Spectrum of the Sturm-Liouville operators with boundary conditions polynomially dependent on the spectral parameter

摘要

In this paper, we consider the operator L generated in L 2 ( R + ) $L_{2}(mathbb{R}_{+})$ by the Sturm-Liouville equation ? y ″ + q ( x ) y = λ 2 y $-y^{primeprime}+q(x)y=lambda^{2}y$ , x ∈ R + = [ 0 , ∞ ) $xin mathbb{R}_{+}= [ 0,infty ) $ , and the boundary condition ( α 0 + α 1 λ + α 2 λ 2 ) y ′ ( 0 ) ? ( β 0 + β 1 λ + β 2 λ 2 ) y ( 0 ) = 0 $( lpha_{0}+lpha_{1}lambda+lpha_{2}lambda^{2} ) y^{prime} ( 0 ) - ( eta_{0}+eta_{1}lambda +eta_{2}lambda^{2} ) y ( 0 ) =0$ , where q is a complex-valued function, α i , β i ∈ C $lpha_{i},eta_{i}inmathbb{C}$ , i = 0 , 1 , 2 $i=0,1,2$ , and λ is an eigenparameter. Under the conditions q , q ′ ∈ AC ( R + ) $q,q^{prime}in operatorname{AC}(mathbb{R}_{+})$ , lim x → ∞ | q ( x ) | + | q ′ ( x ) | = 0 $lim_{xightarrow infty} ert q(x)ert +ert q^{prime}(x) ert =0$ , sup x ∈ R + [ e ε x | q ″ ( x ) | ] 0 $arepsilon>0$ , using the uniqueness theorems of analytic functions, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities.