Analytical Solutions to the D-Dimensional Schrodinger Equation with the Eckart Potential

The analytical solutions to the Schrodinger equation with the Eckart potential in arbitrary dimension D is investigated by using the Nikiforov-Uvarov method, and the centrifugal term is treated approximatively with the scheme of Greene and Aldrich. The discrete spectrum is obtained and the wavefunetion is expressed in terms of the Jacobi polynomial or the hypergeometric function. Some special cases of the Eckart potential are discussed for D=3, and the resulting energy equation agrees well with that obtained by other methods.

Non-Relativistic Bound State Solutions of Modified Quadratic Yukawa plus <i>q</i>-Deformed Eckart Potential

We have obtained approximate bound state solutions of Schr&#246;dinger wave equation with modified quadratic Yukawa plus q-deformed Eckart potential Using Parametric Nikiforov-Uvarov (NU) method. However, we obtained numerical energy eigenvalues and un-normalized wave function using confluent hypergeometric function (Jacobi polynomial). With some modifications, our potential reduces to a well-known potential such as Poschl-Teller and exponential inversely quadratic potential. Numerical bound state energies were carried out using a well-designed Matlab algorithm while the plots were obtained using origin software. The result obtained is in agreement with that of the existing literature.

Nodal Type Bound States for Nonlinear Schrodinger Equations with Decaying Potentials

In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrodinger equation -△u（x）＋V（x）u（x）=｜u｜^p-1 u,x∈R^N,N≥3,where 1 〈 p 〈 （N＋2）/（N-2） and the potential V（x） is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V（x）, Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V（x） as in [1].

Approximate solutions of Schrdinger equation for Eckart potential with centrifugal term

The approximate analytical solutions of the Schrodinger equation for the Eckart potential are presented for the arbitrary angular momentum by using a new approximation of the centrifugal term. The energy eigenvalues and the corresponding wavefunctions are obtained for different values of screening parameter. The numerical examples are presented and the results are in good agreement with the values in the literature. Three special cases, i.e., s-wave, ξ= λ=1, and β=0, are investigated.

Exact bound state solutions of the Klein-Gordon particle in Hulthn potential

In this paper, the Klein-Gordon equation with the spherical symmetric Hulthén potential is turned into a hypergeometric equation and is solved in the framework of function analysis exactly. The corresponding bound state solutions are expressed in terms of the hypergeometric function, and the energy spectrum of the bound states is obtained as a solution to a given equation by boundary constraints.

Approximate Bound State Solutions for Certain Molecular Potentials

We present solutions of the Schrodinger equation with superposition of Manning-Rosen plus inversely Mobius square plus quadratic Yukawa potentials using parametric Nikiforov Uvarov method along with an approximation to the centrifugal term. The bound state energy eigenvalues for any angular momentum quantum number l and the corresponding un-normalized wave functions are calculated. The mixed potential which in some particular cases gives the solutions for different potentials: the Manning-Rosen, the Mobius square, the inversely quadratic Yukawa and the Hulthén potentials along with their bound state energies are obtained.

The possible KK^(*)and D^(*)bound and resonance states by solving the Schrodinger equation

The Schrodinger equation with a Yukawa type of potential is solved analytically.When different boundary conditions are taken into account,a series of solutions are indicated as a Bessel function,the first kind of Hankel function and the second kind of Hankel function,respectively.Subsequently,the scattering processes of K^(*)and D^(*)are investigated.In the K^(*)sector,the f_(1)(1285)particle is treated as a K^(*)bound state,therefore,the coupling constant in the K^(*)Yukawa potential can be fixed according to the binding energy of the f_(1)(1285)particle.Consequently,a K^(*)resonance state is generated by solving the Schrodinger equation with the outgoing wave condition,which lies at 1417-i18 MeV on the complex energy plane.It is reasonable to assume that the K^(*)resonance state at 1417-i18 MeV might correspond to the f_(1)(1420)particle in the review of the Particle Data Group.In the D^(*)sector,since the X(3872)particle is almost located at the D^(*)threshold,its binding energy is approximately equal to zero.Therefore,the coupling constant in the D^(*)Yukawa potential is determined,which is related to the first zero point of the zero-order Bessel function.Similarly to the K^(*)case,four resonance states are produced as solutions of the Schrodinger equation with the outgoing wave condition.It is assumed that the resonance states at 3885~i1 MeV,4029-i108 MeV,4328-i191 MeV and 4772-i267 MeV might be associated with the Zc(3900),the X(3940),theχ_(c1)(4274)andχ_(c1)(4685)particles,respectively.It is noted that all solutions are isospin degenerate.

