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Application of the Frobenius Method to the Schrodinger Equation for a Spherically Symmetric Hyperbolic Potential
Hippolyte Nyengeri, Rénovat Nizigiyima, Eugène Ndenzako, Félix Bigirimana, Dieudonné Niyonkuru, Audace Girukwishaka
Open Access Library Journal (OALib Journal) , 2018, DOI: 10.4236/oalib.1104950
In this paper, an efficient technique for computing the bound state energies and wave functions of the Schrodinger Equation (SE) associated with a new class of spherically symmetric hyperbolic potentials is developed. This technique is based on a recent approximation scheme for the orbital centrifugal term and on the use of the Frobenius method (FM). The bound state eigenvalues are given as zeros of calculable functions. The corresponding eigenfunctions can be obtained by substituting the calculated energies into the recurrence relations for the expanding coefficients of the Frobenius series representing the solution. The excellent performance of this technique is illustrated through numerical results for some special cases like Poschl-Teller potential (PTP), Manning-Rosen potential (MRP) and Poschl-Teller polynomial potential (PTPP), with an application to the Gaussian potential well (GPW). Comparison with other methods is presented. Our results agree noticeably with the previously reported ones.
Schrodinger operators and associated hyperbolic pencils  [PDF]
Sergey A. Denisov
Mathematics , 2009,
Abstract: For a large class of Schrodinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are easier to study. Then, we give some applications to the original Schrodinger operators.
Application of the Frobenius method to the Schrodinger equation for a spherically symmetric potential: anharmonic oscillator  [PDF]
Przemyslaw Koscik,Anna Okopinska
Mathematics , 2005, DOI: 10.1088/0305-4470/38/35/008
Abstract: The power series method has been adapted to compute the spectrum of the Schrodinger equation for central potential of the form $V(r)={d_{-2}\over r^2}+{d_{-1}\over r}+\sum_{i=0}^{\infty} d_{i}r^i$. The bound-state energies are given as zeros of a calculable function, if the potential is confined in a spherical box. For an unconfined potential the interval bounding the energy eigenvalues can be determined in a similar way with an arbitrarily chosen precision. The very accurate results for various spherically symmetric anharmonic potentials are presented.
Resonance asymptotics for Schrodinger operators on hyperbolic space  [PDF]
David Borthwick,Catherine Crompton
Mathematics , 2013,
Abstract: We study the asymptotic distribution of resonances for scattering by compactly supported potentials in hyperbolic space. We first establish an upper bound for the resonance counting function that depends only on the dimension and the support of the potential. We then establish the sharpness of this estimate by proving the a Weyl law for the resonance counting function holds in the case of radial potentials vanishing to some finite order at the edge of the support. As an application of the existence of potentials that saturate the upper bound, we derive additional resonance asymptotics that hold in a suitable generic sense. These generic results include asymptotics for the resonance count in sectors.
用Hyperbolic函数构造高阶Schrodinger方程的辛格式  [PDF]
华侨大学学报(自然科学版) , 1998, DOI: 10.11830/ISSN.1000-5013.1998.01.0006
Abstract: 利用Hyperbolic函数Sinh(x)和tanh(x)构造了高阶Schrodinger方程的任意阶精度的辛格式并讨论了它们的稳定性.
On a solution of the Schrodinger equation with a hyperbolic double-well potential  [PDF]
C. A. Downing
Mathematics , 2012, DOI: 10.1063/1.4811855
Abstract: We report a solution of the one-dimensional Schrodinger equation with a hyperbolic double-well confining potential via a transformation to the so-called confluent Heun equation. We discuss the requirements on the parameters of the system in which a reduction to Heun polynomials is possible, representing the wavefunctions of bound states.
PT/Non-PT Symmetric and Non-Hermitian Poschl-Teller-Like Solvable Potentials via Nikiforov-Uvarov Method  [PDF]
Ozlem Yesiltas
Physics , 2006, DOI: 10.1088/0031-8949/75/1/006
Abstract: The solutions of trigonometric Scarf potential, PT/non-PT-symmetric and non-Hermitian q-deformed hyperbolic Scarf and Manning-Rosen potentials are obtained by solving the Schrodinger equation. The Nikiforov-Uvarov method is used to obtain the real energy spectra and corresponding eigenfunctions.
Supercritical geometric optics for nonlinear Schrodinger equations  [PDF]
Thomas Alazard,Rémi Carles
Mathematics , 2007, DOI: 10.1007/s00205-008-0176-7
Abstract: We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y.Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows.
Intertwining the geodesic flow and the Schrodinger group on hyperbolic surfaces  [PDF]
Nalini Anantharaman,Steve Zelditch
Mathematics , 2010,
Abstract: We construct an explicit intertwining operator $\lcal$ between the Schr\"odinger group $e^{it \frac\Lap2} $ and the geodesic flow $g^t$ on certain Hilbert spaces of symbols on the cotangent bundle $T^* \X$ of a compact hyperbolic surface $\X = \Gamma \backslash \D$. Thus, the quantization Op(\lcal^{-1} a) satisfies an exact Egorov theorem. The construction of $\lcal$ is based on a complete set of Patterson-Sullivan distributions.
High energy eigenfunctions of one-dimensional Schrodinger operators with polynomial potentials  [PDF]
Alexandre Eremenko,Andrei Gabrielov,Boris Shapiro
Physics , 2007,
Abstract: For a class of one-dimensional Schrodinger operators with polynomial potentials that includes Hermitian and PT-symmetric operators, we show that the zeros of scaled eigenfunctions have a limit disctibution in the complex plane as the eigenvalues tend to infinity. This limit distribution depends only on the degree of potential and on the boundary conditions.
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