Multiparameter spectral theory for weakly coupled operator system with unbounded operators by using left definite condition on a Hilbert tensor product space (Q2888267)

Let \(H_d^k\) be a direct sum of the separable Hilbert spaces \(H_j^k\), \(j = 1, 2, \dots, n\), and \(H = \bigotimes_{k=1}^n H_d^k\) be their Hilbert tensor product space. The eigenvalue problem of the weakly coupled operator system \(A_ku_k = \sum_{j=1}^n \lambda_jC_{kj}u_k\), \(k = 1, 2, \dots, n\), is studied under some elliptic condition in this article, with \(C_{kj}:H_d^k \rightarrow H_d^k\) being Hermitian, \(A_k: D(A_k) \subset H_d^k \rightarrow H_d^k\) being self-adjoint and unbounded, \(u_k\) being \(n \times 1\) column vectors, \(i,j,k = 1, 2, \dots, n\). The results is applied to an \(n \times n\), system of second order differential equations.