| Let H:=-Dx+A†A+lf A+lf*A†, be the Hamiltonian for a particle interacting with a harmonic oscillator, where the coupling potential f is a "localized" function of x, A† and A are creation and annihilation operators, and lambda is a parameter to be thought of as large. This operator is a simplification of the polaron Hamiltonian where the quantum field is approximated by a single mode. The large lambda corresponds to large coupling between the electron and the field. Let E0(lambda) be the smallest eigenvalue of H, and let Ep = inf &angl0;y,Hy&angr0; where the infimum is taken over product states for the electron and oscillator, the oscillator function taken as a coherent state. It is a remarkable fact that Epl: =infy∈S &angl0;y,Hy&angr0; is a "good" approximation of E 0(lambda). More specifically, (Ep(lambda) - 1) ≤ E0(lambda) ≤ Ep(lambda) for all lambda, as observed by Lieb. In this thesis we examine this gap Ep(lambda) - E 0(lambda) for various choices of f , showing cases where it closes and cases where it does not. We also provide path integral expressions for E0(lambda), and show the connection with the product state approximation. |