Bound states of the Schrdinger equation for the Pschl-Teller double-ring-shaped Coulomb potential

Poschl-Teller double-ring-shaped Coulomb （PTDRSC） potential, the Coulomb potential surrounded by PSschl- Teller and double-ring-shaped inversed square potential, is put forward. In spherical polar coordinates, PTDRSC potential has supersymmetry and shape invariance in φ,θ and τ coordinates. By using the method of supersymmetry and shape invariance, exact bound state solutions of Schr6dinger equation with PTDRSC potential are presented. The normalized φ,θ angular wave function expressed in terms of Jacobi polynomials and the normalized radial wave function expressed in terms of Laguerre polynomials are presented. Energy spectrum equations are obtained. Wave function and energy spectrum equations of the system are related to three quantum numbers and parameters of PTDRSC potential. The solutions of wave functions and corresponding eigenvalues are only suitable for the PTDRSC potential.

Standing Waves for Discrete Nonlinear Schrodinger Equations with Nonperiodic Bounded Potentials

In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrodinger equations

The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ>0,it holds that sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))and 4πis sharp.However,if we consider the less restrictive constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1,where V(x)is nonnegative and vanishes on an open set in R^(2),it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x)makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))when V is nonnegative and vanishes on an open set in R^(2).As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u)in R^(2),(0.1)where V(x)≥0 and vanishes on an open set of R^(2)and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential)has been the standard assumption when one deals with the existence of solutions to the above Schrodinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.

Bound states resulting from interaction of the non-relativistic particles with the multiparameter potential

In this study, we present the analytical solutions of bound states for the Schrodinger equation with the mulfiparameter potential containing the different types of physical potentials via the asymptotic iteration method by applying the Pekeristype approximation to the centrifugal potential. For any n and l （states） quantum numbers, we derive the relation that gives the energy eigenvalues for the bound states numerically and the corresponding normalized eigenfunctions. We also plot some graphics in order to investigate effects of the multiparameter potential parameters on the energy eigenvalues. Furthermore, we compare our results with the ones obtained in previous works and it is seen that our numerical results are in good agreement with the literature.

Excitonic Eigenstates of Disordered Semiconductor Quantum Wires:Adaptive Wavelet Computation of Eigenvalues for the Electron-Hole Schrodinger Equation

A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle(electron-hole)Schrodinger equation including Coulomb attraction is presented.As an example,we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material.The eigenvalues of the corresponding Schrodinger equation,i.e.,the onedimensional exciton Wannier equation with disorder,correspond to the energies of excitons in the quantum wire.The wavefunctions,in turn,provide information on the optical properties of the wire.We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric,continuous,and coercive.The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers.The numerical solution of this problem is based on adaptive wavelets.Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates.Numerical examples demonstrate the behavior of the smallest eigenvalue,the ground state energies of the exciton,together with the eigenstates depending on the strength and spatial correlation of disorder.

BOUND STATES FOR A STATIONARY NONLINEAR SCHRDINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R^3

We study the following Schrodinger-Poisson system where （Pλ）{-△u＋ V（x）u＋λФ（x）u^p=x∈R^3,-△Ф=u^2,lim│x│→∞Ф（x） =0,u〉0,where λ≥0 is a parameter,1 〈 p 〈 ＋∞, V（x） and Q（x）=1 ,D.Ruiz[19] proved that（Pλ）with p∈ （2, 5） has always a positive radial solution, but （Pλ） with p E （1, 2] has solution only if λ 〉 0 small enough and no any nontrivial solution if λ≥1/4.By using sub-supersolution method,we prove that there exists λ0〉0 such that（Pλ）with p ∈（1＋∞）has alaways a bound state（H^1（R^3）solution for λ∈[0,λ0）and certain functions V（x）and Q（x）in L^∞（R^3）.Moreover,for every λ∈[0,λ0）,the solutions uλ of （Pλ）converges,along a subsequence,to a solution of （P0）in H^1 as λ→0

