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The coupling-dependence of the low-lying spectrum in the Rayleigh-Schrödinger first order (i.e., linear-extrapolation) approximation.
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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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... the logarithmically singular but still positive-definite barrier the unperturbed spectrum is being pushed upwards. Still, due to the immanent weakness of the singularity of the logarithmic type even in the strong-coupling dynamical regime with g ≫ 1, the expected effect of the quasi- degeneracy will get quickly suppressed with the growth of n, i.e., of the excitation. In contrast to the stronger and more common (e.g., power-law) models of the repulsion in the origin, this will make the real influence of the logarithmic barrier restricted to the low-lying spectrum. Table 1: The first ten numerical coefficients E [1] n in Eq. (10). Fig. 2 may be recalled for an explicit quantitative illustration of the latter expectation. Using just the most elementary leading-order-approximation estimates we see there that while the pre- diction of the quasi-degeneracy between the ground (i.e., n = 0) and the first excited (i.e., n = 1) state might still occur near the reasonably small value of coupling g [1] 0,1 = 4.540138798, the next analogous crossing of the first-order energies E [1] 2 and E [1] 3 only takes place near the estimate as large as g [1] 2,3 = 29.13203044, etc. The same Fig. 2 also shows that another first-order crossing may be detected for E [1] 0 and E [1] 2 , emerging even earlier (i.e., at g [1] 0,2 = 23.96744320) and being, obviously, spurious. In other words, for the prediction of the quasidegeneracy in the strong-coupling dynamical regime the knowledge of the mere first-order perturbation corrections must be declared ...
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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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