t to )b x2. This corresponds to a time-dependent mass (TM) Schrödinger equation. We give the specific transformations to a different time-dependent quadratic Schrödinger equation (TQ) and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state 〈x〉 and 〈p〉 (with their classical mo...

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 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...