In this paper,we consider the following quadratic pencil of Schr?dinger operators L(λ)generated in L~2(R~+)by the equation−y″+[p(x)+2λq(x)]y=λ^(2)y,x∈R^(+)=[0,+∞)with the boundary condition y′(0)/y(0)=β_(1)λ+...In this paper,we consider the following quadratic pencil of Schr?dinger operators L(λ)generated in L~2(R~+)by the equation−y″+[p(x)+2λq(x)]y=λ^(2)y,x∈R^(+)=[0,+∞)with the boundary condition y′(0)/y(0)=β_(1)λ+β0-α_(1)λ+α0,where p(x)and q(x)are complex valued functions andα_(0),α_(1),β_(0),β_(1)are complex numbers withα_(0)β_(1)-α_(1)β_(0)≠0.It is proved that L(λ)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^(+)),limx→∞[|p(x)|+|q(x)|+|q′(x)|]=0 and sup 0≤x<+∞{eε√x[|p′(x)|+|q″(x)|]}<+∞hold,whereε>0.展开更多
In this paper we consider the nonselfadjoint (dissipative) Schrodinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint...In this paper we consider the nonselfadjoint (dissipative) Schrodinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrodinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrodinger boundary value problem are given.展开更多
基金the NSF of Shandong Province(Grant Nos.ZR2023MA023,ZR2020QA009,ZR2020QA010)the NNSF of China(Grant No.61973183)the Youth Creative Team Sci-Tech Program of Shandong Universities(Grant No.2019KJI007)。
文摘In this paper,we consider the following quadratic pencil of Schr?dinger operators L(λ)generated in L~2(R~+)by the equation−y″+[p(x)+2λq(x)]y=λ^(2)y,x∈R^(+)=[0,+∞)with the boundary condition y′(0)/y(0)=β_(1)λ+β0-α_(1)λ+α0,where p(x)and q(x)are complex valued functions andα_(0),α_(1),β_(0),β_(1)are complex numbers withα_(0)β_(1)-α_(1)β_(0)≠0.It is proved that L(λ)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^(+)),limx→∞[|p(x)|+|q(x)|+|q′(x)|]=0 and sup 0≤x<+∞{eε√x[|p′(x)|+|q″(x)|]}<+∞hold,whereε>0.
文摘In this paper we consider the nonselfadjoint (dissipative) Schrodinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrodinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrodinger boundary value problem are given.
