Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval
ABSTRACT
We reduce the initial value problem for the generalized Schroedinger equation with piecewise-constant leading coefficient to the system of Volterra type integral equations and construct new useful integral representations for the fundamental solutions of the Schroedinger equation. We also investigate some significant properties of the kernels of these integral representations. The integral representations of fundamental solutions enable to obtain the basic integral equations, which are a powerful tool for solving inverse spectral problems.
We reduce the initial value problem for the generalized Schroedinger equation with piecewise-constant leading coefficient to the system of Volterra type integral equations and construct new useful integral representations for the fundamental solutions of the Schroedinger equation. We also investigate some significant properties of the kernels of these integral representations. The integral representations of fundamental solutions enable to obtain the basic integral equations, which are a powerful tool for solving inverse spectral problems.
KEYWORDS
One Dimensional Schroedinger Equation, Fundamental Solutions, Transformation Operator, Inte-gral Representation, Differential Equation with Discontinues Coefficient, Kernel of an Integral Op-erator, Integral Equation, Method of Successive Approximations
One Dimensional Schroedinger Equation, Fundamental Solutions, Transformation Operator, Inte-gral Representation, Differential Equation with Discontinues Coefficient, Kernel of an Integral Op-erator, Integral Equation, Method of Successive Approximations
Cite this paper
Nabiev, A. and Amirov, R. (2015) Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval. Advances in Pure Mathematics, 5, 777-795. doi: 10.4236/apm.2015.513072.
Nabiev, A. and Amirov, R. (2015) Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval. Advances in Pure Mathematics, 5, 777-795. doi: 10.4236/apm.2015.513072.
References
[1] McHugh, J. (1970) An Historical Survey of Ordinary Linear Differential Equations with a Large Parameter and Turning Points. Archive for History of Exact Sciences, 7, 277-324. [2] Kostyuchenko, A.G. and Shkalikov, A.A. (1983) Selfadjoint Quadratic Operator Pencils and Elliptic Problem. Functional Analysis and Its Applications, 17, 109-128.
http://dx.doi.org/10.1007/BF01083136 [3] Freiling, G. (1988) On the Completeness and Minimality of the Derived Chains of Eigen and Associated Functions of Boundary Eigenvalue Problems Nonlinearly Dependent on the Parameter. Results in Mathematics, 1464-1483. [4] Kaup, D.J. (1975) A Higher-Order Water-Wave Equation and the Method for Solving it. Progress of Theoretical Physics, 54, 396-408.
http://dx.doi.org/10.1143/PTP.54.396 [5] Cornille, H. (1970) Existence and Uniqueness of Crossing Symmetric N/D-Type Equations Corresponding to the Klein-Gordon Equation. Journal of Mathematical Physics, 11, 79-98.
http://dx.doi.org/10.1063/1.1665074 [6] Weiss, R. and Scharf, G. (1971) The Inverse Problem of Potential Scattering According to the Klein-Gordon Equation. Helvetica Physica Acta, 44, 910-929. [7] Jaulent, M. and Jean, C. (1976) The Inverse Problem for the One-Dimensional Schrödinger Equation with an Energy-Dependent Potential I, II. Ann. Inst. H. Poincare Sect. A (N.S.), 25, 105-118, 119-137. [8] Jaulent, M. (1972) On an Inverse Scattering Problem with an Energy-Dependent Potential. Ann. Inst. H. Poincare Sect. A (N.S.), 17, 363-378. [9] Marchenko, V.A. (1977) Sturm-Liouville Operators and Their Applications. Naukova Dumka, Kiev English Trans Birkhauser, Basel. [10] Chadan, K. and Sabatier, P.C. (1989) Inverse Problems in Quantum Scattering Theory, 2nd Edition, Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-3-642-83317-5 [11] Levitan, B.M. (1987) Inverse Surm-Liouville problems. VNU Science Press, Utrecht. [12] Levitan, B.M. and Gasymov, M.G. (1964) Determination of a Differential Equation by Two Spectra. Uspekhi Matematicheskikh Nauk, 192, 3-63.
