New Mexico Tech (June 5, 2014) Heat Trace and Functional Determinant in One Dimension Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA E-mail:
[email protected] arXiv:1406.1499v2 [math-ph] 10 Nov 2014 We study the spectral properties of the Laplace type operator on the cir- cle. 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 ker- nel 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. 1 Introduction The heat kernel of elliptic partial differential operators is one of the most powerful tools in mathematical physics (see, for example, [12, 13, 4, 5, 17, 22] and further references therein). Of special importance are the spectral functions such as the heat trace, the zeta function and the functional determinant that enable one to study the spectral properties of the corresponding operator. The one-dimensional case is a very special one which exhibits an underly- ing symmetry that has deep relations to such diverse areas as integrable systems, infinite-dimensional Hamiltonian systems, isospectrality etc (see [1, 20, 21, 9, 6, 10, 7, 14, 15, 16], for example). Moreover, it has been shown that one can ob- tain closed formulas which express the functional determinant in one dimension in terms of a solution to a particular initial value problem; see, e.g., [19, 11, 18].