[Paper Review] A deformation formula for the heat kernel

Thierry Hargé|arXiv (Cornell University)|Feb 7, 2013

Quantum chaos and dynamical systems5 references3 citations

TL;DR

This paper presents a deformation formula for the heat kernel of a non-autonomous, quadratic time-dependent Schrödinger operator perturbed by a matrix-valued potential. By conjugating the heat kernel with the free heat kernel, it derives an explicit series representation for the conjugated kernel using a deformation matrix tied to the classical Hamiltonian flow, establishing convergence for small time and analytic potentials.

ABSTRACT

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.

Motivation & Objective

• To derive an explicit representation of the heat kernel for a non-autonomous quadratic Schrödinger operator perturbed by a regular matrix-valued potential.
• To extend the deformation formula approach—previously used in autonomous or free cases—to the non-autonomous setting.
• To establish Borel summability of the small-time expansion of the conjugated kernel under suitable analyticity and reality conditions.
• To connect the deformation formula to Wiener and Feynman integrals via a heuristic path integral interpretation.
• To provide a rigorous construction of the conjugated kernel using the solution of a classical Hamiltonian system and a deformation matrix.

Proposed method

• Define the base operator $ P_0 $ as a second-order differential operator with time-dependent coefficients $ A(t), B(t), C(t) $, quadratic in $ \partial_x $ and $ x $.
• Introduce the conjugated kernel $ p_t^{\text{conj}}(x,y) $ via $ p_t(x,y) = p_t^0(x,y) p_t^{\text{conj}}(x,y) $, where $ p_t^0 $ is the free heat kernel.
• Derive a recursive integral representation for $ v_n $, the $ n $-th term in the series expansion of $ p_t^{\text{conj}} $, using iterated integrals over time intervals.
• Construct the deformation matrix $ K_t(s) $ from the classical Hamiltonian flow $ q_t^\natural(s) $, satisfying a propagator equation.
• Use the time derivative of $ K_t(s) $, related to the fundamental solution of the Hamiltonian system, to verify the evolution equation for $ v_n $.
• Prove that the series $ p_t^{\text{conj}} = \sum_{n=0}^\infty v_n $ satisfies the conjugated evolution equation via boundary and interior term analysis.

Experimental results

Research questions

• RQ1 How can the heat kernel of a non-autonomous quadratic Schrödinger operator be represented in terms of a deformation formula?
• RQ2 What is the role of the classical Hamiltonian flow in constructing the deformation matrix for the conjugated kernel?
• RQ3 Under what conditions is the small-time expansion of the conjugated kernel Borel summable?
• RQ4 How does the deformation formula relate to Wiener and Feynman path integrals in the non-autonomous case?
• RQ5 Can the conjugated kernel be rigorously constructed via a series expansion involving iterated integrals and differential operators?

Key findings

• The heat kernel $ p_t(x,y) $ for $ P_0 + c(t,x) $ is represented as $ p_t^0(x,y) p_t^{\text{conj}}(x,y) $, where $ p_t^{\text{conj}} $ admits a convergent series expansion in small time.
• The deformation matrix $ K_t(s) $, derived from the classical Hamiltonian flow $ q_t^\natural(s) $, encodes the time evolution of the system and satisfies a propagator equation.
• The conjugated kernel $ p_t^{\text{conj}} $ is constructed via iterated integrals of products of the potential $ c(s_j, z_j) $, evaluated along the classical trajectory.
• The time derivative of the deformation matrix satisfies $ \partial_t K_t(s,s') = q_t^\flat(s) A(t) {}^\mathfrak{t}q_t^\flat(s') $, linking it to the fundamental solution of the Hamiltonian system.
• The series for $ p_t^{\text{conj}} $ satisfies the conjugated evolution equation $ \partial_t v_n = A(t) \cdot \partial_x^2 v_n + c(t,x) v_{n-1} $ with $ v_n|_{t=0^+} = 0 $, ensuring consistency.
• The solution is valid for $ t \in \mathbb{C} $ with $ \mathcal{R}e(t) \geq 0 $, and under analyticity and reality conditions on $ A(t), B(t), C(t) $, the formula holds in a neighborhood of the origin.