Spectrum And Spectral Singularities Of A Quadratic Pencil Of A Schr(?)dinger Operator With Boundary Conditions Dependent On The Eigenparameter

In this paper,we consider the spectrum of the following quadratic pencil of Schr(?)dinger operators L(?)generated in L2(R+)by the equation-y"+[p(x)+2?q(x)|y=?2y,x ? R+=[0,+?)with the boundary condition y'/y(0)=?1?+?o/?1?+?o where p(x)and q(x)are complex valued functions and ?0,?1,?0,?1 are complex numbers with ?0?1-?1?0?0.It is proved that L(?)has a finite number of eigenvalues and spectral singularities,and each of them is of a finite multiplicity,if the conditions p(x),q'(x)?AC(R+),(?)and#12 hold,where ?>0.In the second chapter of this paper,the series solution of the equation is given by iterative method.In the iterative process,we need to find the conditions that the potential function satisfies.In the third chapter,the properties of analytic functions are mainly used to distinguish complex planes by the zeros of eigenfunctions,so as to clearly describe the spectral structure of the operator.Finally,the conditions of the eigenvalues of the operator and the number of spectral singularities and the finiteness of the multiple number are obtained.