2020-10-01T17:26:05Zhttps://repository.dl.itc.u-tokyo.ac.jp/?action=repository_oaipmh
oai:repository.dl.itc.u-tokyo.ac.jp:000410052020-09-23T04:50:30Z00312:07290:0730000009:00504:07292:07301
Continuous Dependence Problem in an Inverse Spectral Problem for Systems of Ordinary Differential Equations of First Orderenghttp://hdl.handle.net/2261/21192info:doi/10.15083/00040996Departmental Bulletin PaperYamamoto, MasahiroWe 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.Scientific papers of the College of Arts and Sciences, the University of Tokyo3869130198802897520AA10538733application/pdf410https://repository.dl.itc.u-tokyo.ac.jp/?action=repository_action_common_download&item_id=41005&item_no=1&attribute_id=19&file_no=1The University of Tokyo2017-06-14