Some observations on the operator H=−(1/2)d2/dx2 +m2x2/2+g/x2

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Abstract
In the present work we study the differential operator H=−(1/2)d2/dx2 +m2x2/2+g/x2. This operator known as the Hamiltonian of the quantal oscillator has been a matter of study since the beginning of quantum mechanics. Recently, it has become again actual after the paper of Calogero where the correspondent N body problem (developed in many works) is studied. Parisi and the author have used H as Hamiltonian, studying the anomalous dimensions in one‐dimensional quantum field theory. Finally, Klauder, using H as a simple degree of freedom example, has studied some qualitative features of quantum theories with singular interaction potentials. In the following work we are going to study H, showing that H is equivalent to ``half an harmonic oscillator'' for the odd and even eigenspaces separately.

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