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Nonlinear Schr\"odinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of…
In a recent paper, it has been shown the Schr\"{o}dinger equation for the three-dimensional harmonic oscillator can be simplified through the use of an isometric conformal transformation. Here, it is demonstrated that the same…
The classical formalism of the Moment Problem has been combined with a cumulant approach and applied to the extensive many-body problem. This has yielded many new exact results for many-body systems in the thermodynamic limit - for the…
In this paper, we study the Schr\"odinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions and the allowed values of the potential parameters are obtained…
We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schr\"odinger equation with an inverted quasi-exactly…
We propose a simple ansatz that allows to generate new exactly solvable multi-state Landau-Zener models. It is based on a system of two decoupled two-level atoms whose levels vary with time and cross at some moments. Then we consider…
Starting from the integral representation of the three-dimensional Coulomb transition matrix elaborated by us formerly with the use of specific symmetry of the interaction in a four-dimensional Euclidean space introduced by Fock, the…
Friedberg, Lee and Zhao proposed a method for effectively evaluating the eigenenergies and eigen wavefunctions of quantum systems. In this work, we study several special cases to investigate applicability of the method. Concretely, we…
We consider the time evolution of the occupation probabilities for the 2s-2p transition in a hydrogen atom interacting with an external field, V(t). A two-state model and a dipole approximation are used. In the case of degenerate energy…
We analyze a recent pedagogical proposal for an alternative treatment of the angular part of the Schr\"odinger equation with a central potential. We show that the authors' arguments are unclear, unconvincing and misleading.
We use the tridiagonal representation approach to solve the radial Schr\"odinger equation for an inverse power-law potential of a combined quartic and sextic degrees and for all angular momenta. The amplitude of the quartic singularity is…
We generalize Schroedinger's factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach, is the fact that the Hamiltonian is represented…
Bound and scattering state solutions of the effective-mass Klein-Gordon equation are obtained for the Yukawa potential with any angular momentum $\ell$. Energy eigenvalues, normalized wave functions and scattering phase shifts are…
We consider systems of weakly coupled Schr\"odinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and…
We apply a simple transformation method to construct a set of new exactly solvable potentials (ESP) which gives rise to bound state solution of $D$-dimensional Schr\"odinger equation. The important property of such exactly solvable quantum…
Bound state solutions of the Schrodinger Equation for the $-a/r^2$ potential have been presented recently for both the weak and strong coupling cases. However, Shortley in 1931 and Landau and Lifshitz in 1958 claimed that no bound state…
In this paper, we consider the scattering problem for a class of $N$-coupled systems of the cubic nonlinear Schr\"odinger equations in three space dimensions. We prove the scattering of solutions that have a mass-energy quantity less than…
The one-dimensional Schr\"odinger equation with symmetric trigonometric double-well potential (DWP) is exactly solved via angular oblate spheroidal function. The results of stringent analytic calculation for the ground state splitting of…
In the reversible Schrodinger-Newton equation a complex Newton coupling G*exp(-i*alpha) is proposed in place of G. The equation becomes irreversible and all initial one-body states are expected to converge to solitonic stationary states.…
We investigate the Rydberg states generation of Hydrogen atoms with intense laser pulses, by solving the time-dependent Schr\"odinger equation and by means of classical trajectory monte-carlo simulations. Both linearly polarized multi-cycle…