Inverse Nodal Analysis for Vector-Valued Singular Diffusion Operators

We examine a vector-valued diffusion problem given by the differential equation with boundary conditions where p(x) and q(x) denotes a continuous symmetric matrix-valued function on the interval [0,π] and k and K are real symmetric matrices of size dxd. An eigenfunction y(x) is referred to as being of type (CZ) if all isolated zeros in its components coincide with nodal points of the function itself. We demonstrate that in the two-dimensional case d=2 , the existence of infinitely many (CZ)-type eigenfunctions is both necessary and sufficient for the simultaneous diagonalizability of 〖p(x)〗_k and A,B. Consequently, if all but finitely many eigenfunctions possess the (CZ) property, then the coefficient matrix and the associated boundary data can be uniquely reconstructed up to a constant unitary transformation.