The Uniqueness Theorem for Inverse Nodal Problems with a Chemical Potential

Document Type : Research Paper


Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan


In this paper, an inverse nodal problem for a second-order differential equation having a chemical potential on a finite interval is investigated. First, we estimate the nodal points and nodal lengths of differential operator. Then, we show that the potential can be uniquely determined by a dense set of nodes of the eigenfunctions.


Main Subjects

1. R. Kh. Amirov, On system of Dirac differential equations with discontinuity
conditions inside an interval, Ukrainian Math. J. 57 (2005) 712–727.
2. R. Amirov and N. Topsakal, On inverse problem for singular Sturm-Liouville
operator with discontinuity conditions, Bull. Iranian Math. Soc. 40 (2014) 585–
3. W. O. Amrein, A. M. Hinz and D. B. Pearson, Sturmī€­Liouville Theory: Past and
Present, Birkhäuser Verlag, Basel, 2005.
4. R. P. Bell, The Tunnel Effect in Chemistry, Springer-Science, University Press,
Cambridge, 1980.
5. G. Borg, Eine umkehrung der Sturm–Liouvillesehen eigenwertaufgabe, Acta Math.
78 (1945) 1–96.
6. J. P. Boyd, Sturm-Liouville eigenvalue problems with an interior pole, J. Math.
Physics 22 (1981) 1575–1590.
7. Y. H. Cheng, C. K. Law and J. Tsay, Remarks on a new inverse nodal problem, J.
Math. Anal. Appl. 248 (2000) 145–155.
8. W. Eberhard, G. Freiling and K. Wilcken-Stoeber, Indefinite eigenvalue problems
with several singular points and turning points, Math. Nachr. 229 (2001) 51–71.
9. G. Freiling and V. Yurko, On constructing differential equation with singularities
from incomplete spectral information, Inv. Prob. 14 (1998) 1131–1150.
10. G. Freiling and V. Yurko, On the determination of differential equations with
singularities and turning points, Results Math. 41 (2002) 275–290.
11. G. Freiling and V. Yurko, Inverse problems for differential operators with singular
boundary conditions, Math. Nachr. 278 (2005) 1561–1578.
12. I. M. Gelfand and B. M. Levitan, On the determination of a differential equation
from its spectral function, Amer. Math. Soc. Trans. 1 (1951) 253–304.
13. D. M. Haaland and R. T. Meyer, Reaction of exited iodine atoms with methyl iodide
rate constant determinations, Int. J. Chem. Kinetics 6 (1974) 297–308.
14. O. Hald and J. R. McLaughlin, Solutions of inverse nodal problems, Inv. Prob. 5
(1989) 307–347.
15. H. Hellmann, A new approximation method in the problem of many electrons, J.
Chem. Phys. 3 (1935) p. 61.
16. A. Jodayree Akbarfam and A. B. Mingarelli, The canonical product of the solution
of the Sturm-Liouville equation in one turning point case, Canad. Appl. Math.
Quart. 8 (2000) 305–320.
17. K. Jörgens, Spectral theory of second-order ordinary differential operators, Lecture
Notes: Series no.2, Matematisk Institut, Aarhus Universitet, 1962/63.
18. N. Levinson, The inverse Sturm–Liouville problem, Mat. Tidsskr. B. (1949) 25–30.
19. H. R. Marasi and A. Jodeyree Akbarfam, On the canonical solution of indefinite
problem with m turning points of even order, J. Math. Anal. Appl. 332 (2007)
20. V. A. Marchenko, Concerning the theory of a differential operator of the second
order, Dokl. Akad. Nauk. SSSR (N.S.) 72 (1950) 457–460.
21. J. R. McLaughlin, Inverse spectral theory using nodal points as data—a uniqueness
result, J. Diff. Eqs. 73 (1988) 354–362.
22. S. Mosazadeh, Infinite product representation of solution of indefinite Sturm-
Liouville Problem, Iranian J. Math. Chem. 4 (2013) 27–40.
23. S. Mosazadeh, The stability of the solution of an inverse spectral problem with a
singularity, Bull. Iranian Math. Soc. 41 (2015) 1061–1070.
24. J. K. Shaw, A. P. Baronavski, H. D. Ladouceur and W. O. Amrein, Applications of
the Walker method in Spectral Theory and Computational Methods of Sturm-
Liouville problems, Lecture Notes in Pure and Appl. Math. 191 (1997) 377–395.
25. C. T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for
discontinuous boundary value problems, J. Math. Anal. Appl. 347 (2008) 266–272.
26. I. Stemmler, M. Rothe, I. Hense and H. Hepach, Numerical modeling of methyl
iodide in the eastern tropical Atlantic, Biogeosciences 10 (2013) 4211–422.
27. V. Yurko, Recovering singular differential operators on noncompact star-type
graphs from Weyl functions, Tamkang J. Math. 42 (2011) 223–236.