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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...
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Context 1
... 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. ...
Context 2
... our amended approximation (20) we could obtain a qualitatively correct shape of the wave- functions, in principle at least. Naturally, the fully reliable construction of bound states must be performed by the controlled-precision numerical integration of our ordinary differential Schrödinger equations. These results were sampled in Fig. 3 above. What is worth emphasizing is that in this setting the singularity of potential V (x) remains tractable by the standard numerical-integration software, say, of MATHEMATICA or ...
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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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