The low-temperature limit of transfer operators in fixed dimension (Q5956812)

The paper deals with the integral operator \(K\) defined on \(L^2 ({\mathbb R}^m)\) by its kernel \[ K(x, y) = (\beta J)^{m/2}\exp(-\beta V(x)/2) \exp( - \beta J|x-y|^2 /2)\exp(-\beta V(y)/2), \] where \(\beta>0\) is the inverse temperature, \(J>0\) is a coupling constant and \(V\in C^\infty ({\mathbb R}^m)\) is a non-negative potential, which has a finite number of non-degenerate global minima and \(\inf_{|x|>R} V(x) >0\) for some \(R>0\). By means of the WKB-method, the author studies the first eigenvalue and corresponding eigenfunction of \(K\) in the strong-coupling (SC) limit \(J \rightarrow +\infty\) and in the low-temperature (LT) limit \(\beta \rightarrow +\infty\).