An Inverse Spectral Problem For Sturm-Liouville Operators With Frozen Argument

Sturm-Liouville problem for solving solid heat conduction model originated in the early 19th century.Its application has been widely involved in many fields,such as mathematical physics,geophysics,quantum mechanics,meteorological physics,engineering technology and so on.Especially in quantum mechanics,it is a basic mathematical means to describe the state of microscopic particles.The theory of ordinary differential operator represented by Sturm-Liouville operator is formed,and it has gradually become an important theoretical research branch in modern physics and Mathematics.The most complete results in the inverse spectral theory have been obtained for the classical Sturm-Liouville operator[13]and later on for higher order differential operators[2].For operators with frozen argument as well as for other classes of non-local operators the classical methods of the inverse spectral theory do not work.Operators with frozen argument can be classified as a special case of functional differential operators with deviating argument.In recent years,the study of differential operators with frozen argument has made some progress[1,3,4,9,10,28,11,14,20,21,22].The paper concerns the inverse spectral problem of the boundary value problems L=L(q(x),?,?),for the equation:ly=-y"(x)+q(x)y(a)=?y(x),0<x<?.subject to the boundary conditions y(0)=0.y(?)cos?-y'(?)sin?=0.where ? is the spectral parameter,q(x)is a complex-valued function in L2(0,?),and a=?j/k[0,?],??[0,?)with j,k?N.It is also assumed that j and k are always mutually prime.Denote by {?n}n?1 the spectrum of L.The operator l is called the Sturm-Liouville-type operator with frozen argument.We study the following inverse problem:Given {?n}n?1 and ?,?,find the potential function q(x).The paper is organized as follows:In the second chapter,we simplify the inverse prob-lem,obtain the main equation of the inverse problem,study its solvability,and analyze and simplify some elements of the main equation.On this basis,we introduce degenerate and non degenerate cases.In the third chapter,we study the properties of spectrum.In the degenerate case and the non degenerate case,we obtain different forms of potential function according to different a,and obtain the expression of characteristic function.In the fourth chapter,the uniqueness theorem is studied and proved.The algorithm and the necessary and sufficient conditions for solving the inverse problem in degenerate and non degenerate cases are given.In degenerate case,the set of iso-spectrum potentials is described.In the last chapter,a summary and outlook is given.