The inverse problem for pencils of differential operators on the half-line with turning points

V. A. Yurko

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 237-246.

Résumé

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We give a formulation of the inverse problem, establish properties of the spectral characteristics, and prove the uniqueness theorem for the solution of the inverse problem.

Bibliographie

[1] Gasymov M. G., Guseinov G. Sh., “Opredelenie operatora diffuzii po spektralnym dannym”, DAN Azerb. SSR, 37:2 (1981), 19–23 | MR | Zbl

[2] Keldysh M. V., “O sobstvennykh znacheniyakh i sobstvennykh funktsiyakh nekotorykh klassov nesamosopryazhennykh uravnenii”, DAN SSSR, 77 (1951), 11–14 | Zbl

[3] Kostyuchenko A. G., Shkalikov A. A., “Samosopryazhennye kvadratichnye puchki operatorov i ellipticheskie zadachi”, Funkts. analiz i ego pril., 17:2 (1983), 38–61 | MR | Zbl

[4] Levitan B. M., Obratnye zadachi Shturma–Liuvillya, Nauka, M., 1984 | MR

[5] Levitan B. M., Sargsyan I. S., Vvedenie v spektralnuyu teoriyu, Nauka, M., 1970 | MR | Zbl

[6] Marchenko V. A., Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR

[7] markin Ya. D., O nekotorykh obschikh zadachakh teorii obyknovennykh lineinykh differentsialnykh uravnenii, Petrograd, 1917

[8] Yurko V. A., “O kraevykh zadachakh s parametrom v kraevykh usloviyakh”, Izvestiya AN Arm. SSR. Ser. matem., 19:5 (1984), 398–409 | MR | Zbl

[9] Yurko V. A., “Obratnaya zadacha dlya sistem differentsialnykh uravnenii s nelineinoi zavisimostyu ot spektralnogo parametra”, Differents. uravn., 33:3 (1997), 390–395 | MR | Zbl

[10] Yurko V. A., “Obratnaya zadacha dlya puchkov differentsialnykh operatorov”, Mat. sb., 191:10 (2000), 137–160 | MR | Zbl

[11] Conway J. B., Functions of One Complex Variable, vol. I. 2nd ed., Springer, New York, 1995 | MR

[12] Daho K., Langer H., “Sturm–Liouville operators with an indefinite weight functions”, Proc. Roy. Soc. Edinburgh Sect. A, 78, 1977, 161–191 | MR | Zbl

[13] Eberhard W., Freiling G., “An expansion theorem for eigenvalue problems with several turning points”, Analysis, 13 (1993), 301–308 | MR | Zbl

[14] Freiling G., “On the completeness and minimality of the derived chains of eigen and associated functions of boundary eigenvalue problems nonlinearly dependent on the parameter”, Results Math., 14 (1988), 64–83 | MR | Zbl

[15] Freiling G., Yurko V. A., “Inverse problems for differential equations with turning points”, Inverse Problems, 13 (1997), 1247–1263 | DOI | MR | Zbl

[16] Freiling G., Yurko V. A., “Inverse spectral problems for differential equations on the half-line with turning points”, J. Differential Equations, 154 (1999), 419–453 | DOI | MR | Zbl

[17] Freiling G., Yurko V. A., Inverse Sturm–Liouville Problems and Their Applications., NOVA Science Publishers, New York, 2001 | MR | Zbl

[18] McHugh J., “An historical survey of ordinary linear differential equations with a large parameter and turning points”, Arch. History Exact Sci., 7 (1970), 277–324 | DOI | MR

[19] Mennicken R., Moeller M., Non-Self-Adjoint Boundary Eigenvalue Problems, Elsevier, Amsterdam, 2003

[20] Regge T., “Construction of potentials from resonance parameters”, Nuovo Cimento; X. Ser, 9, 1958, 491–503

[21] Rykhlov V. S., “Asymptotical formulas for solutions of linear differential systems of the first order”, Results Math., 36:3–4 (1999), 342–353 | MR | Zbl

[22] Wasow W., Linear Turning Point Theory, Springer, Berlin, 1985 | MR | Zbl

[23] Yamamoto M., “Inverse eigenvalue problem for a vibration of a string with viscous drag”, J. Math. Anal. Appl., 152:1 (1990), 20–34 | DOI | MR | Zbl

[24] Yurko V. A., Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach, Amsterdam, 2000 | MR | Zbl

[25] Yurko V. A., Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 2002 | MR | Zbl