arXiv math-ph/06050792006-05math-phmath.MP

整数上的热核展开与Toda格族

Heat kernel expansions on the integers and the Toda lattice hierarchy

Plamen Iliev

备注：: corrected typos
发表信息：: Selecta Math. (N.S.) 13 (2007), no. 3, 497--530
DOI：: 10.1007/s00029-007-0046-4

摘要：

考虑热方程$u_t=Lu$，其中$L$是离散变量$n$的二阶差分算子.基本解是以虚宗量的贝塞尔函数展开的。该展开式中的系数$α_k(n,m)$类似于（连续）薛定谔算子的Hadamard系数。我们推导出一个明确的公式$α_k$的波和伴随的波函数的Toda格族。作为这个结果的第一个应用程序，我们证明了这些系数的值的对角线$n=m$和$n=m+1$定义的微分差分方程是等价的户田格族的层次。利用这一事实以及差分算子交换环与代数曲线的对应关系，我们证明了基本解可以被求和，给出了一个仅涉及时变$t$中两个多项式系数Bessel函数的有限公式，当且仅当该算子$L$属于文[18]中构造的双谱算子族.

英文摘要：

We consider the heat equation $u_t=Lu$ where $L$ is a second-order difference operator in a discrete variable $n$. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients $α_k(n,m)$ in this expansion are analogs of Hadamard's coefficients for the (continuous) Schrodinger operator. We derive an explicit formula for $α_k$ in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals $n=m$ and $n=m+1$ define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable $t$, if and only if the operator $L$ belongs to the family of bispectral operators constructed in [18].

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