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
By cognitive radio,the low Earth orbit（LEO） satellites may prefer to operate in the unlicensed spectrum which is open to all the users,and compete for the limited resources with terrestrial cognitive radio networks...By cognitive radio,the low Earth orbit（LEO） satellites may prefer to operate in the unlicensed spectrum which is open to all the users,and compete for the limited resources with terrestrial cognitive radio networks（CRNs）.The competition can be regarded as a game and analyzed with game theory.This particular unlicensed spectrum sharing problem is modeled here,and the special properties of ＂spatially-distinguished-interference＂ and the short period of the interactions between satellites and terrestrial CRNs are explored.Then,the problem is formulated as a ＂partially-blind＂ finitely repeated prisoner＇s dilemma by game theory.Finally,we begin with two promising spectrum sharing schemes,which can be used to enforce the frequency reuse among the remotely located terrestrial CRN players as well as to overcome the observation noise.By analysis and comparison,it is proposed that the novel refreshing-contrite-tit-for-tat（R-CTFT） is the optimal spectrum sharing scheme.Simulation results verify that it can be used to utilize the spectrum most efficiently.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By cognitive radio,the low Earth orbit（LEO） satellites may prefer to operate in the unlicensed spectrum which is open to all the users,and compete for the limited resources with terrestrial cognitive radio networks（CRNs）.The competition can be regarded as a game and analyzed with game theory.This particular unlicensed spectrum sharing problem is modeled here,and the special properties of ＂spatially-distinguished-interference＂ and the short period of the interactions between satellites and terrestrial CRNs are explored.Then,the problem is formulated as a ＂partially-blind＂ finitely repeated prisoner＇s dilemma by game theory.Finally,we begin with two promising spectrum sharing schemes,which can be used to enforce the frequency reuse among the remotely located terrestrial CRN players as well as to overcome the observation noise.By analysis and comparison,it is proposed that the novel refreshing-contrite-tit-for-tat（R-CTFT） is the optimal spectrum sharing scheme.Simulation results verify that it can be used to utilize the spectrum most efficiently.
