We consider eigenvalue problems in quantum mechanics in one dimension. Hami
ltonians contain a class of double well potential tel ms, x(6) + alphax(2),
for example. The space coordinate is continued to a complex plane and the
connection problem of fundamental system of solutions is considered. A hidd
en U-q((gl) over cap (2\1)) structure arises in "fusion relations" of Stoke
s multipliers. With this observation, we derive coupled nonlinear integral
equations which characterize the spectral properties of both +/- alpha pote
ntials simultaneously.