Asymptotics of determinants of discrete Schr\"odinger operators

We consider the asymptotics of the determinants of large discrete Schr"odinger operators, i.e. “discrete Laplacian $+$ diagonal”: $$T_n(f)=-[{\delta }_{j,j+1}+{\delta }_{j+1,j}]+diag(f\left(\frac{1}{n}\right),f\left(\frac{2}{n}\right),\dots ,f\left(\frac{n}{n}\right))$$ We extend a result of M. Kac, who found a formula for $$\underset{n\to \mathrm{\infty }}{\lim }\frac{\det (T_n(f))}{G(f{)}^n}$$ in terms of the values of $f$, where $G(f)$ is a constant. We extend this result in two ways: First, we consider shifting the index: Let $$T_n(f;\epsilon )=-[{\delta }_{j,j+1}+{\delta }_{j+1,j}]+diag(f\left(\frac{\epsilon }{n}\right),f\left(\frac{1+\epsilon }{n}\right),\dots ,f\left(\frac{n-1+\epsilon }{n}\right))$$ We calculate $\lim \det T_n(f;\epsilon )\mathrm{/}G(f{)}^n$ and show that this limit can be any positive number by shifting $\epsilon$, even though the asymptotic eigenvalue distribution of $T_n(f;\epsilon )$ does not depend on $\epsilon$. Secondly, we derive a formula for the asymptotics of $\det T_n(f)/G(f)^n$when$f$ has jump discontinuities. In this case the asymptotics depend on the fractional part of $cn$, where $c$ is a point of discontinuity.