Abstract
Here we wish to study the exact eigenvalues of a family of one-dimensional, positive, symmetric potentials with two turning points, which includes the simple harmonic oscillator. The eigenvalues are shown to depend on branch point singularities in the coupling constant by both Bohr–Sommerfeld and canonical quantization procedures, thus ruling out convergent power series expansions in the coupling constant and severely restricting the analytic continuation of the coupling constant g in the complex plane. In particular, analytic continuation of the eigenvalues from positive to negative values in the usual inertial Hilbert space is not possible because of the branch point singularities as Dyson argued long ago.
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