arXiv subjects
Publications and source records attributed to Thierry Harge.
We prove that the partition function associated to a perturbation of the semi-classical harmonic oscillator is the Borel sum of its semi-classical expansion.
In previous works, we used a so-called deformation formula in order to study, in particular, the Borel summability of the heat kernel of some operators. A goal of this paper is to collect miscellaneous remarks related to these works. Here the complex setting plays an important role. Moreover, the deformation formula provides a solution of the heat equation in "unusual" cases. We also give a uniqueness statement concerning these cases.
Let us consider a time-dependent differential operator quadratic with respect to the phase variables. Let us consider a multiplication operator defined with the help of a "small" matrix-valued function. Under suitable conditions, we give an "explicit" expression of "the" heat kernel associated to the sum of the two previous operators for small time.
We study the Borel summability of the small time expansion of the heat kernel associated to a first order perturbation of a Laplacian. An explicit formula for this kernel plays a central role. As a consequence, we get a Poisson formula on the torus.
We study the Borel summability of the small time expansion of the heat kernel for the Schrödinger operator in several dimensions. On the torus, we prove a Poisson formula.