  The vacuum expectation values of the so-called Q-operators of certain
integrable quantum field theories have recently been identified with spectral
determinants of particular Schrodinger operators. In this paper we extend the
correspondence to the T-operators, finding that their vacuum expectation values
also have an interpretation as spectral determinants. As byproducts we give a
simple proof of an earlier conjecture of ours, proved by another route by
Suzuki, and generalise a problem in PT symmetric quantum mechanics studied by
Bender and Boettcher. We also stress that the mapping between Q-operators and
Schrodinger equations means that certain problems in integrable quantum field
theory are related to the study of Regge poles in non-relativistic potential
scattering.
