On the isospectrality of the scalar energy-dependent Schr odinger problems
On the isospectrality of the scalar energy-dependent Schr odinger problems
In this study, we discuss the inverse spectral problem for the energy-dependent Schr odinger equation on a nite interval. We construct an isospectrality problem and obtain some relations between constants in boundary conditions of the problems that constitute isospectrality. Above all, we obtain degeneracy of K ( x;t )
___
- [1] Adamjan V, Pivovarchik V, Tretter C. On a class of non-self-adjoint quadratic matrix operator pencils arising in elasticity theory. J Operat Theory 2002; 47: 325-341.
- [2] Buterin SA, Yurko VA. Inverse spectral problem for pencils of differential operators on a nite interval. Vestn Bashkir Univ 2006; 4: 8-12.
- [3] Chelkak D, Korotyaev E. Parametrization of the isospectral set for the vector-valued Sturm{Liouville problem. J Func Anal 2006; 241: 359-373.
- [4] Chern HH. On the construction of isospectral vectorial Sturm-Liouville differential equations. arXiv preprint math/9902041 1999.
- [5] Gasymov MG, Guseinov GSh. Determination of a diffusion operator from spectral data. Dokl Akad Nauk Azerb SSSR 1981; 37: 19-23.
- [6] Gesztesy F, Simon B, Teschl G. Spectral deformations of one-dimensional Schrodinger operators. J Anal Mat 1996; 70.1: 267-324.
- [7] Gesztesy F, Simon B. Connectedness of the isospectral manifold for one-dimensional half-line Schrodinger opera- tors. J Stat Phys 2004; 116: 361-365.
- [8] Ghanbari K, Mirzaei H. On the isospectral sixth order Sturm-Liouville equation. J Lie Theory 2013; 23: 921-935.
- [9] Gohberg IC, Krein MG. Introduction to the Theory of Linear Nonselfadjoint Operators. Am Math Soc Colloq Publ: Providence, RI, USA, 1969.
- [10] Gottlieb HPW. Iso-spectral operators: some model examples with discontinuous coefficients. J Math Anal Appl 1988; 132: 123-137.
- [11] Guseinov GSh. On the spectral analysis of a quadratic pencil of Sturm-Liouville operators. Dokl Akad Nauk SSR 1985; 285: 1292-1296; English translation, Soviet Math Dokl 1985; 32: 859-862.
- [12] Guseinov GSh. Inverse spectral problems for a quadratic pencil of Sturm-Liouville operators on a nite interval. Spec Theo Oper Appl Elm, Baku, Azerbaijan, 1986; 51-101.
- [13] Guseinov GSh. On construction of a quadratic Sturm-Liouville operator pencil from spectral data. Proc Inst Math Mech Natl Acad Sci Azerb 2014; 40: 203-214.
- [14] Guseinov IM, Nabiev IM. An inverse spectral problem for pencils of differential operators. Mat Sb 2007; 198: 47-66.
- [15] Hochstadt H. The inverse Sturm-Liouville problem. Commun Pur Appl Math 1973; 26: 715-729.
- [16] Hryniv R, Pronska N. Inverse spectral problem for energy-dependent Sturm-Liouville equation. Inverse Probl 2012; 28: 085008.
- [17] Isaacson EL, McKean HP, Trubowitz E. The inverse Sturm-Liouville problem II. Commun Pur Appl Math 1984; 37: 1-11.
- [18] Jaulent M. On an inverse scattering problem with an energy- dependent potential. Ann Inst H Poincare Sect A (NS) 1972; 17: 363-378.
- [19] Jaulent M, Jean C. The inverse s -wave scattering problem for a class of potentials depending on energy. Commun Math Phys 1972; 28: 177-220.
- [20] Jaulent M, Jean C. The inverse problem for the one-dimensional Schrodinger equation with an energy-dependent potential I. Ann Inst H Poincare Sect A (NS) 1976; 25: 105-118.
- [21] Jaulent M, Jean C. The inverse problem for the one-dimensional Schrodinger equation with an energy-dependent potential II. Ann Inst H Poincare Sect A (NS) 1976; 25: 119-137.
- [22] Jodeit M, Levitan BM. The isospectrality problem for the classical Sturm-Liouville equation. Adv Differential Equ 1997; 2: 297-318.
- [23] Jodeit M, Levitan BM. A characterization of some even vector-valued Sturm-Liouville problems. J Math Phys Anal Geo 1998; 5: 166-181.
- [24] Jodeit M, Levitan BM. Isospectral vector-valued Sturm-Liouville problems. Lett Math Phys 1998; 43: 117-122.
- [25] Koyunbakan H, Panakhov ES. Half-inverse problem for diffusion operators on the nite interval. J Math Anal Appl 2007; 326: 1024-1030.
- [26] Koyunbakan H, Ylmaz E. Reconstruction of the potential function and its derivatives for the diffusion operator. Z Naturforsch A 2008; 63: 127-130.
- [27] Krall AM, Bairamov E, Cakar O. Spectrum and spectral singularities of a quadratic pencil of a Schrodinger operator with a general boundary condition. J Differ Equations 1999; 151: 252-267.
- [28] Levitan BM. On the determination of the Sturm-Liouville operator from one and two spectra. Math USSR Izv 1978; 12: 179-193.
- [29] Nabiev IM. Inverse spectral problem for diffusion operator on an interval. J Math Phys Anal Geo 2004; 11: 302-313.
- [30] Pronska N. Reconstruction of energy-dependent Sturm{Liouville equations from two spectra. Integr Equat Oper Th 2013; 76: 403-419.
- [31] Ralston J, Trubowitz E. Isospectral sets for boundary value problems on the unit interval. Ergod Theor Dyn Syst 1988; 8: 301-358.
- [32] Sat M, Panakhov ES. Spectral problem for diffusion operator. Appl Anal 2014; 93: 1178-1186.
- [33] Yamamoto M. Inverse eigenvalue problem for a vibration of a string with viscous drag. J Math Anal Appl 1990; 152: 20-34.
- [34] Yang CF, Zettl A. Half inverse problems for quadratic pencils of Sturm-Liouville operators. Taiwan J Math 2012; 16: 1829-1846.
- [35] Yurko VA. An inverse problem for differential operator pencils. Sbornik: Matem 2000; 191: 137-160; English transl, Sb Math 2000; 191: 1561-1586.