Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator $-d^2/dx^2$ on $L^2[-a,a]$, $a>0$, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging to the first group (energy of ground state strictly positive) has an infinite sequence of supersymmetric partners, such that the $ell$-th order partner differs in one energy level from both the $(ell-1)$-th and the $(ell+1)$-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of $-d^2/dx^2$ come from a transcendental equation and are all infinite. For the case under our study, we determine the eigenvalues, which are also infinite, {all the extensions have a purely discrete spectrum,} and their respective eigenfunctions for all of its $ell$-th supersymmetric partners of each extension.

The Schrodinger particle on the half-line with an attractive $delta$-interaction: bound states and resonances

In this paper we provide a detailed description of the eigenvalue $ E_{D}(x_0)leq 0$ (respectively $ E_{N}(x_0)leq 0$) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution $-lambda delta(x-x_0)$ for any fixed value of the magnitude of the coupling constant. We also investigate the $lambda$-dependence of both eigenvalues for any fixed value of $x_0$. Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of the resolvent. These results will be connected with the study of the ground state energy of two remarkable three-dimensional self-adjoint operators, studied in depth in Albeverios monograph, perturbed by an attractive $delta$-distribution supported on the spherical shell of radius $r_0$.

A discussion on the approximate solutions of first order systems of non-linear ordinary equations

We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the exact solution. We study the precision, in terms of the local error, of the method by applying it to different well known examples. The advantage of the method over others widely used lies on the simplicity of its implementation.

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

We study a non-relativistic particle subject to a three-dimensional spherical potential consisting of a finite well and a radial $delta$-$delta$ contact interaction at the well edge. This contact potential is defined by appropriate matching conditions for the radial functions, thereby fixing a self adjoint extension of the non-singular Hamiltonian. Since this model admits exact solutions for the wave function, we are able to characterize and calculate the number of bound states. We also extend some well-known properties of certain spherically symmetric potentials and describe the resonances, defined as unstable quantum states. Based on the Woods-Saxon potential, this configuration is implemented as a first approximation for a mean-field nuclear model. The results derived are tested with experimental and numerical data in the double magic nuclei $^{132}$Sn and $^{208}$Pb with an extra neutron.

Spherical harmonics and rigged Hilbert spaces

This paper is devoted to study discrete and continuous bases for spaces supporting representations of SO(3) and SO(3,2) where the spherical harmonics are involved. We show how discrete and continuous bases coexist on appropriate choices of rigged Hilbert spaces. We prove the continuity of relevant operators and the operators in the algebras spanned by them using appropriate topologies on our spaces. Finally, we discuss the properties of the functionals that form the continuous basis.

On the spectrum of the Schrodinger Hamiltonian of the one-dimensional conic oscillator perturbed by a point interaction

We decorate the one-dimensional conic oscillator $frac{1}{2} left[-frac{d^{2} }{dx^{2} } + left|x right| right]$ with a point impurity of either $delta$-type, or local $delta$-type or even nonlocal $delta$-type. All the three cases are exactly solvable models, which are explicitly solved and analysed, as a first step towards higher dimensional models of physical relevance. We analyse the behaviour of the change in the energy levels when an interaction of the type $-lambda,delta(x)$ or $-lambda,delta(x-x_0)$ is switched on. In the first case, even energy levels (pertaining to antisymmetric bound states) remain invariant with $lambda$ although odd energy levels (pertaining to symmetric bound states) decrease as $lambda$ increases. In the second, all energy levels decrease when the form factor $lambda$ increases. A similar study has been performed for the so called nonlocal $delta$ interaction, requiring a coupling constant renormalization, which implies the replacement of the form factor $lambda$ by a renormalized form factor $beta$. In terms of $beta$, even energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of $beta$ the energy of each odd level, with the natural exception of the first one, becomes lower than the constant energy of the previous even level. Finally, we consider an interaction of the type $-adelta(x)+bdelta(x)$, and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.

Resonances and antibound states of Poschl-Teller potential: Ladder operators and SUSY partners

We analyze the one dimensional scattering produced by all variations of the Poschl-Teller potential, i.e., potential well, low and high barriers. We show that the Poschl-Teller well and low barrier potentials have no resonance poles, but an infinite number of simple poles along the imaginary axis corresponding to bound and antibound states. A quite different situation arises on the Poschl-Teller high barrier potential, which shows an infinite number of resonance poles and no other singularities. We have obtained the explicit form of their associated Gamow states. We have also constructed ladder operators connecting wave functions for bound and antibound states as well as for resonance states. Finally, using wave functions of Gamow and antibound states in the factorization method, we construct some examples of supersymmetric partners of the Poschl-Teller Hamiltonian.

Two-point one-dimensional $delta$-$delta^prime$ interactions: non-abelian addition law and decoupling limit

In this contribution to the study of one dimensional point potentials, we prove that if we take the limit $qto 0$ on a potential of the type $v_0delta({y})+{2}v_1delta({y})+w_0delta({y}-q)+ {2} w_1delta({y}-q)$, we obtain a new point potential of the type ${u_0} delta({y})+{2 u_1} delta({y})$, when $ u_0$ and $ u_1$ are related to $v_0$, $v_1$, $w_0$ and $w_1$ by a law having the structure of a group. This is the Borel subgroup of $SL_2({mathbb R})$. We also obtain the non-abelian addition law from the scattering data. The spectra of the Hamiltonian in the exceptional cases emerging in the study are also described in full detail. It is shown that for the $v_1=pm 1$, $w_1=pm 1$ values of the $delta^prime$ couplings the singular Kurasov matrices become equivalent to Dirichlet at one side of the point interaction and Robin boundary conditions at the other side.

Point-form dynamics of quasistable states

We present a field theoretical model of point-form dynamics which exhibits resonance scattering. In particular, we construct point-form Poincare generators explicitly from field operators and show that in the vector spaces for the in-states and out-states (endowed with certain analyticity and topological properties suggested by the structure of the $S$-matrix) these operators integrate to furnish differentiable representations of the causal Poincare semigroup, the semidirect product of the semigroup of spacetime translations into the forward lightcone and the group of Lorentz transformations. We also show that there exists a class of emph{irreducible} representations of the Poincare semigroup defined by a complex mass and a half-integer spin. The complex mass characterizing the representation naturally appears in the construction as the square root of the pole position of the propagator. These representations provide a description of resonances in the same vein as Wigners unitary irreducible representations of the Poincare group provide a description of stable particles.

Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space

In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantum system with purely continuous spectrum: the quantum free particle in a Lobachevski space with constant negative curvature. The SGA contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the Hamiltonian among themselves. In our case, the geometrical symmetry algebra is $frak{so}(3,1)$ and the SGA is $frak{so}(4,2)$. We start with a representation of $frak{so}(4,2)$ by functions on a realization of the Lobachevski space given by a two sheeted hyperboloid, where the Lie algebra commutators are the usual Poisson-Dirac brackets. Then, introduce a quantized version of the representation in which functions are replaced by operators on a Hilbert space and Poisson-Dirac brackets by commutators. Eigenfunctions of the Hamiltonian are given and naive ladder operators are identified. The previously defined naive ladder operators shift the eigenvalues by a complex number so that an alternative approach is necessary. This is obtained by a non self-adjoint function of a linear combination of the ladder operators which gives the correct relation among the eigenfunctions of the Hamiltonian. We give an eigenfunction expansion of functions over the upper sheet of two sheeted hyperboloid in terms of the eigenfunctions of the Hamiltonian.