This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas fo...This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.展开更多
In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.
The three-dimensional Klein-Gordon equation is solved for the case of equal vector and scalar second Poschl-Teller potential by proper approximation of the centrifugal term within the framework of the asymptotic itera...The three-dimensional Klein-Gordon equation is solved for the case of equal vector and scalar second Poschl-Teller potential by proper approximation of the centrifugal term within the framework of the asymptotic iteration method. Energy eigenvalues and the corresponding wave function are obtained analytically. Eigenvalues are computed numerically for some values of n and It is found that the results are in good agreement with the findings of other methods for short-range potential.展开更多
The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such op...The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.展开更多
The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with pi...The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenvalue parameter on the boundary and transmission conditions. The authors suggest their own approach for finding asymptotic approximations formulas for eigenvalues and eigenfunctions of such discontinuous problems.展开更多
We consider the inverse spectral problem for a singular Sturm-Liouville operator with Coulomb potential. In this paper, we give an asymptotic formula and some properties for this problem by using methods of Trubowitz ...We consider the inverse spectral problem for a singular Sturm-Liouville operator with Coulomb potential. In this paper, we give an asymptotic formula and some properties for this problem by using methods of Trubowitz and Poschel.展开更多
Letλf(n)be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and Pc(x):={p≤x|[pc]prime},c∈R+.In this paper,we show that for all 0<c<1 the mean value ofλf(n)in Pc(x...Letλf(n)be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and Pc(x):={p≤x|[pc]prime},c∈R+.In this paper,we show that for all 0<c<1 the mean value ofλf(n)in Pc(x)is x log-A x assuming the Riemann Hypothesis.Unconditionally,in the sense of Lebesgue measure,it holds for almost all c∈(ε,1-ε).展开更多
文摘This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.
文摘In present paper, using some methods of approximation theory, the trace formulas for eigenvalues of a eigenvalue problem are calculated under the periodic condition and the decaying condition at x∞.
文摘The three-dimensional Klein-Gordon equation is solved for the case of equal vector and scalar second Poschl-Teller potential by proper approximation of the centrifugal term within the framework of the asymptotic iteration method. Energy eigenvalues and the corresponding wave function are obtained analytically. Eigenvalues are computed numerically for some values of n and It is found that the results are in good agreement with the findings of other methods for short-range potential.
文摘The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.
文摘The purpose of this paper is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenvalue parameter on the boundary and transmission conditions. The authors suggest their own approach for finding asymptotic approximations formulas for eigenvalues and eigenfunctions of such discontinuous problems.
文摘We consider the inverse spectral problem for a singular Sturm-Liouville operator with Coulomb potential. In this paper, we give an asymptotic formula and some properties for this problem by using methods of Trubowitz and Poschel.
基金Supported by National Natural Science Foundation of China(Grant Nos.11771256,11971476)。
文摘Letλf(n)be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and Pc(x):={p≤x|[pc]prime},c∈R+.In this paper,we show that for all 0<c<1 the mean value ofλf(n)in Pc(x)is x log-A x assuming the Riemann Hypothesis.Unconditionally,in the sense of Lebesgue measure,it holds for almost all c∈(ε,1-ε).
