On the matrix spectral function of a generalized second-order differentialoperator in a ramified space

The existence of a unique n x n matrix spectral function is shown for a sel fadjoint operator A in a Hilbert space L-alpha(2)(m). This Hilbert space is a subspace of the product of spaces L-2(m(i)) with measures m(i), i = 1,.. ., n, having support in [0, infinity). The inner product in L-alpha(2)(m) i s the weighted sum of the inner products in the L-2(m(i)), i.e., (f, g)(m,a lpha) = Sigma alpha(i)(f(i), g(i))m(i), f = (f(1),...,f(n)), g = (g(1),..., g(n)) epsilon L-alpha(2)(m), with positive constants alpha(i), i = i,..., n . The operator A is given by (Af)(i) = -D-mi D-x(+) f(i) with generalized s econd order derivatives DmiDx+. The elements of the domain of A have contin uous representatives satisfying f(i)(0) = f(i)(0), i, j = 1,..., n, and an additional gluing condition at 0.