http://dx.doi.org/10.1070/rm1964v019n02abeh001145 [13] Pöschel, J. and Trubowitz, E. (1987) Inverse Spectral Theory. Academic Press, New York. [14] Deift, P. and Trubowitz, E. (1979) Inverse Scattering on the Line. Communications on Pure and Applied Mathematics, 32, 121-251.
http://dx.doi.org/10.1002/cpa.3160320202 [15] Freiling, G. and Yurko, V. (2001) Inverse Sturm-Liouville Problems and Their Applications. Nova Science Publishers, Inc., Huntington. [16] McLaughlin, J.R. (1986) Analytical Methods for Recovering Coefficients in Differential Equations from Spectral Data. SIAM Review, 28, 53-57.
http://dx.doi.org/10.1137/1028003 [17] Gasymov, M.G. and Guseinov, G.S. (1981) Determination of a Diffusion Operator from Spectral Data. Akademiya Nauk Azerbaijani SSR Doklady, 37, 19-23. [18] Guseinov, G.S. (1985) On the Spectral Analysis of a Quadratic Pencil of Sturm-Liouville Operators. Doklady Akademii nauk SSSR, 285, 1292-1296. [19] Guseinov, I.M. and Nabiev, I.M. (2007) An Inverse Spectral Problem for Pencils of Differential Operators. Matematicheskii Sbornik, 198, 47-66.
http://dx.doi.org/10.1070/sm2007v198n11abeh003897 [20] Nabiev, I.M. (2004) The Inverse Spectral Problem for the Diffusion Operator on an Interval. Matematicheskaya Fizika, Analiz, Geometriya, 11, 302-313. [21] Yurko, V.A. (2000) An Inverse Problem for Pencils of Differential Operators. Sbornik: Mathematics, 191, 1561-1586.
http://dx.doi.org/10.1070/SM2000v191n10ABEH000520 [22] Hryniv, R. and Pronska, N. (2012) Inverse Spectral Problem for Energy-Dependent Sturm-Liouville Equation. Inverse Problems, 28, Article ID: 085008.
http://dx.doi.org/10.1088/0266-5611/28/8/085008 [23] Pronska, N. (2013) Reconstruction of Energy-Dependent Sturm-Liouville Operators from Two Spectra. Integral Equations and Operator Theory, 76, 403-419. [24] Sattinger, D.H. and Szmigielski, J. (1995) Energy Dependent Scattering Theory. Differential Integral Equations, 8, 945-959. [25] Tsutsumi, M. (1981) On the Inverse Scattering Problem for the One-Dimensional Schrödinger Equation with an Energy Dependent Potential. Journal of Mathematical Analysis and Applications, 83, 316-350.
http://dx.doi.org/10.1016/0022-247X(81)90266-3 [26] van der Mee, C. and Pivovarchik, V. (2001) Inverse Scattering for a Schrodinger Equation with Energy Dependent Potential. Journal of Mathematical Physics, 42, 158-181.
http://dx.doi.org/10.1063/1.1326921 [27] Aktosun, T. and van der Mee, C. (1991) Scattering and Inverse Scattering for the1-D Schrödinger Equation with Energy-Dependent Potentials. Journal of Mathematical Physics, 32, 2786-2801.
http://dx.doi.org/10.1063/1.529070 [28] Nabiev, A.A. (2006) Inverse Scattering Problem for the Schrödinger-Type Equation with a Polynomial Energy-Dependent Potential. Inverse Problems, 22, 2055-2068.
http://dx.doi.org/10.1088/0266-5611/22/6/009 [29] Nabiev, A.A. and Guseinov, I.M. (2006) On the Jost Solutions of the Schrödinger-Type Equations with a Polynomial Energy-Dependent Potential. Inverse Problems, 22, 55-67.
http://dx.doi.org/10.1088/0266-5611/22/1/004 [30] Kamimura, Y. (2007) An Inversion Formula in Energy Dependent Scattering. Journal of Integral Equations and Applications, 19, 473-512.
