Higher-order heat equation and the Gelfand-Dickey hierarchy

TL;DR

This paper explores the heat kernel for higher-order differential operators, linking its properties to the Gelfand-Dickey hierarchy and algebraic geometry, and characterizing operators with finitely many heat kernel terms.

Contribution

It provides explicit formulas for Hadamard's coefficients, relates them to the Gelfand-Dickey hierarchy, and characterizes operators with finitely many heat kernel terms via algebraic and spectral properties.

Findings

Heat kernel has finitely many terms iff the operator belongs to a rank-one commutative ring with a rational spectral curve.

Explicit formulas for Hadamard's coefficients are derived using pseudo-differential operators.

Operators with finitely many heat kernel terms are characterized as rational solutions of the Gelfand-Dickey hierarchy.

Abstract

In this paper we analyze the heat kernel of the equation $\partial_tv =\pm\mathcal{L} v$, where $\mathcal{L}=\partial_x^N+u_{N-2}(x)\partial_x^{N-2}+\cdots+u_0(x)$ is an $N$-th order differential operator and the $\pm$ sign on the right-hand side is chosen appropriately. Using formal pseudo-differential operators, we derive an explicit formula for Hadamard's coefficients in the expansion of the heat kernel in terms of the resolvent of $\mathcal{L}$. Combining this formula with soliton techniques and Sato's Grassmannian, we establish different properties of Hadamard's coefficients and relate them to the Gelfand-Dickey hierarchy. In particular, using the correspondence between commutative rings of differential operators and algebraic curves due to Burchnall-Chaundy and Krichever, we prove that the heat kernel consists of finitely many terms if and only if the operator $\mathcal{L}$ belongs to a rank-one commutative ring of differential operators whose spectral curve is rational with only one cusp-like singular point, and the coefficients $u_j(x)$ vanish at $x=\infty$. We also characterize these operators $\mathcal{L}$ as the rational solutions of the Gelfand-Dickey hierarchy with coefficients $u_j $ vanishing at $x=\infty$, or as the rank-one solutions of the bispectral problem vanishing at $\infty$.