Boundary regularity for the Ricci equation, geometric convergence, and Gel'fand's inverse boundary problem

Discrete Determinants and the Gel'fand-Yaglom formula

I present a partly pedagogic discussion of the Gel'fand-Yaglom formula for the functional determinant of a one-dimensional, second order difference operator, in the simplest settings. The formula is a textbook one in discrete Sturm-Liouville theory and orthogonal polynomials. A two by two matrix approach is developed and applied to Robin boundary conditions. Euler-Rayleigh sums of eigenvalues are computed. A delta potential is introduced as a simple, non-trivial example and extended, in an appendix, to the general case. The continuum limit is considered in a non--rigorous way and a rough compa…

通过 Gel'fand--Levitan--Marchenko 方程对若干负阶非线性波动方程的反散射方法

Inverse scattering method via Gel'fand--Levitan--Marchenko equation for some negative order nonlinear wave equations

本文利用三方程 Lax 层级构造负阶 AKNS 方程，并借助 Gel'fand--Levitan--Marchenko 方程的反散射方法，给出耦合标量场的非线性 Klein--Gordon 方程与负阶 mKdV 方程的 N 孤子解。

Gel'fand's inverse problem for the graph Laplacian

We study the discrete Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: $X=B\cup G$, where $B$ is called the set of the boundary vertices and $G$ is called the set of the interior vertices. We consider the case where the vertices in the set $G$ and the edges connecting them are unknown. Assume that we are given the set $B$ and the pairs $(λ_j,ϕ_j|_B)$, where $λ_j$ are the eigenvalues of the graph Laplacian and $ϕ_j|_B$ are the values of the corresponding eigenfunctions at the verti…