http://dx.doi.org/10.1216/jiea/1192628620 [31] Yurko, V. (2006) Inverse Spectral Problems for Differential Pencils on the Half-Line with Turning Points. Journal of Mathematical Analysis and Applications, 320, 439-463.
http://dx.doi.org/10.1016/j.jmaa.2005.06.085 [32] Hald, O. (1984) Discontinuous Inverse Eigenvalue Problems. Communications on Pure and Applied Mathematics, 37, 539-577.
http://dx.doi.org/10.1002/cpa.3160370502 [33] Carlson, R. (1994) An Inverse Spectral Problem for Sturm-Liouville Operators with Discontinuous Coefficients. Proceedings of the American Mathematical Society, 120, 475-484.
http://dx.doi.org/10.1090/S0002-9939-1994-1197532-5 [34] Yurko, V. (2000) Integral Transforms Connected with Discontinuous Boundary Value Problems. Integral Transforms and Special Functions, 10, 141-164.
http://dx.doi.org/10.1080/10652460008819282 [35] Yurko, V.A. (2000) On Boundary Value Problems with Discontinuity Conditions inside an Interval. Differential Equations, 36, 1266-1269.
http://dx.doi.org/10.1007/BF02754199 [36] Amirov, R.K. (2006) On Sturm-Liouville Operators with Discontinuity Conditions inside an Interval. Journal of Mathematical Analysis and Applications, 317, 163-176.
http://dx.doi.org/10.1016/j.jmaa.2005.11.042 [37] Kobayashi, M. (1989) A Uniqueness Proof for Discontinuous Inverse Sturm-Liouville Problems with Symmetric Potentials. Inverse Problems, 5, 767-781.
http://dx.doi.org/10.1088/0266-5611/5/5/007 [38] Guseinov, I.M. and Pashaev, R.T. (2002) On an Inverse Problem for a Second-Order Differential Equation. Russian Mathematical Surveys, 57, 597-598.
http://dx.doi.org/10.1070/RM2002v057n03ABEH000517 [39] Akhmedova, E.N. (2002) On Representation of Solution of Sturm-Liouville Equation with Discontinuous Coefficients. Proceedings of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 16, 208. [40] Akhmedova, E. and Huseynov, H.M. (2003) On Eigenvalues and Eigenfunctions of One Class of Sturm-Liouville Operators with Discontinuous Coefficients. Transactions of National Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, 23, 7-18. [41] Mamedov, K.R. (2010) On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition, Boundary Value Problems, 2010, Article ID: 171967.
http://dx.doi.org/10.1155/2010/171967 [42] Nabiev Adiloglu, A. and Amirov, R.K. (2013) On the Boundary Value Problem for the Sturm-Liouville Equation with the Discontinuous Coefficient. Mathematical Methods in the Applied Sciences, 36, 1685-1700.
http://dx.doi.org/10.1002/mma.2714
[1] McHugh, J. (1970) An Historical Survey of Ordinary Linear Differential Equations with a Large Parameter and Turning Points. Archive for History of Exact Sciences, 7, 277-324. [2] Kostyuchenko, A.G. and Shkalikov, A.A. (1983) Selfadjoint Quadratic Operator Pencils and Elliptic Problem. Functional Analysis and Its Applications, 17, 109-128.
http://dx.doi.org/10.1007/BF01083136 [3] Freiling, G. (1988) On the Completeness and Minimality of the Derived Chains of Eigen and Associated Functions of Boundary Eigenvalue Problems Nonlinearly Dependent on the Parameter. Results in Mathematics, 1464-1483. [4] Kaup, D.J. (1975) A Higher-Order Water-Wave Equation and the Method for Solving it. Progress of Theoretical Physics, 54, 396-408.
http://dx.doi.org/10.1143/PTP.54.396 [5] Cornille, H. (1970) Existence and Uniqueness of Crossing Symmetric N/D-Type Equations Corresponding to the Klein-Gordon Equation. Journal of Mathematical Physics, 11, 79-98.
