本論文的目的在研究微分方程的反問題。論文的第一部分，作者討
論向量型的微分束算子,當 P(x)為對角矩陣時,可以找到 Weyl 矩陣
來唯一決定 P(x), Q(x), 和邊界條件。第二部分則是利用史特姆劉維方程式的固有函數的內結點集和一些條件,來唯一決定位勢函
數ｑ(x) 。

The main focus of the thesis is about Inverse Problems on differential pencils and
Sturm-Liouville equations. In first part of the thesis, the author discusses vectorial
differential pencil. When P(x) is diagonal matrix, Weyl’s Matrix can be used to uniquely
determine P(x), Q(x) and boundary conditions. In second part of the thesis, the author uses
Sturm-Liouville equations’ interior nodal subset and certain conditions to uniquely
determine q(x).

Contents i
1 Introduction 1
2 Spectral Theory for Vectorial Differential pencils 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Weyl’s matrix and a uniqueness theorem . . . . . . . . . . . . . . . . 8
3 Inverse nodal problems 14
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Inverse problems for the boundary value problem with the interior
nodal subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 An improved inverse nodal problem . . . . . . . . . . . . . . . . . . 25
References 32

[1] V.A. Ambarzumian, Uber eine frage der eigenwerttheori ¨ , Zeitschrift f¨ur Physik
53, 690–695 (1929).
[2] G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta
Math., 78, pp. 1–96 (1946).
[3] S.A. Buterin, C.T. Shieh, Inverse nodal problem for differential pencils, Appl.
Math. Lett. 22, 1240–1247 (2009).
[4] Y.H. Cheng, C.K. Law and Jhishen Tsay, Remarks on a New Inverse Nodal
Problem, J. Math. Anal. Appl. 248, 145–155 (2000).
[5] H.H. Chern, C.K. Law, and H.J. Wang, Extension of Ambarzumyan’s theorem
to general boundary conditions, J. Mathematical Analysis and Applications,
263: 333–342, 2001. Corrigendum, 309: 764–768, 2005.
[6] A. Chernozhukova, G. Freiling, A uniqueness theorem for the boundary value
problems with non-linear dependence on the spectral parameter in the boundary
conditions, Inverse Probl. Sci. Eng. 17, 777–785 (2009).
[7] G. Freiling, V.A. Yurko, Inverse problems for Sturm-Liouville equations with
boundary conditions polynomially dependent on the spectral parameter, Inverse
Prob. 26, 055003 (2010).
[8] F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on
the potential, II. The case of discrete spectrum, Trans. Am. Math. Soc. 352,
2765–2787 (2000).
[9] Y.X. Guo, G.S. Wei, Inverse problems: dense nodal subset on an interior subinterval,
J. Differ. Equ. 255, 2002–2017 (2013).
[10] H. Koyunbakan, A new inverse problem for the diffusion operator, Appl. Math.
Lett., 19, pp. 995–999 (2006).
[11] J.R. McLaughlin, Inverse spectral theory using nodal points as data-a uniqueness
result, J. Differential Equations 73, 354–362 (1988).
32
[12] C.L. Shen, On the nodal sets of the eigenfunctions of the string equations, SIAM
J. Math. Anal. 19, 1419–1424 (1988).
[13] C.L. Shen, Some eigenvalue problems for the vectorial Hill’s equation, Inverse
Problems 16, No.3, 749–783 (2000).
[14] C.T. Shieh, V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous
boundary value problems, J. Math. Anal. Appl. 347, 266–272 (2008).
[15] C.F. Yang, Solution to open problems of Yang concerning inverse nodal problems,
Isr. J. Math. 204, 283–298 (2014).
[16] X.F. Yang, A new inverse nodal problem, J. Differential Equations 169, 633–653
(2001).
[17] C.F. Yang, Z.Y. Huang, X.P. Yang, Ambarzumians theorems for vectorial
Sturm-Liouville systems with coupled boundary conditions, Taiwan. J. Math.
14 (4), 1429–1437.
[18] C.F. Yang, X.P. Yang, Inverse nodal problems for the Sturm-Liouville equation
with polynomially dependent on the eigenparameter, Inverse Prob. Sci. Eng. 19,
951–961 (2011).
[19] V.A. Yurko, Inverse problems for the matrix Sturm-Liouville equation on a
finite interval, Inverse Problems 22, 1139–1149 (2006).
[20] V.A. Yurko, Inverse nodal problems for Sturm-Liouville operators on star-type
graphs, Journal of Inverse and Ill-Posed Problems 16, 715–722 (2008).
[21] V.A. Yurko, On Ambarzumian-type theorems, Appl. Math. Lett. 26, 506–509
(2013).
[22] Y.P. Wang, Uniqueness Theorems for Sturm-Liouville Operators with Boundary
Conditions Polynomially Dependent on the Eigenparameter from Spectral Data,
Results in Mathematics 63, 3-4, 1131–1144 (2013).
[23] Y.P. Wang, Inverse problems for a class of Sturm-Liouville operators with the
mixed spectral data, Operators and Matrices, Volume 11, Number 1, 89–99
(2017).
33
[24] Y.P. Wang, C.T. Shieh and H.Y. Miao, Inverse transmission eigenvalue problems
with the twin-dense nodal subset, J. Inverse Ill-Posed Probl. 25 (2), 237–249
(2017).
[25] Y.P. Wang and V.A. Yurko, On the inverse nodal problems for discontinuous
Sturm-Liouville operators, J. Differential Equations 260, 4086–4109 (2016).