Asymptotic solution for first and second order integro-differential equations

TL;DR

This paper develops an asymptotic solution method for first and second order integro-differential equations with arbitrary kernels, extending beyond traditional tauberian theorems, and applies it to quantum Lindblad equations to analyze density matrix positivity.

Contribution

It introduces a novel asymptotic solution approach for integro-differential equations that surpasses existing tauberian theorem limitations, with applications to quantum dynamics.

Findings

Provides asymptotic solutions for integro-differential equations with arbitrary kernels.

Demonstrates conditions under which the density matrix remains positive.

Highlights the importance of eigenvalue structure of the Liouvillian in quantum positivity.

Abstract

This paper addresses the problem of finding an asymptotic solution for first and second order integro-differential equations containing an arbitrary kernel, by evaluating the corresponding inverse Laplace and Fourier transforms. The aim of the paper is to go beyond the tauberian theorem in the case of integral-differential equations which are widely used by the scientific community. The results are applied to the convolute form of the Lindblad equation setting generic conditions on the kernel in such a way as to generate a positive definite density matrix, and show that the structure of the eigenvalues of the correspondent liouvillian operator plays a crucial role in determining the positivity of the density matrix.