Splitting of the lowest energy levels of the Schr�dinger equation and asymptotic behavior of the fundamental solution of the equation hut=h2?u/2?V(x)u

1991 ◽  
Vol 87 (3) ◽  
pp. 561-599 ◽  
Author(s):  
S. Yu. Dobrokhotov ◽  
V. N. Kolokol'tsov ◽  
V. P. Maslov
Keyword(s):  
Asymptotic Behavior ◽  
Fundamental Solution ◽  
Energy Levels ◽  
Dinger Equation

scholarly journals The Quantum Black Hole as a Gravitational Hydrogen Atom

2019 ◽  
Author(s):  
Christian Corda ◽  
Fabiano Feleppa
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Energy Spectrum ◽  
Hydrogen Atom ◽  
Gravitational Potential ◽  
Particle System ◽  
Energy Levels ◽  
Planck Scale ◽  
Dinger Equation ◽  
The One

In this paper only one basic assumption has been made: if we try to describe black holes, their behavior should be understood in the same language as the one we use for particles; black holes should be treated just like atoms. They must be quantum forms of matter, moving in accordance with Schrödinger equations just like everything else. In particular, Rosen’s quantization approach to the gravitational collapse is applied in the simple case of a pressureless “star of dust” by finding the gravitational potential, the Schrödinger equation and the solution for the collapse’s energy levels. By applying the constraints for a Schwarzschild black hole (BH) and by using the concept of BH effective state, previously introduced by one of the authors (CC), the BH quantum gravitational potential, Schrödinger equation and the BH energy spectrum are found. Remarkably, such an energy spectrum is in agreement (in its absolute value) with the one which was conjectured by Bekenstein in 1974 and consistent with other ones in the literature. This approach also permits to find an interesting quantum representation of the Schwarzschild BH ground state at the Planck scale. Moreover, two fundamental issues about black hole quantum physics are addressed by this model: the area quantization and the singularity resolution. As regards the former, a result similar to the one obtained by Bekenstein, but with a different coefficient, has been found. About the latter, it is shown that the traditional classical singularity in the core of the Schwarzschild BH is replaced, in a full quantum treatment, by a two-particle system where the two components strongly interact with each other via a quantum gravitational potential. The two-particle system seems to be non-singular from the quantum point of view and is analogous to the hydrogen atom because it consists of a “nucleus” and an “electron”.

Size and Piezoelectric Effects on Optical Properties of Self-Assembled InGaAs/GaAs Quantum Dots

2006 ◽  
Author(s):  
M. K. Kuo ◽  
T. R. Lin ◽  
K. B. Hong
Keyword(s):  
Quantum Dots ◽  
Optical Properties ◽  
Quantum Dot ◽  
Optical Transition ◽  
Energy Levels ◽  
Three Dimensional ◽  
Dinger Equation ◽  
Element Analysis ◽  
Piezoelectric Effects ◽  
Self Assembled

Size effects on optical properties of self-assembled quantum dots are analyzed based on the theories of linear elasticity and of strain-dependent k-p with the aid of finite element analysis. The quantum dot is made of InGaAs with truncated pyramidal shape on GaAs substrate. The three-dimensional steady-state effective-mass Schro¨dinger equation is adopted to find confined energy levels as well as wave functions both for electrons and holes of the quantum-dot nanostructures. Strain-induced as well as piezoelectric effects are taken into account in the carrier confinement potential of Schro¨dinger equation. The optical transition energies of quantum dots, computed from confined energy levels for electrons and holes, are significantly different for several quantum dots with distinct sizes. It is found that for QDs with the the larger the volume of QD is, the smaller the values of the optical transition energy. Piezoelectric effect, on the other hand, splits the p-like degeneracy for the electron first excited state about 1~7 meV, and leads to anisotropy on the wave function.

Note on the fast decay property of steady Navier–Stokes flows in the whole space

Analysis ◽  
2018 ◽  
Vol 38 (2) ◽  
pp. 81-89
Author(s):  
Tomoyuki Nakatsuka
Keyword(s):  
Asymptotic Behavior ◽  
Fundamental Solution ◽  
Unique Solution ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Asymptotic Behavior Of Solutions ◽  
Fast Decay ◽  
Rapid Decay ◽  
Navier Stokes Equation ◽  
Whole Space

Abstract We investigate the pointwise asymptotic behavior of solutions to the stationary Navier–Stokes equation in {\mathbb{R}^{n}} ( {n\geq 3} ). We show the existence of a unique solution {\{u,p\}} such that {|\nabla^{j}u(x)|=O(|x|^{1-n-j})} and {|\nabla^{k}p(x)|=O(|x|^{-n-k})} ( {j,k=0,1,\ldots} ) as {|x|\rightarrow\infty} , assuming the smallness of the external force and the rapid decay of its derivatives. The solution {\{u,p\}} decays more rapidly than the Stokes fundamental solution.

scholarly journals A STOCHASTIC DERIVATION OF THE GEODESIC RULE

2006 ◽  
Vol 21 (07) ◽  
pp. 1493-1502 ◽  
Author(s):  
NIKOS KALOGEROPOULOS
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Fundamental Solution ◽  
Heat Kernel ◽  
Continuum Limit ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
The Continuum ◽  
Fokker Planck

We argue that the geodesic rule, for global defects, is a consequence of the randomness of the values of the Goldstone field ϕ in each causally connected volume. As these volumes collide and coalescence, ϕ evolves by performing a random walk on the vacuum manifold [Formula: see text]. We derive a Fokker–Planck equation that describes the continuum limit of this process. Its fundamental solution is the heat kernel on [Formula: see text], whose leading asymptotic behavior establishes the geodesic rule.

THREE-DIMENSIONAL ISOTROPIC PSEUDO-GAUSSIAN OSCILLATORS

2009 ◽  
Vol 20 (07) ◽  
pp. 1103-1111 ◽  
Author(s):  
ION I. COTĂESCU ◽  
PAUL GRĂVILĂ ◽  
MARIUS PAULESCU
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Methods ◽  
Harmonic Oscillator ◽  
Energy Levels ◽  
Three Dimensional ◽  
Finite Energy ◽  
Energy Spectra ◽  
Isotropic Harmonic Oscillator ◽  
Quantum Models

A family of isotropic three-dimensional quantum models governed by isotropic pseudo-Gaussian potentials is proposed. These potentials are defined to have a Gaussian asymptotic behavior but approaching to the potential of the isotropic harmonic oscillator when x → 0. These models may have finite energy spectra with approximately equidistant energy levels that can be calculated using efficient numerical methods based on generating functionals.

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