  We study the spectral properties of the Laplace type operator on the circle.
We discuss various approximations for the heat trace, the zeta function and the
zeta-regularized determinant. We obtain a differential equation for the heat
kernel diagonal and a recursive system for the diagonal heat kernel
coefficients, which enables us to find closed approximate formulas for the heat
trace and the functional determinant which become exact in the limit of
infinite radius. The relation to the generalized KdV hierarchy is discussed as
well.
