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Frobenius Series Solutions of the Schrodinger Equation with Various Types of Symmetric Hyperbolic Potentials in One Dimension
Hippolyte Nyengeri, René Simbizi, Audace Girukwishaka, Rénovat Nizigiyimana, Eugène Ndenzako
Open Access Library Journal (OALib Journal) , 2018, DOI: 10.4236/oalib.1104728
Abstract:
The Schrodinger equation (SE) for a certain class of symmetric hyperbolic potentials is solved with the aid of the Frobenius method (FM). The bound state energies are given as zeros of a calculable function. The calculated bound state energies are successively substituted into the recurrence relations for the expanding coefficients of the Frobenius series representing even and odd solutions in order to obtain wave functions associated with even and odd bound states. As illustrative examples, we consider the hyperbolic Poschl-Teller potential (HPTP) which is an exactly solvable potential, the Manning potential (MP) and a model of the Gaussian potential well (GPW). In each example, the bound state energies obtained by means of the FM are presented and compared with the exact results or the literature ones. In the case of the HPTP, we also make a comparison between exact bound state wave functions and the eigenfunctions obtained by means of the present approach. We find that our results are in good agreement with those given by other methods considered in this work, and that our class of potentials can be a perfect candidate to model the GPW.
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.
A numerical study of the Schrodinger-Newton equation 1: Perturbing the spherically-symmetric stationary states  [PDF]
R. Harrison,I. Moroz,K. P. Tod
Mathematics , 2002,
Abstract: We consider the linear stability of the spherically-symmetric stationary solutions of the Schrodinger-Newton equations. We find that the ground state is linearly stable, with only imaginary eigenvalues, while the n-th excited state has n quadruples of complex eigenvalues as well as purely imaginary ones and so is linearly unstable.
A Simple General Solution of the Radial Schrodinger Equation for Spherically Symmetric Potentials  [PDF]
H. H. Erbil
Physics , 2003,
Abstract: By using a simple procedure the general solution of the time-independent radial Schrodinger Equation for spherical symmetric potentials was made without making any approximation. The wave functions are always periodic. It appears two diffucilties: one of them is the solution of the equation E= U(r), where E and U(r) are the total an effective potential energies, respectively, and the other is the calculation of the integral of the square root of U(r). If analytical calculations are not possible, one must apply numerical methods. To find the energy wave function of the ground state, there is no need for the calculation of this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(r).
Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds  [PDF]
Thierry Daude,Damien Gobin,Fran?ois Nicoleau
Physics , 2013,
Abstract: In this paper, we adapt the well-known \emph{local} uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schr{\"o}dinger equation to prove \emph{local} uniqueness results in the setting of inverse \emph{metric} problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients $T(\lambda,n)$ and reflection coefficients $L(\lambda,n)$ and $R(\lambda,n)$ of a Dirac wave having a fixed energy $\lambda$ and angular momentum $n$. For instance, the reflection coefficients $L(\lambda,n)$ correspond to the scattering experiment in which a wave is sent from the \emph{left} end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed $\lambda \not=0$, the knowledge of the reflection coefficients $L(\lambda,n)$ (resp. $R(\lambda,n)$) - up to a precise error term of the form $O(e^{-2nB})$ with $B\textgreater{}0$ - determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude $B$ of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.
A short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric Schrodinger--Poisson--Slater system  [PDF]
Xuanchun Dong
Physics , 2011, DOI: 10.1016/j.jcp.2011.07.026
Abstract: In a recent paper we proposed and compared various approaches to compute the ground state and dynamics of the Schr\"{o}dinger--Poisson--Slater (SPS) system for general external potential and initial condition, concluding that the methods based on sine pseudospectral discretization in space are the best candidates. This note is concerned with the case that the external potential and initial condition are spherically symmetric. For the SPS system with spherical symmetry, via applying a proper change of variables into the reduced quasi-1D model we simplify the methods proposed for the general 3D case such that both the memory and computational load are significantly reduced.
Axially and spherically symmetric solitons in warm plasma  [PDF]
Maxim Dvornikov
Physics , 2010, DOI: 10.1017/S002237781100016X
Abstract: We study the existence of stable axially and spherically symmetric plasma structures on the basis of the new nonlinear Schrodinger equation (NLSE) accounting for nonlocal electron nonlinearities. The numerical solutions of NLSE having the form of spatial solitions are obtained and their stability is analyzed. We discuss the possible application of the obtained results to the theoretical description of natural plasmoids in the atmosphere.
Stability of Spherically Symmetric Wave Maps  [PDF]
Joachim Krieger
Mathematics , 2005,
Abstract: We study Wave Maps from R^{2+1} to the hyperbolic plane with smooth compactly supported initial data which are close to smooth spherically symmetric ones with respect to some H^{1+\mu}, \mu>0. We show that such Wave Maps don't develop singularities and stay close to the Wave Map extending the spherically symmetric data with respect to all H^{1+\delta}, \delta<\mu_{0}(\mu). We obtain a similar result for Wave Maps whose initial data are close to geodesic ones. This generalizes a theorem of Sideris for this context.
On spherically symmetric structures in GR  [PDF]
Angelo Loinger,Tiziana Marsico
Physics , 2007,
Abstract: We reconsider some subtle points concerning the relativistic treatment of the gravitational fields generated by spherically symmetric structures.
Spherically symmetric dynamical horizons  [PDF]
Robert Bartnik,Jim Isenberg
Physics , 2005, DOI: 10.1088/0264-9381/23/7/020
Abstract: We determine sufficient and necessary conditions for a spherically symmetric initial data set to satisfy the dynamical horizon conditions in the spacetime development. The constraint equations reduce to a single second order linear master equation, which leads to a systematic construction of all spherically symmetric dynamical horizons (SSDH) satisfying certain boundedness conditions. We also find necessary and sufficient conditions for a given spherically symmetric spacetime to contain a SSDH.
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