一类具有耦合转移条件Sturm-Liouville问题的逆问题

考虑了具有耦合转移条件Sturm-Liouville(简称S-L)问题的逆问题,在一定条件下,通过利用S-L方程右边的函数f_j(x)确定方程的解,并由数据{u_j(x_0)}_j~∞=i或{p(x_0)(du_j(x_0))-(dx)}_j~∞=1唯一确定S-L算子中的系数p(x)和q(x).其中u_j(x)满足S-L方程,分离边界条件和耦合转移条件,而{f_j(x)}_(j-i)~∞构成L~2(I)的一个基.     

Inverse spectral problems for the Sturm-Liouville operator on a d-star graph

In this work, we consider inverse spectral problems for the Sturm-Liouville differential operator on a d-star-type graph with standard matching conditions in the internal vertex, where the integer d?2. By using the Yurko's method (Yurko (2008) [27], Yurko (2009) [28]) we show that

(1)

if the potential function qj(x) on a fixed edge ej is prescribed on the interval , then the reciprocal of d of the spectrum suffices to determine qj(x) on the whole interval [0,π];

(2)

the 2 over d of the spectrum suffices to determine qj(x) on a fixed edge ej.

     

边界条件含特征参数的Sturm-Liouville问题的特征值比率

给出修正Prfer变换的几何意义以及相关性质,并利用上述性质讨论Sturm-Liouville问题-(py′)′+gy=λωy,x∈[0,1],参数边界条件为y(0)=0与((py′)(1))/(y(1))=aλ+b,得到了当a>0,q≥0的特征值比率上界,这把Ashbaugh等人的结果推广到边界条件含参数的情形.     

On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator

On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.     

Inverse nodal problems for the Sturm-Liouville differential operators on a star-type graph

We study inverse nodal problems for the second order differential operators on a star-type graph satisfying the standard matching conditions at the interior vertex. We prove uniqueness theorems and obtain a constructive solution to the inverse problems of this class. Original Russian Text Copyright ? 2009 Yurko V. A. The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00003) and the National Science Council of Taiwan (Grant 07-01-92000-NSC-a). __________ Saratov. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 469–475, March–April, 2009.     

Spectral Components of Operators with Spectrum on a Curve

Trace class perturbations of normal operators with spectrum on a curve and spectral components of such operators are studied. We establish duality relations for the spectral components of an operator and its adjoint. The generalized Sz.-Nagy–Foia–Naboko functional model introduced in the paper is a basic tool for this theorem. The results have applications in nonself-adjoint scattering theory and to extreme factorizations of J-contraction-valued functions (J-inner-outer and A-regular-singular factorizations).     

The Inverse Spectral Problem for the Sturm-Liouville Operators with Discontinuous Coefficients

We study the inverse spectral problem for the Sturm–Liouville operator whose piecewise constant coefficient A(x) has discontinuity points x _k , k=1,...,n, and jumps A _k =A(x _k +0)/A(x _k -0). We show that if the discontinuity points x ₁,...,x _n are noncommensurable, i.e., none of their linear combinations with integer coefficients vanishes; then the spectral function of the operator determines all discontinuity points x _k and jumps A _k uniquely. We give an algorithm for finding x _k and A _k in finitely many steps.     

N维向量Sturm-Liouville算子的正则迹及其在反谱问题中的应用

研究了赋予一般分离型边界条件的N维向量Sturm-Liouville方程的特征值问题,获得了该系统的一个正则迹公式.迹公式不仅形式美观,而且它在反谱理论中具有重要的作用.     

板模型具广义边界条件的迁移算子的谱

研究具广义边界条件、非均匀介质、各向异性和连续能量的板模型迁移算子A的谱．证明了K=A-B的相对紧性，在L^1空间研究算子A的谱，以及占优本征值和严格占优本征值．     

Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients

We consider a non-self-adjoint Schrödinger operator describing the motion of a particle in a one-dimensional space with an analytic potential iV (x) that is periodic with a real period T and is purely imaginary on the real axis. We study the spectrum of this operator in the semiclassical limit and show that the points of its spectrum asymptotically belong to the so-called spectral graph. We construct the spectral graph and evaluate the asymptotic form of the spectrum. A Riemann surface of the particle energy-conservation equation can be constructed in the phase space. We show that both the spectral graph and the asymptotic form of the spectrum can be evaluated in terms of integrals of the pdx form (where x ∈31 ?/T? and p ∈, ? are the particle coordinate and momentum) taken along basis cycles on this Riemann surface. We use the technique of Stokes lines to construct the asymptotic form of the spectrum.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.     

A trace formula of a boundary value problem for the operator Sturm-Liouville equation

We obtain a regularized trace formula for the operator Sturm-Liouville equation with a boundary condition depending on a spectral parameter.     

一类半正奇异Sturm-Liouville边值问题的正解

本文讨论了一类半正奇异Sturm-Liouville边值问题正解的存在性,其中非线性项f(t,u)关于t=0,1和u=0奇异.在非线性项可取负值且下方无界的情形下,利用不动点指数理论以及线性算子的特征值理论得到了该问题正解存在性结果.     

Spectrum of a Model Operator in the Perturbation Theory of the Essential Spectrum

We consider a model operator acting in a subspace of a Fock space and obtain a symmetrized analogue of the Faddeev equation. For the operator considered, we describe the position and the structure of its essential spectrum.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 544–554, September, 2005.     

A Sturm-Liouville Inverse Spectral Problem with Boundary Conditions Depending on the Spectral Parameter

We consider a boundary value problem generated by the Sturm-Liouville equation on a finite interval. Both the equation and the boundary conditions depend quadratically on the spectral parameter. This boundary value problem occurs in the theory of small vibrations of a damped string. The inverse problem, i.e., the problem of recovering the equation and the boundary conditions from the given spectrum, is solved.     

偶阶非对称微分算子离散谱准则

本文研究了由2n阶复系数J-对称微分算式生成的J-自伴微分算子谱的离散性,分别得到了一类J-自伴微分算子谱离散的充分条件与必要条件,为判断一类微分算子谱的离散性提供了若干准则．     

Calculating the eigenvalues of the Sturm-Liouville problem with a fractal indefinite weight

An efficient method is proposed for calculating the eigenvalues of the boundary value problem ?y″ ? λρy = 0, y(0) = y(1) = 0, where ρ ? \(W_2^{^\circ - 1} \) [0, 1] is the generalized derivative of a self-similar function P ? L ₂[0, 1].     

一类具指数系数的对称微分算子谱的离散性

研究了具指数函数系数的2n阶实系数微分算式生成的对称微分算子,利用算子的直和分解法及不等式估计得到此类微分算子谱是离散的充分条件.