The Eigenvalue Properties And The Inverse Spectral Problem For Vectorial Sturm-Liouville Operator With Dirichlet Conditions

Let Q(x) be a square integrable second-ordered real symmetric matrix-valued function defined on the interval [0,1], LQ=-d2/dx2+Q(x) be a second-order Sturm-Liouville oper-ator with Dirichlet boundary condition. In this paper, we generalize the nonlinear analysis method of scalar inverse spectral problem(Poschel J, Trubowitz E[24]) to the case of the vector inverse spectral problem, and obtain the following results:(1) If Y and Z are the fundamental solutions of the eigen-equation LQy = λy, then Y and Z are continuously differentiable with respect to λ and Q, the derivatives are also given in this paper;(2)The Dirichlet spectrum of LQ is a set of monotonically increasing unbounded se-quence of real numbers, and the simple eigenvalues must appear consecutively;(3)Every eigenvalue is a compact real analytical mapping of potential function and the partial derivative function matrix is given by using the implicit function theorem;(4)The asymptotic behaviour of Dirichlet spectrum of vector Sturm-Liouville operator is given by the nonlinear analysis method, this conclusion removed the conditions that each Dirichlet eigenvalue is of multiplicity 2(Chao-Liang Shen[29]). The result in this paper is more general because we extend the potential function space to square integrable second-ordered real symmetric matrix-valued function space.(5)With two different methods of the orthogonal techniques on Hilbert space and nonlin-ear analysis techniques we prove the conclusions:if Q is even of1/2 in the interval [0, 1],i.e.Q(1 x)= Q(x), and each of the Dirichlet eigenvalues of the operator LQ is of multiplicity 2, then the Dirichlet spectrum can determine the potential function.