Exactly solvable model of a particle interacting with a field: The origin of a quantum-mechanical divergence

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Abstract
The dynamically exactly solvable system of a local oscillator coupled harmonically to a finitely extended one-dimensional string (continuum transmission line field) is studied in detail. An earlier recognized quantum-mechanical logarithmic ultraviolet divergence in the oscillator’s kinetic fluctuations is shown to originate in the infinitely strong (rigid) coupling limit. For any finite coupling strength the fluctuations are finite. Inter alia attention is given to such aspects as mode stability, nonorthogonality of the eigenfunctions, the essential singular nature of the eigenmatrix, completeness relations, the initial-value problem, infinite-system limit, quantum exponential decay, coherent-state evolution, and the thermal string model.