arXiv subjects

Fabian Brau

Publications and source records attributed to Fabian Brau.

42 records · Page 3Linked to original sources

arXiv2004

Lower limit in semiclassical form for the number of bound states in a central potential

We identify a class of potentials for which the semiclassical estimate $N^{\text{(semi)}}=\frac{1}π\int_0^\infty dr\sqrt{-V(r)θ[-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\ge {N^{\text{(semi)}}}$, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential $V(r)$ with the effective $\ell$-wave potential $V_\ell^{\text{(eff)}}(r)=V(r)+\frac{\ell(\ell+1)}{r^2}$. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.

math-ph↗

arXiv2004

On the decrease of the number of bound states with the increase of the angular momentum

For the class of central potentials possessing a finite number of bound states and for which the second derivative of $r V(r)$ is negative, we prove, using the supersymmetric quantum mechanics formalism, that an increase of the angular momentum $\ell$ by one unit yields a decrease of the number of bound states of at least one unit: $N_{\ell+1}\le N_{\ell}-1$. This property is used to obtain, for this class of potential, an upper limit on the total number of bound states which significantly improves previously known results.

math-ph↗

arXiv2004

A class of ($\ell$-dependent) potentials with the same number of ($\ell$-wave) bound states

We introduce and investigate the class of central potentials $$V_{\text{CIC}}(g^{2},μ^{2},\ell,R;r)=-\frac{g^{2}}{R^{2}} (\frac{r}{R})^{4\ell} {[ 1+(\frac{1}{2\ell+1}) (\frac{r}{R})^{2\ell+1}]^{2}-1+μ^{2}}^{-2}$$, which possess, in the context of nonrelativistic quantum mechanics, a number of $\ell$-wave bound states given by the ($\ell$-independent !) formula $$N_{\ell}^{\text{(CIC)}}(g^{2},μ^{2}) ={{\frac{1}π\sqrt{g^{2}+μ^{2}-1} (\sqrt{μ^{2}-1})^{-1} \arctan(\sqrt{μ^{2}-1})}}$$. Here $g$ and $μ$ are two arbitrary real parameters, $\ell$ is the angular momentum quantum number, and the double braces denote of course the integer part. An extension of this class features potentials that possess the same number of $\ell$-wave bound states and behave as $(a/r)^{2}$ both at the origin ($r\to 0^{+}$) and at infinity ($r\to \infty$), where $a$ is an additional free parameter.

math-ph↗

arXiv2004

Upper limit on the number of bound states of the spinless Salpeter equation

We obtain, using the Birman-Schwinger method, upper limits on the total number of bound states and on the number of $\ell$-wave bound states of the semirelativistic spinless Salpeter equation. We also obtain a simple condition, in the ultrarelativistic case ($m=0$), for the existence of at least one $\ell$-wave bound states: $C(\ell,p/(p-1))$ $\int_0^{\infty}dr r^{p-1} |V^-(r)|^p\ge 1$, where $C(\ell,p/(p-1))$ is a known function of $\ell$ and $p>1$.

math-ph↗

arXiv2004

Necessary and sufficient conditions for existence of bound states in a central potential

We obtain, using the Birman-Schwinger method, a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics. These conditions yield a monotonic series of lower limits on the "critical" value of the strength of the potential (for which a first bound state appears) which converges to the exact critical strength. We also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential.

math-ph↗

arXiv2004

Upper and lower limits on the number of bound states in a central potential

In a recent paper new upper and lower limits were given, in the context of the Schrödinger or Klein-Gordon equations, for the number $N_{0}$ of S-wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity. In this paper these results are extended to the number $N_{\ell}$ of bound states for the $\ell$-th partial wave, and results are also obtained for potentials that are not monotonic and even somewhere positive. New results are also obtained for the case treated previously, including the remarkably neat \textit{lower} limit $N_{\ell}\geq \{\{[σ/(2\ell+1)+1]/2\}\}$ with $% σ=(2/π) \underset{0\leq r<\infty}{\max}[r| V(r)| ^{1/2}]$ (valid in the Schrödinger case, for a class of potentials that includes the monotonically nondecreasing ones), entailing the following \textit{lower} limit for the total number $N$ of bound states possessed by a monotonically nondecreasing central potential vanishing at infinity: $N\geq \{\{(σ+1)/2\}\} {(σ+3)/2\} \}/2$ (here the double braces denote of course the integer part).

math-ph↗