http://dx.doi.org/10.1063/1.1665074 [6] Weiss, R. and Scharf, G. (1971) The Inverse Problem of Potential Scattering According to the Klein-Gordon Equation. Helvetica Physica Acta, 44, 910-929. [7] Jaulent, M. and Jean, C. (1976) The Inverse Problem for the One-Dimensional Schrödinger Equation with an Energy-Dependent Potential I, II. Ann. Inst. H. Poincare Sect. A (N.S.), 25, 105-118, 119-137. [8] Jaulent, M. (1972) On an Inverse Scattering Problem with an Energy-Dependent Potential. Ann. Inst. H. Poincare Sect. A (N.S.), 17, 363-378. [9] Marchenko, V.A. (1977) Sturm-Liouville Operators and Their Applications. Naukova Dumka, Kiev English Trans Birkhauser, Basel. [10] Chadan, K. and Sabatier, P.C. (1989) Inverse Problems in Quantum Scattering Theory, 2nd Edition, Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-3-642-83317-5 [11] Levitan, B.M. (1987) Inverse Surm-Liouville problems. VNU Science Press, Utrecht. [12] Levitan, B.M. and Gasymov, M.G. (1964) Determination of a Differential Equation by Two Spectra. Uspekhi Matematicheskikh Nauk, 192, 3-63.
http://dx.doi.org/10.1070/rm1964v019n02abeh001145 [13] Pöschel, J. and Trubowitz, E. (1987) Inverse Spectral Theory. Academic Press, New York. [14] Deift, P. and Trubowitz, E. (1979) Inverse Scattering on the Line. Communications on Pure and Applied Mathematics, 32, 121-251.
http://dx.doi.org/10.1002/cpa.3160320202 [15] Freiling, G. and Yurko, V. (2001) Inverse Sturm-Liouville Problems and Their Applications. Nova Science Publishers, Inc., Huntington. [16] McLaughlin, J.R. (1986) Analytical Methods for Recovering Coefficients in Differential Equations from Spectral Data. SIAM Review, 28, 53-57.
http://dx.doi.org/10.1137/1028003 [17] Gasymov, M.G. and Guseinov, G.S. (1981) Determination of a Diffusion Operator from Spectral Data. Akademiya Nauk Azerbaijani SSR Doklady, 37, 19-23. [18] Guseinov, G.S. (1985) On the Spectral Analysis of a Quadratic Pencil of Sturm-Liouville Operators. Doklady Akademii nauk SSSR, 285, 1292-1296. [19] Guseinov, I.M. and Nabiev, I.M. (2007) An Inverse Spectral Problem for Pencils of Differential Operators. Matematicheskii Sbornik, 198, 47-66.
http://dx.doi.org/10.1070/sm2007v198n11abeh003897 [20] Nabiev, I.M. (2004) The Inverse Spectral Problem for the Diffusion Operator on an Interval. Matematicheskaya Fizika, Analiz, Geometriya, 11, 302-313. [21] Yurko, V.A. (2000) An Inverse Problem for Pencils of Differential Operators. Sbornik: Mathematics, 191, 1561-1586.
http://dx.doi.org/10.1070/SM2000v191n10ABEH000520 [22] Hryniv, R. and Pronska, N. (2012) Inverse Spectral Problem for Energy-Dependent Sturm-Liouville Equation. Inverse Problems, 28, Article ID: 085008.
http://dx.doi.org/10.1088/0266-5611/28/8/085008 [23] Pronska, N. (2013) Reconstruction of Energy-Dependent Sturm-Liouville Operators from Two Spectra. Integral Equations and Operator Theory, 76, 403-419. [24] Sattinger, D.H. and Szmigielski, J. (1995) Energy Dependent Scattering Theory. Differential Integral Equations, 8, 945-959. [25] Tsutsumi, M. (1981) On the Inverse Scattering Problem for the One-Dimensional Schrödinger Equation with an Energy Dependent Potential. Journal of Mathematical Analysis and Applications, 83, 316-350.
http://dx.doi.org/10.1016/0022-247X(81)90266-3 [26] van der Mee, C. and Pivovarchik, V. (2001) Inverse Scattering for a Schrodinger Equation with Energy Dependent Potential. Journal of Mathematical Physics, 42, 158-181.
