Exact solutions of the Schrödinger equation for zero energy

Some new exact solutions of the Schrödinger equation for zero energy are presented for certain nontrivial model potentials. Exact expressions for the different scattering lengths are derived and their differences and similarities are worked out. In particular, the respective distributions of the zeros and poles of the scattering lengths are characterized in detail.     

Exact solution of N-dimensional radial Schrödinger equation for the fourth-order inverse-power potential

Radial Schrödinger equation in N-dimensional Hilbert space with the potential V(r)=ar^-1+br^-2+cr^-3+dr^-4 is solved exactly by power series method via a suitable ansatz to the wave function with parameters those also exist in the potential function possibly for the first time. Exact analytical expressions for the energy spectra and potential parameters are obtained in terms of linear combinations of known parameters of radial quantum number n, angular momentum quantum number l, and the spatial dimensions N. Expansion coefficients of the wave function ansatz are generated through the two-term recursion relation for odd/even solutions.     

Eigen solutions of the D-dimensional Schrӧdinger equation with inverse trigonometry scarf potential and Coulomb potential

The approximate analytical solutions of the D-dimensional space of the Schrӧdinger equation is studied with a newly proposed potential model. The proposed potential is a combination of Coulomb potential and inverse trigonometry scarf-type potential. The energy equation and the corresponding wave function are obtained using parametric Nikiforov–Uvarov method. The energy equations for Coulomb potential and inverse trigonometry scarf-type potential are respectively obtained by changing the numerical values of the potential strengths. It is found that the results obtained are equivalent to that previously obtained for Hellmann potential which is a combination of Coulomb potential and Yukawa potential. It is also found that the results for inverse trigonometry scarf potential are equivalent to the results previously obtained for Yukawa potential. Also, the Onicescus information energy of a system under the influence of the newly proposed potential is investigated in detail.     

Exact solution of the discrete Schrödinger equation for ferromagnetic chains

It is shown that Bethe's exact solution for the finite Heisenberg ferromagnetic chain can be obtained via a direct Fourier transform. Unlike the Bethe ansatz approach, the latter is not confined to one space dimension and opens the possibility of obtaining exact solutions in higher dimensions.     

Exact and approximate solutions of Schrödinger’s equation for a class of trigonometric potentials

The asymptotic iteration method is used to find exact and approximate solutions of Schrödinger’s equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using a coordinate transformation to reduce the Schrödinger equation to a second-order differential equation with an appropriate form. The asymptotic iteration method is also employed indirectly to obtain the terms in perturbation expansions, both for the energies and for the corresponding eigenfunctions.     

Exact solution of D-dimensional Schrödinger equation generated from certain central power-law potentials

By applying Extended Transformation method we have generated exact solution of D-dimensional radial Schrödinger equation for a set of power-law multi-term potentials taking singular potentials $V(r) = ar^{ - \tfrac{1} {2}} + br^{ - \tfrac{3} {2}}$ , $V(r) = ar^{\tfrac{2} {3}} + br^{ - \tfrac{2} {3}} + cr^{ - \tfrac{4} {3}}$ , V(r) = ar + br ^?1 + cr ² and V(r) = ar ²+br ^?2+cr ^?4+dr ^?6 as input reference. The restriction on the parameters of the given potentials and angular momentum quantum number ? are obtained. The multiplet structure of the generated exactly solvable potentials are also shown.     

Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients

A broad class of exact self-similar solutions to the nonlinear Schr?dinger equation (NLSE) with distributed dispersion, nonlinearity, and gain or loss has been found. Appropriate solitary wave solutions applying to propagation in optical fibers and optical fiber amplifiers with these distributed parameters have also been studied. These solutions exist for physically realistic dispersion and nonlinearity profiles in a fiber with anomalous group velocity dispersion. They correspond either to compressing or spreading solitary pulses which maintain a linear chirp or to chirped oscillatory solutions. The stability of these solutions has been confirmed by numerical simulations of the NLSE.     

Rapidly decaying solutions of the nonlinear Schrödinger equation

We consider global solutions of the nonlinear Schrödinger equation

Exact solutions to the Schrödinger equation for potentials with Coulomb and harmonic oscillator terms

Exact solutions to the Schrödinger equation for potentials containing Coulomb (∼1/r) plus harmonic oscillator (∼r²) terms are found, subject to constraints on the ratio of the strengths of the Coulomb and harmonic oscillator terms. The solutions have the simple form of a product of exponential and polynomial functions.     

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrödinger equation. Firstly, the generalized nonlinear Schrödinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.     

Solutions of the Schrödinger equation for Dirac delta decorated linear potential

We have studied bound states of the Schrödinger equation for a linear potential together with any finite number (P) of Dirac delta functions. Forx<-0, the potential is given as

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where 0<f; 0<x ₁<x ₂<...<x _P , theσ _i are arbitrary real numbers, and the potential is infinite forx<0.

     

Focusing and multi-focusing solutions of the nonlinear Schrödinger equation

A method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic Schrödinger equations in two and three dimensions and describe their universal properties. The same method is applied to simulate the multi-focusing phenomena produced by simple models of saturating nonlinearities.     

Quantization of the potential amplitude in the one-dimensional Schrödinger equation

A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ?_n(x) and eigenvalues α_n corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ?_n(x) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.     

Approximate solutions of Schrdinger equation for Eckart potential with centrifugal term

The approximate analytical solutions of the Schrdinger 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.     

Lyapunov exponents of the Schrödinger equation with quasi-periodic potential on a strip

We prove that all the non-negative Lyapunov exponents of difference Schrödinger equation

Solutions of the D-dimensional Schrdinger equation with Killingbeck potential:Lie algebraic approach

Algebraic solutions of the D-dimensional Schr o¨dinger equation with Killingbeck potential are investigated using the Lie algebraic approach within the framework of quasi-exact solvability.The spectrum and wavefunctions of the system are reported and the allowed values of the potential parameters are obtained through the sl(2) algebraization.     

Properties of weakly collapsing solutions to the nonlinear Schrödinger equation

It is shown that one of the conditions for a weakly collapsing solution with zero energy produces an infinite number of functionals I _N identically vanishing on the regular solutions to the corresponding differential equation. On the parameter plane {A, C₁}, there are at least two singular lines. Along one of these lines (A/C₁=1/6), are located weakly collapsing solutions with zero energy. It is assumed that, along the second line (A/C₁=α_c), another family of weakly collapsing solutions with zero energy is located. In the domain of large values of the parameters C₁, α=A/C₁, there exists a domain of an intermediate asymptotic form, where the amplitude of oscillations of the function U grows in a large domain relative to the ξ coordinate.