Related papers: Quantum Energy Eigenvalue Spectra for Very Flat Po…
Starting from the semi-classical spectrum of Schr\"odinger operators $-h^2\Delta+V$ (on $\mathbb{R}^n$ or on a Riemannian manifold) it is possible to detect critical levels of the potential $V$. Via micro-local methods one can express…
This is to review some recent progress in PDE. The emphasis is on (energy) supercritical nonlinear Schr\"odinger equations. The methods are applicable to other nonlinear equations.
We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schr\"odinger equation for a free particle, finite and infinite fuzzy wells. We show that all results reduce to the appropriate commutative limits.…
An differential equation for wave functions is proposed, which is equivalent to Schr\"{o}dinger's wave equation and can be used to determine energy-level gaps of quantum systems. Contrary to Schr\"{o}dinger's wave equation, this equation is…
The Schr\"{o}dinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schr\"{o}dinger equation into a second order differential equation by using an appropriate coordinate transformation. The…
$D$-dimensional Schr\"{o}dinger equation is addressed for square root power law potential. Bound state unnormalized eigenfunctions and the energy eigenvalues are obtained using wave function ansatz method. Some special cases are studied at…
In this work we calculate the exact Green's function for arbitrary rectangular potentials. Specifically we focus on Green's function for rectangular quantum wells enlarging the knowledge of exact solutions for Green's functions and also…
Eigenvalue problems are critical to several fields of science and engineering. We present a novel unsupervised neural network for discovering eigenfunctions and eigenvalues for differential eigenvalue problems with solutions that…
For quasiexactly solvable (QES) potentials a certain number of wave functions and energy levels can be analytically calculated. The complexity of an explicit calculation of the energy levels grows with the dimension of the QES sector. For a…
We suppose: (1) that the ground-state eigenvalue E = F(v) of the Schroedinger Hamiltonian H = -Delta + vf(x) in one dimension is known for all values of the coupling v > 0; and (2) that the potential shape can be expressed in the form f(x)…
In this paper, a quantum dot mathematical model based on a two-dimensional Schr\"odinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a…
Experimental results from literature show equidistant energy levels in thin Bi films on surfaces, suggesting a harmonic oscillator description. Yet this conclusion is by no means imperative, especially considering that any measurement only…
The spherical wave functions of charge-dyon bounded system in a rectangular spherical quantum dot of infinitely and finite height are calculated. The transcendent equations, defining the energy spectra of the systems are obtained. The…
In a large class of factorizing scattering models, we construct candidates for the local energy density on the one-particle level starting from first principles, namely from the abstract properties of the energy density. We find that the…
In the broad range of studies related to quantum graphs, quantum graph spectra appear as a topic of special interest. They are important in the context of diffusion type problems posed on metric graphs. Theoretical findings suggest that…
We determine approximate eigenvalues and eigenfunctions shapes for bound states in the $3D$ shallow spherical ultrarelativistic well. Existence thresholds for the ground state and first excited states are identified, both in the purely…
In this paper, we implement a quantum algorithm -on IBM quantum devices, IBM QASM simulator and PPRC computer cluster -to find the energy values of the ground state and the first excited state of a particle in a finite square-well…
We explore the spectral properties and behaviour of confining superexponential potentials. Several prototypes of these highly nonlinear potentials are analyzed in terms of the eigenvalues and eigenstates of the underlying stationary…
The optical properties of wide Quantum Wells are considered, taking into account the screened electron-hole interaction potential and parabolic confinement potentials, different for the electrons and for the holes. The role of the…
A new supersymmetry method for the generation of the quasi-exactly solvable (QES) potentials with two known eigenstates is proposed. Using this method we obtained new QES potentials for which we found in explicit form the energy levels and…