http://dx.doi.org/10.1063/1.1326921 [27] Aktosun, T. and van der Mee, C. (1991) Scattering and Inverse Scattering for the1-D Schrödinger Equation with Energy-Dependent Potentials. Journal of Mathematical Physics, 32, 2786-2801.
http://dx.doi.org/10.1063/1.529070 [28] Nabiev, A.A. (2006) Inverse Scattering Problem for the Schrödinger-Type Equation with a Polynomial Energy-Dependent Potential. Inverse Problems, 22, 2055-2068.
http://dx.doi.org/10.1088/0266-5611/22/6/009 [29] Nabiev, A.A. and Guseinov, I.M. (2006) On the Jost Solutions of the Schrödinger-Type Equations with a Polynomial Energy-Dependent Potential. Inverse Problems, 22, 55-67.
http://dx.doi.org/10.1088/0266-5611/22/1/004 [30] Kamimura, Y. (2007) An Inversion Formula in Energy Dependent Scattering. Journal of Integral Equations and Applications, 19, 473-512.
http://dx.doi.org/10.1216/jiea/1192628620 [31] Yurko, V. (2006) Inverse Spectral Problems for Differential Pencils on the Half-Line with Turning Points. Journal of Mathematical Analysis and Applications, 320, 439-463.
http://dx.doi.org/10.1016/j.jmaa.2005.06.085 [32] Hald, O. (1984) Discontinuous Inverse Eigenvalue Problems. Communications on Pure and Applied Mathematics, 37, 539-577.
http://dx.doi.org/10.1002/cpa.3160370502 [33] Carlson, R. (1994) An Inverse Spectral Problem for Sturm-Liouville Operators with Discontinuous Coefficients. Proceedings of the American Mathematical Society, 120, 475-484.
http://dx.doi.org/10.1090/S0002-9939-1994-1197532-5 [34] Yurko, V. (2000) Integral Transforms Connected with Discontinuous Boundary Value Problems. Integral Transforms and Special Functions, 10, 141-164.
http://dx.doi.org/10.1080/10652460008819282 [35] Yurko, V.A. (2000) On Boundary Value Problems with Discontinuity Conditions inside an Interval. Differential Equations, 36, 1266-1269.
http://dx.doi.org/10.1007/BF02754199 [36] Amirov, R.K. (2006) On Sturm-Liouville Operators with Discontinuity Conditions inside an Interval. Journal of Mathematical Analysis and Applications, 317, 163-176.
http://dx.doi.org/10.1016/j.jmaa.2005.11.042 [37] Kobayashi, M. (1989) A Uniqueness Proof for Discontinuous Inverse Sturm-Liouville Problems with Symmetric Potentials. Inverse Problems, 5, 767-781.
http://dx.doi.org/10.1088/0266-5611/5/5/007 [38] Guseinov, I.M. and Pashaev, R.T. (2002) On an Inverse Problem for a Second-Order Differential Equation. Russian Mathematical Surveys, 57, 597-598.
http://dx.doi.org/10.1070/RM2002v057n03ABEH000517 [39] Akhmedova, E.N. (2002) On Representation of Solution of Sturm-Liouville Equation with Discontinuous Coefficients. Proceedings of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 16, 208. [40] Akhmedova, E. and Huseynov, H.M. (2003) On Eigenvalues and Eigenfunctions of One Class of Sturm-Liouville Operators with Discontinuous Coefficients. Transactions of National Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, 23, 7-18. [41] Mamedov, K.R. (2010) On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition, Boundary Value Problems, 2010, Article ID: 171967.
http://dx.doi.org/10.1155/2010/171967 [42] Nabiev Adiloglu, A. and Amirov, R.K. (2013) On the Boundary Value Problem for the Sturm-Liouville Equation with the Discontinuous Coefficient. Mathematical Methods in the Applied Sciences, 36, 1685-1700.
http://dx.doi.org/10.1002/mma.2714