  We introduce a general method for transforming the equations of motion
following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into
a boundary value problem in one-dimensional quantum mechanics. For the
particular case of a one-dimensional chain of interacting NxN Hermitean
matrices, the corresponding large N boundary value problem is mapped into a
linear Fredholm equation with Hilbert-Schmidt type kernel. The equivalence of
this kernel, in special cases, to a second order differential operator allows
us recover all previously known explicit solutions for the matrix eigenvalues.
In the general case, the distribution of eigenvalues is formally derived
through a series of saddle-point approximations. The critical behaviour of the
system, including a previously observed Kosterlitz-Thouless transition, is
interpreted in terms of the stationary points. In particular we show that a
previously conjectured infinite series of sub-leading critical points are due
to expansion about unstable stationary points and consequently not realized.
