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The shapes of the perturbed square-well potentials (8) at g = 1/4 (the lowest, light spike), g = 1/2 (the intermediate spike) and g = 1 (the upper spike). Horizontal lines mark the first two energy levels for g = 0. Attached to them, the picture also displays the shapes of the unperturbed wave functions.
Source publication
Singular repulsive barrier V (x) = −gln(|x|) inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction ℒeff(x) = −gln[ψ∗(x)ψ(x)] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small g or after a...
Contexts in source publication
Context 1
... spike-shaped logarithmic barrier is unbounded, repulsive and centrally symmetric here -cf. Fig. 1 where the shape of the potential is displayed at g = 0.25, 0.5 and 1. It is worth adding that although we are choosing here (i.e., in Eq. (6)) the first nontrivial nodal number N = 1 for the sake of simplicity, Eq. (9) describes all of the bound states generated by the linearized toy-model interaction (8). These states are numbered by a different, lower-case index n. Naturally, one would have to set, for the sake of consistency, n = N at the end of the analysis and, in principle at least, before an ultimate return to the initial nonlinear-equation ...
Context 2
... long as these bound states are determined by the ordinary differential Schrödinger Eq. (9), there exists a number of methods of their construction. The choice of the method may be inspired by the inspection of Figs. 1 and 2. This indicates that the influence of our singular logarithmic potential (8) is felt, first of all, by the low-lying bound states and/or in the strong-coupling regime. Directly, this may be demonstrated by the routine numerical construction of the wavefunctions (sampled in Fig. 3) and by the routine numerical evaluation of the energies (sampled in Table 2). In both cases, due attention must be paid to the singular nature of our potential (8) in the origin. This is a challenging aspect of the numerical calculations which will be discussed and illustrated by some examples in what follows. ...
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Citations
... Finally, one could mention that potentials of type (24) were studied, albeit in the context of a linear Schrödinger equation, in Refs. [53,54]. ...
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