The analytic solution of the radial Schrodinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely ana...The analytic solution of the radial Schrodinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schrodinger equation is V（r） =α1r^8 ＋α2r^3 ＋ α3r^2 ＋β3r^-1 ＋β2r^-3 ＋β1r6-4. Generally speaking, there is only an approximate solution, but not analytic solution for SchrSdinger equation with several potentials＇ superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schrodinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r → ∞ and r →0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions of the stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial SchrSdinger equation; and lastly, they discuss the solutions and make conclusions.展开更多
The optimization of the network throughput and transmission range is one of the most important issues in cognitive relay networks （CRNs）. Existing research has focused on the dual-hop network, which cannot be extend...The optimization of the network throughput and transmission range is one of the most important issues in cognitive relay networks （CRNs）. Existing research has focused on the dual-hop network, which cannot be extended to a triple-hop network due to its shortcomings, including the limited transmission range and one-way communication. In this paper, a novel, triple-hop relay scheme is proposed to implement time-division duplex （TDD） transmission among secondary users （SUs） in a three-phase transmission. Moreover, a superposition coding （SC） method is adopted for handling two-receiver cases in triple-hop networks with a cognitive relay. We studied a joint optimization of time and power allocation in all three phases, which is formulated as a nonlinear and concave problem. Both analytical and numerical results show that the proposed scheme is able to improve the throughput of SUs, and enlarge the transmission range of primary users （PUs） without increasing the number of hops.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.10575140the Basic Research of Chongqing Education Committee under Grant No.KJ060813
文摘The analytic solution of the radial Schrodinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schrodinger equation is V（r） =α1r^8 ＋α2r^3 ＋ α3r^2 ＋β3r^-1 ＋β2r^-3 ＋β1r6-4. Generally speaking, there is only an approximate solution, but not analytic solution for SchrSdinger equation with several potentials＇ superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schrodinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r → ∞ and r →0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions of the stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial SchrSdinger equation; and lastly, they discuss the solutions and make conclusions.
基金Project supported by the Zhejiang Provincial National Natural Science Foundation （No. LQ14F020005）, the National Natural Science Foundation of China （Nos. 61379123 and 61402414）, and the Research Program of the Educational Commission of Zhej iang Province, China （No. Y201431815）
文摘The optimization of the network throughput and transmission range is one of the most important issues in cognitive relay networks （CRNs）. Existing research has focused on the dual-hop network, which cannot be extended to a triple-hop network due to its shortcomings, including the limited transmission range and one-way communication. In this paper, a novel, triple-hop relay scheme is proposed to implement time-division duplex （TDD） transmission among secondary users （SUs） in a three-phase transmission. Moreover, a superposition coding （SC） method is adopted for handling two-receiver cases in triple-hop networks with a cognitive relay. We studied a joint optimization of time and power allocation in all three phases, which is formulated as a nonlinear and concave problem. Both analytical and numerical results show that the proposed scheme is able to improve the throughput of SUs, and enlarge the transmission range of primary users （PUs） without increasing the number of hops.
