Hyperspherical δ-δ′ potentials

Abstract

The spherically symmetric potential a δ(r − r 0 ) + b δ (r − r 0 ) is generalised for the ddimensional 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 d ≥ 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 δ-δ potential for arbitrary a > 0 admits a bound state with zero angular momentum.

An approximation to the Woods–Saxon potential based on a contact interaction

“…where the equality follows from standard properties of the Bessel functions [59] and the second relation from the Turan-type inequalities [50,70]:…”

“…(15). These requirements are given by matching conditions relating the function u (r ) and its first derivative at the limit values of R. They can be written in terms of a SL(2, R) matrix as [14,15,50,51]…”

Exact results for nonequilibrium dynamics in Wigner phase space

“…We are going to consider now an extension of the previous study of two interacting particles that takes into account the presence of an extra point-like interaction term in the potential, proportional to δ . This type of point or zerorange potentials are a subject of recent study in differents contexts [30][31][32][33]. The Hamiltonian of the light particle with mass m is now given for t < 0 by…”

The Energy of the Ground State of the Two-Dimensional Hamiltonian of a Parabolic Quantum Well in the Presence of an Attractive Gaussian Impurity

“…In addition, the one-dimensional Laplacian can be equipped with four one-parameter families of point potentials [6], which provide a wide range of interesting examples from the mathematical as well as from the physical point of view, which are easily constructed via matching conditions on the wave functions of their domains [6] (this is not the case for one-dimensional Salpeter Hamiltonians, namely semirelativistic Hamiltonians with kinetic term given by (p 2 + m 2 ) 1 2 , for which delta interactions may only be added after a regularisation [7][8][9]). Some attempts have been made to extend the formalism to systems in two or three dimensions, where contact potentials have been defined over circles (two-dimensional case) [10], surfaces like hollow spheres (three-dimensional case) [11] or points [12][13][14], or even to a non-linear Schrödinger Equation [15]. In two or three dimensions, the construction of a self-adjoint Hamiltonian with a point potential requires either the use of the theory of self-adjoint extensions of symmetric operators or the procedure known as coupling constant renormalisation.…”