On the Eigenvalues of Certain Canonical Higher-Order Ordinary Differential Operators

From Toeplitz Eigenvalues through Green's Kernels to Higher-Order Wirtinger-Sobolev Inequalities

The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary differential operators, the norm of the Green kernels of these operators, the best constant in a Wirtinger-Sobolev inequality, and the conditioning of a special least squares problem. The main result of the paper gives the asymptotics of this sequence.

关于某些有限阶常微分方程的多项式解

On polynomial solutions of certain finite order ordinary differential equations

本文研究有限阶微分算子双谱问题的多项式解，在特定条件下显式求出特征函数与特征值，并给出线性变换后序列成为新算子特征函数序列的必要条件，最后应用于经典埃尔米特多项式。

The estimates for the number of the eigenvalues of abstract and differential operator functions

We consider an operator function (F(λ)) for (λ\in(σ,τ)\subseteq\mathbb R) whose values are semibounded selfadjoint operators in Hilbert space (\mathfrak H). Our main goal is to estimate the number (\mathcal N_F(α,β)) of the eigenvalues of (F(λ)) on a segment ([α,β)\Subset(σ,τ)). In particular, we prove the estimates (\mathcal N_F(α,β)\geqslant ν_F(β)-ν_F(α)) and (\mathcal N_F(α,β)= ν_F(β)-ν_F(α)) where (ν(ξ)) is the number of the negative eigenvalues of the operator (F(ξ)), (ξ\in(σ,τ)). The obtained results are applied for the functions of ordinary differential operators on a finite interval.

An inverse problem for a class of lacunary canonical systems with diagonal Hamiltonian

Hamiltonians are 2-by-2 positive semidefinite real symmetric matrix-valued functions satisfying certain conditions. In this paper, we solve the inverse problem for which recovers a Hamiltonian from the solution of a first-order system attached to a given Hamiltonian, consisting of ordinary differential equations parametrized by a set of complex numbers, under certain conditions for the solutions. This inverse problem is a generalization of the inverse problem for two-dimensional canonical systems.