Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSome recent results on contact or point supported potentials
125
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We introduced some contact potentials that can be written as a linear combination of the Dirac delta and its first derivative, the $delta$-$delta$ interaction. After a simple general presentation in one dimension, we briefly discuss a one dimensional periodic potential with a $delta$-$delta$ interaction at each node. The dependence of energy bands with the parameters (coefficients of the deltas) can be computed numerically. We also study the $delta$-$delta$ interaction supported on spheres of arbitrary dimension. The spherical symmetry of this model allows us to obtain rigorous conclusions concerning the number of bound states in terms of the parameters and the dimension. Finally, a $delta$-$delta$ interaction is used to approximate a potential of wide use in nuclear physics, and estimate the total number of bound states as well as the behaviour of some resonance poles with the lowest energy.
rate research
Read More
We first review the problem of a rigorous justification of Kubos formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubos formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.
The spherically symmetric potential $a ,delta (r-r_0)+b,delta (r-r_0)$ is generalised for the $d$-dimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum of negative, zero and positive energy states is studied in $dgeq 2$, providing numerical results for the expectation value of the radius as a function of the free parameters of the potential. Remarkably, only if $d=2$ the $delta$-$delta$ potential for arbitrary $a>0$ admits a bound state with zero angular momentum.
Based on a method that produces the solutions to the Schrodinger equations of partner potentials, we give two conditionally exactly solvable partner potentials of exponential type defined on the half line. These potentials are multiplicative shape invariant and each of their linearly independent solution includes a sum of two hypergeometric functions. Furthermore we calculate the scattering amplitudes and study some of their properties.
We give two conditionally exactly solvable inverse power law potentials whose linearly independent solutions include a sum of two confluent hypergeometric functions. We notice that they are partner potentials and multiplicative shape invariant. The method used to find the solutions works with the two Schrodinger equations of the partner potentials. Furthermore we study some of the properties of these potentials.
We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S^d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x-y|^(-s) with d-2 <= s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on S^d is determined. The special case s = d-2 yields interesting phenomena, which we investigate in detail. A weak* asymptotic analysis is provided as s goes to (d-2)^+.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
