The most general combination of couplings of fermions with external potentials in 1+1 dimensions, must include vector, scalar and pseudoscalar potentials. We consider such a mixing of potentials in a PT-symmetric time-independent Dirac equation. The Dirac equation is mapped into an effective PT-symmetric Schrödinger equation. Despite the non-hermiticity of the effective potential, we find real ...

We investigate quantum mechanical Hamiltonians with explicit time dependence. We find a class of models in which an analogue of the time independent Schrödinger equation exists. Among the models in this class is a new exactly soluble model, the harmonic oscillator with frequency inversely proportional to time.

Since Schrödinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrödinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrödinger equation is in principle the differential equation and difficult to solve, b...

We consider two dimensional Grushin Schr\"odinger equation posed on a finite cylinder $\Omega=(-1,1)_x\times \T_y$ with Dirichlet boundary condition. obtain the sharp observability by any horizontal strip, optimal time $T_*>0$ depending size of strip. Consequently, we prove exact controllability for equation. By exploiting concentration eigenfunctions harmonic oscillator at $x=0$, also show tha...

We investigate the solutions for a time dependent potential by considering two scenarios fractional Schr\"odinger equation. The first scenario analyzes influence of in absence kinetic term. obtain analytical and numerical this case Caputo derivative, which extends Rabi's model. In second scenario, we incorporate term equation consider spatial derivatives. For case, analyze spreading Gaussian wa...

We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger eq...

We explore a novel approach to the study of large-scale structure formation in which self-gravitating cold dark matter (CDM) is represented by a complex scalar field whose dynamics are governed by coupled Schrödinger and Poisson equations. We show that, in the quasi-linear regime, the Schrödinger equation can be reduced to the free-particle Schrödinger equation. We advocate using the free-parti...

In this paper we have solved analytically the Schrödinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. Based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square ...