Quantum mechanics of one-dimensional motion in a field with the singularity λ|x|−ν

Gostev, V. B.; Frenkin, A. R.

doi:10.1007/bf01886488

Cited by 5 publications

(2 citation statements)

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“…We study the zeros of g b pηq´g b pη ω q further on. (5) implies that any function ϕ k orthogonal to ϕ kω obeys g b pηq " g b pη ω q so all functions in B ω are either proportional or orthogonal to each other. By construction, B ω is maximal, because any function orthogonal to ϕ kω belongs to it; there cannot be any other eigenfunction in D ω corresponding to a bound state , so tφ e P D ω L e P S ω u Ď B ω .…”

Section: Description Of a Self-adjoint Extensionmentioning

confidence: 99%

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Incompatible Coulomb hamiltonian extensions

Abramovici¹

2020

Sci Rep

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We revisit the resolution of the one-dimensional Schrödinger hamiltonian with a Coulomb λ{|x| potential. We examine among its self-adjoint extensions those which are compatible with physical conservation laws. In the one-dimensional semi-infinite case, we show that they are classified on a U p1q circle in the attractive case and on pR,`8q in the repulsive one. In the one-dimensional infinite case, we find a specific and original classification by studying the continuity of eigenfunctions. In all cases, different extensions are incompatible one with the other. For an actual experiment with an attractive potential, the bound spectrum can be used to discriminate which extension is the correct one.

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Section: Description Of a Self-adjoint Extensionmentioning

confidence: 99%

“…The Coulomb problem addresses the non-relativistic Schrödinger equation with a 3-dimensional Coulomb potential, restricted to one dimension; it has inspired a vast corpus of scientific literature for the last seventy years [1][2][3][4][5][6][7][8][9][10][11] . Some results have been much debated.…”

Section: Introductionmentioning

confidence: 99%