On the Matrix Spectral Function of a Generalized Second‐Order Differential Operator in a Ramified Space

The existence of a unique n × n matrix spectral function is shown for a selfadjoint operator A in a Hilbert space L 2 α (m). This Hilbert space is a subspace of the product of spaces L 2 (m i ) with measures m 1 , i = 1, n , having support in [0,∞]. The inner product in L 2 α (m) is the weighted sum of the inner products in the L 2 ( m i ), i.e., ( f, g ) m ,α =∑α i ( f i ,g i ) m i , f =( f 1 ,… f n ), g =( g 1 ,… g n )∈ L α 2 (m), with positive constants α i = 1, …, n . The operator A is given by (A f ) i = − D m i D x + f , with generalized second order derivatives D mi D + x . The elements of the domain of A have continuous representatives satisfying f 1 (0) f j (0), i, j = 1, n , and an additional gluing condition at 0.