Department of Mathematics, College of Arts and Sciences, University of Tokyo
抄録
We consider an eigenvalue problem (1)-(2) : (1) $\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} du(x)/dx +P(x)u(x)=\lambda u(x)$ $(0\leqq x\leqq 1 ; u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}$. (2) $u_2()+hu_1(0)=0, u_2(1)+Hu_1(1)=0. $ Here $P=\begin{pmatrix} a & b \\ (p_1) & (p_2) \end{pmatrix} \in {C^1[1,1]}^4 : $ : real-valued and h, H are real constants, and λ corresponds to an eigenvalue. We denote the set of eigenvalues of (1)-(2) by ${\lambda _n)P,h,H)}_(n\in Z)$ under an appropriate numbering. For $Q=\begin{pmatrix} a & b \\ (q_1) & (q_2) \end{pmatrix} \in {C^1[0,1]}^4$ and $h, H, H^*, J, J^* \in R\setminus {-1,1} (H\
eq H^*, J\
eq J^*)$, we obtain the following result on continuous dependence of coefficients and boundary conditions upon eigenvalues : If $\delta _0\equiv \sum _n^\infty =_(-\infty )(\mid \lambda _n(Q,h,J)-\lambda _n(P,h,H)\mid + (\mid \lambda _n(Q,h,J^*)-\lambda _n(P,h,H^*)\mid $ is sufficientrly small, then $\parallel Q-P\parallel _{[c^1[0,1]]^4}\leqq +\mid J-H\mid +\mid J^*-H^*\mid \leqq M\delta _0$ for some constant M>0. Moreover we get $\parallel Q-P\parallel _{[c^1[0,1]]^4}\leqq M$ . We show also that for given $\mu _n, \mu _n ^* \in C (n\in Z)$, there exists a unique $(Q,h,J^*)\in {C^1[0,1]}^4 \times (R\setminus {-1,1})^2$ satisfying $\lambda _n(Q, h, J)=\mu _n$ and $\mu _n(Q,h,j^*)=\mu _n^*$ under appropriate assumptions on $\mu _n, \mu _n^*(n\in Z)$ $(e.g. \sum _n^\infty =_(-\infty )(\mid \mu _n - \lambda _n(P,h,H^*)\mid)$ is sufficiently small.). We prove these results by the principle of contraction mappings and, in order to apply the principle, we establish a priori estimates of solutions to some hyperbolic systems and results on perturbation of Riesz bases.
雑誌名
Scientific papers of the College of Arts and Sciences, the University of Tokyo
Continuous Dependence Problem in an Inverse Spectral Problem for Systems of Ordinary Differential Equations of First Order
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