Schroedinger Operator: Heat Kernel and Its Applications

TL;DR

This paper explores the geometry of Schroedinger operators using Hamiltonian and Lagrangian formalisms, constructs heat kernels with various coefficient matrices, and applies these to solve Riccati equations and generalize classical results.

Contribution

It introduces a new method for constructing heat kernels for Schroedinger operators with diagonal and non-diagonal coefficients, and extends classical operator results.

Findings

Constructed heat kernels for Schroedinger operators with complex coefficients

Obtained a globally closed solution to matrix Riccati equations

Generalized classical results on celebrated operators

Abstract

In this paper, we study the geometry associated with Schroedinger operator via Hamiltonian and Lagrangian formalism. Making use of a multiplier technique, we construct the heat kernel with the coefficient matrices of the operator both diagonal and non-diagonal. For applications, we compute the heat kernel of a Schroedinger operator with terms of lower order, and obtain a globally closed solution to a matrix Riccati equations as a by-product. Besides, we finally recover and generalise several classical results on some celebrated operators.