Spectral Characterizations For Vectorial Sturm-Liouville Operators With Eigenparameter Boundary Conditions And Their Inverse Spectral Problems

In this paper, we discuss the multiplicities of the eigenvalues and the inverse spectral problems of the two-dimensional vectorial Sturm-Liouville operators where y(x)= col(y1(x),y2(x)), I2 is a unitary matrix of 2 x 2, A, B, C,D and H are scalar matrices of 2 x 2, the 2 x 2 matrix Q(x) is nonnegative definite in the interval [0, n]. Vectorial Sturm-Liouville equations can be used to describe propagation of seismic and electromag-netic waves, and play a role in the quantum-mechanics. Firstly, this paper shows that all eigenvalues of the operators L(Q;A, B) and L(Q;H, C, D) are real, and the algebraic multi-plicity (the order as a zero of the characteristic function) is equal to its geometric multiplic-ity (the dimension of eigen-subspace), respectively; secondly, by the theory of Hadamard’s factorization, we also prove that the characteristic function ωA,B(λ, Q)(ωH,C,D(λ,Q)) of the operator L(Q;A,B)(L(Q;H,C,D)) is uniquely determined by its spectrum (including their multiplicities), respetively; finally, we consider the inverse spectral problems of the opera-tors L(Q; A, B) and L(Q; H, C, D), and show that the potential function Q(x) can be uniquely determined by three spectrum from different boundary conditions.