## Taiwanese Journal of Mathematics

### HALF INVERSE PROBLEMS FOR QUADRATIC PENCILS OF STURM-LIOUVILLE OPERATORS

#### Abstract

Generally, the coefficients $p(x)$ and $q(x)$ of quadratic pencils of Sturm-Liouville operators are uniquely determined by two spectra or one spectrum and norming constants. In the present paper we show if $% p(x)$ and $q(x)$ are known on half of the domain interval, then one spectrum suffices to determine them uniquely on the other half.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 5 (2012), 1829-1846.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406800

Digital Object Identifier
doi:10.11650/twjm/1500406800

Mathematical Reviews number (MathSciNet)
MR2970688

Zentralblatt MATH identifier
1256.34013

#### Citation

Yang, Chuan-Fu; Zettl, Anton. HALF INVERSE PROBLEMS FOR QUADRATIC PENCILS OF STURM-LIOUVILLE OPERATORS. Taiwanese J. Math. 16 (2012), no. 5, 1829--1846. doi:10.11650/twjm/1500406800. https://projecteuclid.org/euclid.twjm/1500406800

#### References

• S. A. Buterin and C. T. Shieh, Inverse nodal problem for differential pencils, Applied Math. Letters, 22 (2009), 1240-1247.
• J. B. Conway, Functions of One Complex Variable, 2nd ed., Vol. I, Springer-Verlag, New York, 1995.
• M. G. Gasymov and G. Sh. Guseinov, Determination of diffusion operator on spectral data, Dokl. Akad. Nauk Azerb. SSR, 37(2) (1981), 19-23.
• F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, II: The case of discrete spectrum, Trans. Amer. Math. Soc., 352(6) (2000), 2765-2787.
• R. P. Gilbert, A method of ascent for solving boundary value problems, Bull. Amer. Math. Soc., 75 (1969), 1286-1289.
• O. H. Hald, Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., XXXVII (1984), 539-577.
• H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34 (1978), 676-680.
• O. R. Hryniv and Y. V. Mykytyuk, Half-inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 20(5) (2004), 1423-1444.
• H. Koyunbakan, Inverse problem for a quadratic pencil of Sturm-Liouville operator, J. Math. Anal. Appl., 378 (2011), 549-554.
• H. Koyunbakan and E. S. Panakhov, Half-inverse problem for diffusion operators on the finite interval, J. Math. Anal. Appl., 326 (2007), 1024-1030.
• M. M. Malamud, Uniqueness questions in inverse problems for systems of differential equations on a finite interval, Trans. Moscow Math. Soc., 60 (1999), 204-262.
• V. A. Marchenko, Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977; English transl.: Birkhäser, 1986.
• O. Martinyuk and V. Pivovarchik, On the Hochstadt-Lieberman theorem, Inverse Problems, 26 (2010), 035011, (6 pages).
• W. Rundell and P. E. Sacks, Reconstruction of a radially symmetric potential from two spectral sequences, J. Math. Anal. Appl., 264 (2001), 354-381.
• L. Sakhnovich, Half inverse problems on the finite interval, Inverse Problems, 17 (2001), 527-532.
• I. Trooshin and M. Yamamoto, Hochstadt-Lieberman type theorem for a non-symmetric system of first-order ordinary differential operators, Recent development in theories & numerics: International Conference on Inverse Problems, Hong Kong, China, 9-12, January, 2002.
• C. F. Yang, New trace formulae for a quadratic pencil of the Schr$\ddot{o}$inger operator, Journal of Mathematical Physics, 51 033506, 2010.
• C. F. Yang and Y. X. Guo, Determination of a differential pencil from interior spectral data, J. Math. Anal. Appl., 375 (2011), 284-293.
• C. F. Yang, A half-inverse problem for the coefficients for a diffusion equation, Chinese Annals of Math. Ser. A, 32 (2011), 89-96.
• V. A. Yurko, Inverse Spectral Problems for Linear Differential Operators and Their Applications, Gordon and Breach, New York, 2000.
• A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, Vol. 121, American Mathematical Society, 2005.