Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator (Q2751495)

Let \(\Delta^{(m)}\) be the Laplacian in \(L^2(\mathbb{R}^m)\). The authors investigate the semiclassical Kac operator (transfer operator) NEWLINE\[NEWLINEK^{(m)}= e^{-V^{(m)}/2} e^{h^2\Delta^{(m)}} e^{-V^{(m)}/2},NEWLINE\]NEWLINE where the potential \(V^{(m)}\) can be written as NEWLINE\[NEWLINEV^{(m)}(x)= \sum_{j\in\mathbb{Z}/m\mathbb{Z}} v(x_j)+ w(x_j- x_{j+1}).NEWLINE\]NEWLINE Let \(\mu^{(m)}_1(h)\) denote the highest eigenvalue of \(K^{(m)}\) and \(\Lambda(h)=-\lim_{m\to\infty} {\ln\mu^{(m)}_1(h)\over m}\). Under suitable assumptions, which are satisfied by many examples from statistical mechanics, a formal asymptotic expansion for \(\Lambda(h)\) in powers of \(h\) is constructed. This expansion leads to precise estimates of \(\Lambda(h)\).NEWLINENEWLINEFor the entire collection see [Zbl 0966.00028].