Quantum Tunneling and Caustics under Inverse Square Potential

Abstract

Quantization of a harmonic oscillator with inverse square potential $V(x)=(m{\omega }^2/2)x^2+g/x^2$ on the line $-\infty <x<\infty$ is re-examined. It is shown that, for $0<g<3{\hslash }^2/(8m)$, the system admits a U(2) family of inequivalent quantizations allowing for quantum tunneling through the potenatial barrier at $x=0$. In the family is a distinguished quantization which reduces smoothly to the harmonic oscillator as $g\to 0$, in contrast to the conventional quantization applied to the Calogero model which prohibits the tunneling and has no such limit. The tunneling renders the classical caustics anomalous at the quantum level, leading to the possibility of copying an arbitrary state from one side $x>0$, say, to the other $x<0$.

13 pages, 1 figure, written by PlainTeX, Ver.4.0. Title changed, typos corrected, a new reference and a comment on quantum cloning added

References in corpus (3)

• Near-Horizon Conformal Structure of Black Holes
• Moebius Structure of the Spectral Space of Schroedinger Operators with Point Interaction
• Black holes and conformal mechanics

Cited by in corpus (10)

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• Supersymmetry in quantum mechanics with point interactions
• Comment on "Supersymmetry and Singular Potentials" by Das and Pernice [Nucl. Phys. B 561 (1999) 357]
• Quantum Abacus
• Brane-Antibrane Systems at Finite Temperature and Phase Transition near the Hagedorn Temperature
• On the spectrum-generating superalgebras of the deformed one-dimensional quantum oscillators
• A three-dimensional superconformal quantum mechanics with $sl(2|1)$ dynamical symmetry
• Higher Spin Symmetries and Deformed Schrödinger Algebra in Conformal Mechanics
• Quantum tunneling of a singular potential