Related papers: On the asymptotic decay of the Schr\"odinger--Newton ground state

On the asymptotic decay of the Schr\"odinger--Newton ground state

URL: http://arxiv.org/abs/2101.01296v4
Date: Sun, 7 Mar 2021 17:44:56 GMT
Title: On the asymptotic decay of the Schr\"odinger--Newton ground state
Authors: Michael K.-H. Kiessling
Abstract summary: The ground state $u(r)$ of the Schr"odinger--Newton equation in $mathbbR3$ was determined by V. Moroz and J. van Schaftingen. Asymptotic results are proposed for the Schr"odinger--Newton equation with external $sim - K/r$ potential, and for the related Hartree equation of a bosonic atom or ion.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: The asymptotics of the ground state $u(r)$ of the Schr\"odinger--Newton equation in $\mathbb{R}^3$ was determined by V. Moroz and J. van Schaftingen to be $u(r) \sim A e^{-r}/ r^{1 - \|u\|_2^2/8\pi}$ for some $A>0$, in units that fix the exponential rate to unity. They left open the value of $\|u\|_2^2$, the squared $L^2$ norm of $u$. Here it is rigorously shown that $2^{1/3}3\pi^2\leq \|u\|_2^2\leq 2^{3}\pi^{3/2}$. It is reported that numerically $\|u\|_2^2\approx 14.03\pi$, revealing that the monomial prefactor of $e^{-r}$ increases with $r$ in a concave manner. Asymptotic results are proposed for the Schr\"odinger--Newton equation with external $\sim - K/r$ potential, and for the related Hartree equation of a bosonic atom or ion.

Related papers

Quantum charges of harmonic oscillators [55.2480439325792]
We show that the energy eigenfunctions $psi_n$ with $nge 1$ are complex coordinates on orbifolds $mathbbR2/mathbbZ_n$. We also discuss "antioscillators" with opposite quantum charges and the same positive energy.
arXiv Detail & Related papers (2024-04-02T09:16:18Z)
The SUSY partners of the QES sextic potential revisited [0.0]
We study the SUSY partner Hamiltonians of the quasi- solvable (QES) sextic potential $Vrm qes(x) = nu, x6 + 2, nu, mu,x4 + left[mu2-(4N+3)nu right], x2$, $N in mathbbZ+$.
arXiv Detail & Related papers (2023-11-10T18:38:02Z)
The effect of a $\delta$ distribution potential on a quantum mechanical particle in a box [0.0]
We obtain the limit of the eigenfunctions of the time independent Schr"odinger equation as $alphanearrow+infty$ and as $alphasearrow-infty$. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking $|alpha|toinfty$.
arXiv Detail & Related papers (2023-11-05T09:56:48Z)
Quantum connection, charges and virtual particles [65.268245109828]
A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation. We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
arXiv Detail & Related papers (2023-10-10T10:27:09Z)
Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
arXiv Detail & Related papers (2023-06-21T10:11:28Z)
Quantization Condition of the Bound States in $n$th-order Schr\"{o}dinger equations [2.5822051639377137]
We will prove a general approximate quantization rule $% int_L_ER_Ek. The only hypothesis is that all exponentially growing components are negligible, which is appropriate for not narrow wells.
arXiv Detail & Related papers (2023-04-03T12:07:34Z)
Ground state solution of a Kirchhoff type equation with singular potentials [0.0]
We study the existence and blow-up behavior of minimizers for $E(b)=infBigmathcalE_b(u),|, uin H1(R2), |u|_L2=1Big,$ here $mathcalE_b(u)$ is the Kirchhoff energy functional defined by $mathcalE_b(u)= int_R2.
arXiv Detail & Related papers (2022-12-15T16:40:48Z)
Some Remarks on the Regularized Hamiltonian for Three Bosons with Contact Interactions [77.34726150561087]
We discuss some properties of a model Hamiltonian for a system of three bosons interacting via zero-range forces in three dimensions. In particular, starting from a suitable quadratic form $Q$, the self-adjoint and bounded from below Hamiltonian $mathcal H$ can be constructed. We show that the threshold value $gamma_c$ is optimal, in the sense that the quadratic form $Q$ is unbounded from below if $gammagamma_c$.
arXiv Detail & Related papers (2022-07-01T10:01:14Z)
From quartic anharmonic oscillator to double well potential [77.34726150561087]
It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction $Psi_ao(u)$, obtained recently, it is possible to get highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.
arXiv Detail & Related papers (2021-10-30T20:16:27Z)
Anharmonic oscillator: a solution [77.34726150561087]
The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
arXiv Detail & Related papers (2020-11-29T22:13:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.