The 1/r expansion for the critical multiple well problem

Abstract

We consider the critical multiple well problem $$H = - \Delta + \sum\limits_{i = 1}^n {V(x - rx_i )} ,$$ where −Δ+V(x) has a zero energy resonance. We prove that all eigenvalues and resonances ofH tending to zero as 1/r 2 are analytic in 1/r. We give an explicit equation for the lowest nonvanishing coefficient in the 1/r expansion for any of these eigenvalues or resonances and observe thatH has infinitely many resonances tending to zero. Forn=2 andn=3, we compute the coefficients explicitly and forn=2, we also give the next coefficient in the 1/r expansion.