Reply to comment on Inverse problem from the discrete spectrum in the D = 2 dimensional space (2014 Phys. Scr. 989 085201)

International audienceReply to comment on Inverse problem from the discrete spectrum in the D = 2 dimensional space (2014 Phys. Scr. 989 085201

Inverse problem from the discrete spectrum in the D = 2 dimensional space

International audienceConsidering the Schrödinger equation in the D = 2 dimensional space, we propose a method to determine a circular symmetric potential from its discrete spectrum. The approach is based on the relationships between the moments of the ground state density and the lowest excitation energy of each angular momentum. The required condition for a unique answer is the knowledge of all the lowest eigenvalues. In principle, it means an infinite number of moments to be known. As we shall show, reasonable accuracy can be reached in practice with a finite set of moments. Two illustrative examples are presented

Energy-dependent potentials and the problem of the equivalent local potential

The properties of the wave equation are studied in the case of energy-dependent potentials for bound sates. The nonlinearity induced by the energy dependence requires modification of the standard rules of quantum mechanics. These modifications are briefly recalled. Analytical and numerical solutions are given in the three-dimensional space for power-law radial shape potentials with a linear energy dependence. This last is chosen since it allows the construction of a coherent theory. Among the results, we stress the saturation of the spectrum observed for confining potentials: as the quantum numbers increase, the eigenvalues reach an upper limit. Finally, the problem of the equivalent local potential is discussed. The existence of analytical solutions presents a good opportunity to tackle this problem in detail

A pseudo Coulombian potential in D=1 dimensional space

In the D=1 dimensional space, we study the bound state solutions of the potential V(x) = -\frac{e}{x} + \frac{b}{x^2} (e, b>0). They occur on the right half-plane xin[0, ∞[. In the limit b→0, we recover the spectrum of the D=1 Coulomb potential. Supersymmetric properties are briefly discussed. The model is extended by considering complex coupling constants. Nonlinear effects are also treated by considering a linear energy dependence of the e coupling constant

The inverse problem in the case of bound states

We investigate the inverse problem for bound states in the D = 3 dimensional space. The potential is assumed to be local and spherically symmetric. The present method is based on relationships connecting the moments of the ground state density to the lowest energy of each state of angular momentum ℓ. The reconstruction of the density ρ(r) from its moments is achieved by means of the series expansion of its Fourier transform F(q). The large q-behavior is described by Padé approximants. The accuracy of the solution depends on the number of known moments. The uniqueness is achieved if this number is infinite. In practice, however, an accuracy better than 1% is obtained with a set of about 15 levels

The rotational spectrum and the attractive delta-shell potential

The spectrum of the attractive delta-shell potential is investigated in the D-dimensional space (D≥2). A compact expression for the eigenvalues is derived in the large coupling constant limit for odd dimensions D. With respect to the ground state energy, the spectrum is rotational, being proportional to L(L+D-2), where L is the grand orbital momentum. Extension to even D is achieved on the grounds of the Hellmann–Feynman theorem. Simulating a delta-shell potential by a normalized Gaussian, we discuss finite range effects

Applying supersymmetry to energy dependent potentials

We investigate the supersymmetry properties of energy dependent potentials in the D=1 dimensional space. We show the main aspects of supersymmetry to be preserved, namely the factorization of the Hamiltonian, the connections between eigenvalues and wave functions of the partner Hamiltonians. Two methods are proposed. The first one requires the extension of the usual rules via the concept of local equivalent potential. In this case, the superpotential becomes depending on the state. The second method, applicable when the potential depends linearly on the energy, is similar to what has been already achieved by means of the Darboux transform

Complex Potentials with Real Eigenvalues and the Inverse Problem

International audienceThe existence of complex potentials with real eigenvalues rises twoquestions. The first one concerns the determination of the local real equivalent potentials. The second one underlines the fact that any even potential with respect to x inthe one-dimensional (D = 1) space has at least two complex partners with the samespectrum. The present paper illustrates this situation with a few examples

Observables of Complex PT-Symmetric “Shifted” Potentials

International audienceWe study complex PT-symmetric potentials, with real eigenvalues, corresponding to a complex coordinate shift (x+ic2) of a real even potential. In this case,the rules to achieve a coherent quantum mechanics are known. They allow the calculation of observables, which are found to be independent on c. This result is illustratedby few analytical or semi analytical examples. On the other hand, trying to test thisproperty numerically faces problems linked to the difficulty of finding the proper solutions of the Schrodinger equation. In particular, the large distance behaviour of the ¨wave functions generates instabilities. As an example, we have studied the (x +ic2)4potential