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A vectorial inverse nodal problem


Authors: Yan-Hsiou Cheng, Chung-Tsun Shieh and C. K. Law
Journal: Proc. Amer. Math. Soc. 133 (2005), 1475-1484
MSC (2000): Primary 34B24, 34C10
DOI: https://doi.org/10.1090/S0002-9939-04-07679-8
Published electronically: November 19, 2004
MathSciNet review: 2111948
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Abstract: Consider the vectorial Sturm-Liouville problem:

\begin{displaymath}\left\{\begin{array}{l} -{\mathbf y}''(x)+P(x){\mathbf y}(x) ... ...hbf y}(1)+I_{d}{\mathbf y}'(1)={\mathbf 0} \end{array}\right. \end{displaymath}

where $P(x)=[p_{ij}(x)]_{i,j=1}^{d}$ is a continuous symmetric matrix-valued function defined on $[0,1]$, and $A$ and $B$ are $d\times d$ real symmetric matrices. An eigenfunction ${\mathbf y}(x)$ of the above problem is said to be of type (CZ) if any isolated zero of its component is a nodal point of ${\mathbf y}(x)$. We show that when $d=2$, there are infinitely many eigenfunctions of type (CZ) if and only if $(P(x), A, B)$ are simultaneously diagonalizable. This indicates that $(P(x), A, B)$ can be reconstructed when all except a finite number of eigenfunctions are of type (CZ). The results supplement a theorem proved by Shen-Shieh (the second author) for Dirichlet boundary conditions. The proof depends on an eigenvalue estimate, which seems to be of independent interest.


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  • 1. Z.S. Agranovich and V.A. Marchenko, The Inverse Problem of Scattering Theory, Gordon and Breach, New York, 1963. MR 0162497 (28:5696)
  • 2. R. Carlson, Large eigenvalues and trace formulas for matrix Sturm-Liouville problems, SIAM J. Math. Anal. 30 (1999), No. 5, 949-962. MR 1709782 (2000f:34167)
  • 3. O.H. Hald and J.R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems 5 (1989), 307-347. MR 0999065 (90c:34015)
  • 4. Y.T. Chen, Y.H. Cheng, C.K. Law and J. Tsay, $L^{1}$convergence of the reconstruction formula for the potential function, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2319-2324. MR 1896415 (2002k:34021)
  • 5. H.H. Chern, C.K. Law, and H.J. Wang, Extensions of Ambarzumyan's theorem to general boundary conditions, J. Math. Anal. Appl. 263 (2001), no. 2, 333-342. MR 1866051 (2002g:34195)
  • 6. I.C. Gohberg and M.G. Kre1n, Theory and Applications of Volterra Operators in Hilbert Space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, 1970. MR 0264447 (41:9041)
  • 7. C.K. Law, C.L. Shen and C.F. Yang, The inverse nodal problem on the smoothness of the potential function, Inverse Problems 15 (1999), 253-263; Errata, 17(2001), 361-364. MR 1675348 (2000a:34020)
  • 8. J.R. McLaughlin, Inverse spectral theory using nodal points as data - a uniqueness result, J. Diff. Eqns. 73 (1988), 354-362. MR 0943946 (89f:34035)
  • 9. C.L. Shen, On the nodal sets of the eigenfunctions of the string equation, SIAM J. Math. Anal. 19 (1988), No. 6, 1419-1424. MR 0965261 (89j:34035)
  • 10. C.L. Shen and C.T. Shieh, Two inverse eigenvalue problems for vectorial Sturm-Liouville equations, Inverse Problems 14 (1998), no. 5, 1331-1343. MR 1654647 (99j:34016)
  • 11. C.L. Shen and C.T. Shieh, On the multiplicity of eigenvalues of a vectorial Sturm-Liouville differential equation and some related spectral problems, Proc. Amer. Math. Soc. 127 (1999), no. 10, 2943-2952. MR 1622977 (2000a:34165)
  • 12. C.L. Shen and C.T. Shieh, An inverse nodal problem for vectorial Sturm-Liouville equations, Inverse Problems 16 (2000), 349-356. MR 1766766 (2001e:34024)
  • 13. X.F. Yang, A solution of the inverse nodal problem, Inverse Problems 13 (1997), 203-213. MR 1435878 (98c:34017)

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Additional Information

Yan-Hsiou Cheng
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Republic of China
Email: jengyh@math.nsysu.edu.tw

Chung-Tsun Shieh
Affiliation: Department of Mathematics, Tamkang University, Tamsui, Taipei County, Taiwan 251, Republic of China
Email: ctshieh@math.tku.edu.tw

C. K. Law
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, Republic of China
Email: law@math.nsysu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-04-07679-8
Received by editor(s): August 27, 2003
Received by editor(s) in revised form: February 4, 2004
Published electronically: November 19, 2004
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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