Class of Nonanalytic Perturbations in Quantum Mechanics

Abstract

It is shown that the resolvent operator for the Hamiltonian which is the sum of the harmonic oscillator Hamiltonian H₀ and a polynomial perturbation gP(x) of degree exceeding two (g is a coupling constant) is not expressible as a convergent power series in g. The source of this nonanalyticity is the failure of the anharmonic perturbation operator to be small in norm compared to H₀. The nature of the singularity at g = 0 is conjectured. The result makes clear that the divergence of the Ward‐Hurst‐Thirring model has nothing to do with the ``difficulties'' of field theory which are related to the infinite number of degrees of freedom of a field.