A Statistical Mechanical Analysis on the Bound State Solution of an Energy-Dependent Deformed Hulthén Potential Energy

In this article, we investigate the bound state solution of the Klein Gordon equation under mixed vector and scalar coupling of an energy-dependent deformed Hulthén potential in D dimensions. We obtain a transcendental equation after we impose the boundary conditions. We calculate energy spectra in four different limits and in arbitrary dimension via the Newton-Raphson method. Then, we use a statistical method, namely canonical partition function, and discuss the thermodynamic properties of the system in a comprehensive way. We find out that some of the thermodynamic properties overlap with each other, some of them do not.

Solution of the Schrodinger equation for a particular form of Morse potential using the Laplace transform

In this paper, we have solved the Schrdinger equation for a particular kind of Morse potential and find its normalized eigenfunctions and eigenvalues, exactly. Our work is based on the Laplace transform technique which reduces the second-order differential equation to a first-order.

Aharonov-Bohm effect for bound states on spin-0 massive charged particles in a Gödel-type space-time with Coulomb potential

In this paper,we investigate the relativistic quantum dynamics of spin-0 massive charged particles in a G?del-type space-time with electromagnetic interactions.We derive the radial wave equation of the Klein-Gordon equation with an internal magnetic flux field and Coulombtype potential in the Som-Raychaudhuri space-time with cosmic string.We solve this equation and analyze the analog effect in relation to the Aharonov-Bohm effect for bound states.

On the Role of Differentiation Parameter in a Bound State Solution of the Klein-Gordon Equation

Recently, the bound state solutions of a confined Klein-Gordon particle under the mixed scalar-vector generalized symmetric Woods-Saxon potential in one spatial dimension have been investigated. The obtained results reveal that in the spin symmetric limit discrete spectrum exists, while in the pseudo-spin symmetric limit it does not.In this manuscript, new insights and information are given by employing an analogy of the variational principle. The role of the difference of the magnitudes of the vector and scalar potential energies, namely the differentiation parameter,on the energy spectrum is examined. It is observed that the differentiation parameter determines the measure of the energy spectrum density by modifying the confined particle's mass-energy in addition to narrowing the spectrum interval length.

Approximate Analytical Solutions for Scattering States of D-dimensional Hulthen Potentials

Using the exponential function transformation approach along with an approximation for the centrifugal potential, the radial Schr6dinger equation of D-dimensional Hulthen potential is transformed to a hypergeometric differential equation. The approximate analytical solutions of scattering states are attained. The normalized wave functions expressed in terms of hypergeometrie functions of scattering states on the ＂k/2π scale＂ and the calculation formula of phase shifts are given. The physical meaning of the approximate analytical solutions is discussed.

Dynamics analysis of higher-order soliton solutions for the coupled mixed derivative nonlinear Schrodinger equation

In this paper,the dynamics of the higher-order soliton solutions for the coupled mixed derivative nonlinear Schrodinger equation¨are investigated via generalized Darboux transformation.Given a pair of linearly independent solutions of the Lax pair,the oneto three-soliton solutions are obtained via algebraic iteration.Furthermore,two and three solitons are respectively displayed via numerical simulation.Moreover,the dynamics of solitons are illustrated with corresponding evolution plots,such as elastic collisions,inelastic collisions,and bound states.It is found that there are some novel phenomena of interactions among solitons,which may provide a theoretical basis for studying optical solitons in experiments.

Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator Potential

A new ring-shaped non-harmonic oscillator potential is proposed. The precise bound solution of Dirac equation with the potential is gained when the scalar potential is equal to the vector potential. The angular equation and radial equation are obtained through the variable separation method. The results indicate that the normalized angle wave function can be expressed with the generalized associated-Legendre polynomial, and the normalized radial wave function can be expressed with confluent hypergeometric function. And then the precise energy spectrum equations are obtained. The ground state and several low excited states of the system are solved. And those results are compared with the non-relativistic effect energy level in Phys. Lett. A 340 （2005） 94. The positive energy states of system are discussed and the conclusions are made properly.