Supersymmetric embedding of a model of a quantum harmonic oscillator interacting with infinitely many bosons

A mathematically rigorous analysis is given on supersymmetric embedding of a model of a one-dimensional quantum harmonic oscillator interacting with infinitely many bosons moving in the s-dimensional space R/sup s/. The model is exactly soluble. By a rigorous explicit construction of supersymmetric quantum field theories, it was proven that the Hamiltonian of the model is supersymmetrically embeddable if it is non-negative. The index of the Dirac--Kaehler-type operators associated with the supercharges of the supersymmetric quantum field theories is computed