zero exponentially fast in
1/\epsilon. Modulo errors of exponentially small order in 1/\epsilon this
eigenvalue is given as the inverse of the expected metastable relaxation time.
The eigenstate is highly concentrated in the basin of attraction of the
corresponding trap.
