Negatif Yoğunluk Fonksiyonuna Sahip Kendine Eşlenik Olmayan Schrödinger Operatörü Üzerine Bir Çalışma

Yıl 2023, Cilt: 28 Sayı: 1, 220 - 229, 30.04.2023

Nimet Coskun

https://doi.org/10.53433/yyufbed.1139044

Öz

Bu çalışmada kendine eşlenik olmayan, singüler ve standard dışı bir ağırlık fonksiyonuyla birlikte tanımlanmış operatörün spektral özellikleri ele alınacaktır. Bir boyutlu, zamana bağımlı Schrödinger tipli diferansiyel denklem
-y^''+q(x)y=μ^2 ρ(x)y,x∈[0,∞),
y(0)=0,
başlangıç koşulu ve tamamen negatif olarak tanımlı
ρ(x)=-1,
yoğunluk fonksiyonuyla birlikte göz önüne alınsın. Pozitif değerli sürekli ve süreksiz yoğunluk fonksiyonuna sahip operatörler için literatürde çok sayıda çalışma bulunmaktadır. Yoğunluk fonksiyonunun yapısı operatörün analitik özelliklerini ve çözümlerin gösterimini etkilemektedir. Klasik literatürden farklı olarak, bu çalışmada hiperbolik tipli temel çözümler operatörün spektrumunu belirlemek için kullanılmıştır. Buna ek olarak, özdeğerlerin ve spektral tekilliklerin sonluluğu için gerekli koşullar elde edilmiştir. Böylece, Naimark ve Pavlov koşulları, negatif yoğunluk fonksiyonuna sahip operatör durumunda çözülmüştür.

Anahtar Kelimeler

Negatif yoğunluk fonksiyonu, Spektral analiz, Spektral tekillikler

A Study on the Non-selfadjoint Schrödinger Operator with Negative Density Function

Öz

This study focuses on the spectral features of the non-selfadjoint singular operator with an out-of-the-ordinary type weight function. Take into consideration the one-dimensional time-dependent Schrödinger type differential equation
-y^''+q(x)y=μ^2 ρ(x)y,x∈[0,∞),
holding the initial condition
y(0)=0,
and the density function defined with a completely negative value as
ρ(x)=-1.
There is an enormous number of the papers considering the positive values of ρ(x) for both continuous and discontinuous cases. The structure of the density function affects the analytical properties and representations of the solutions of the equation. Unlike the classical literature, we use the hyperbolic type representations of the equation’s fundamental solutions to obtain the operator’s spectrum. Additionally, the requirements for finiteness of eigenvalues and spectral singularities are addressed. Hence, Naimark’s and Pavlov’s conditions are adopted for the negative density function case.

Anahtar Kelimeler

Negative density function, Spectral analysis, Spectral singularities

Kaynakça

• Adıvar, M., & Akbulut, A. (2010). Non-self-adjoint boundary-value problem with discontinuous density function. Mathematical Methods in the Applied Sciences, 33(11), 1306-1316. doi:10.1002/mma.1247
• Amrein, W. O., Hinz, A. M., & Pearson, D. B. (2005). Sturm-Liouville Theory: Past and Present. Basel; Boston, USA: Birkhäuser. doi:10.1007/3-7643-7359-8
• Bairamov, E., Cakar, Ö. & Krall, A. M. (1999). An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities. Journal of Differential Equations, 151(2) 268-289. doi:10.1006/jdeq.1998.3518
• Bairamov, E., Cakar, Ö. & Krall, A. M. (2001). Non-selfadjoint difference operators and Jacobi matrices with spectral singularities. Mathematische Nachrichten, 229(1), 5-14. doi:10.1002/1522-2616(200109)229:1%3C5::AID-MANA5%3E3.0.CO;2-C
• Bairamov, E., Aygar, Y., & Olgun, M. (2010). Jost solution and the spectrum of the discrete Dirac systems. Boundary Value Problems, 2010, 1-11. doi:10.1155/2010/306571
• Bairamov, E., Erdal, I., & Yardimci, S. (2018). Spectral properties of an impulsive Sturm–Liouville operator. Journal of Inequalities and Applications, 2018(1), 1-16. doi:10.1186/s13660-018-1781-0
• Chadan, K., & Sabatier, P. C. (1977). Inverse Problems in Quantum Scattering Theory. New York, USA: Springer-Verlag, New York Inc. doi:10.1007/978-3-662-12125-2
• Darwish, A. A. (1993). On a non-self adjoint singuluar boundary value problem. Kyungpook Mathematical Journal, 33(1), 1-11.
• Dolzhenko, E. P. (1979). Boundary value uniqueness theorems for analytic functions. Mathematical notes of the Academy of Sciences of the USSR, 25, 437-442. doi:10.1007/BF01230985
• El-Raheem, Z. F., & Nasser, A. H. (2014). On the spectral investigation of the scattering problem for some version of one-dimensional Schrödinger equation with turning point. Boundary Value Problems, 2014(1), 1-12. doi:10.1186/1687-2770-2014-97
• El-Raheem, Z. F., & Salama, F. A. (2015). The inverse scattering problem of some Schrödinger type equation with turning point. Boundary Value Problems, 2015(1), 1-15. doi:10.1186/s13661-015-0316-6
• Gasymov, M. G., & El-Reheem, Z. F. A. (1993). On the theory of inverse Sturm-Liouville problems with discontinuous sign-alternating weight. Doklady Akademii Nauk Azerbaidzana, 48(50), 13-16.
• Guseinov, I. M. O., & Pashaev, R. T. O. (2002). On an inverse problem for a second-order differential equation. Russian Mathematical Surveys, 57(3), 597. doi:10.1070/RM2002v057n03ABEH 000517
• Koprubasi, T., & Yokus, N. (2014). Quadratic eigenparameter dependent discrete Sturm–Liouville equations with spectral singularities. Applied Mathematics and Computation, 244, 57-62. doi:10.1016/j.amc.2014.06.072
• Koprubasi, T. (2021). A study of impulsive discrete Dirac system with hyperbolic eigenparameter, Turkish Journal of Mathematics, 45(1), 540-548. doi:10.3906/mat-2010-29
• Koprubasi, T., & Aygar Küçükevcilioğlu, Y. (2022). Discrete impulsive Sturm-Liouville equation with hyperbolic eigenparameter. Turkish Journal of Mathematics, 46(2), 377-396. doi:10.3906/mat-2104-97
• Levitan, B. M. (1987). Inverse Sturm-Liouville Problems. Berlin, Germany; Boston, USA: Walter de Gruyter GmbH & Co KG. doi:10.1515/9783110941937
• Lyantse, V. E. (1968). The spectrum and resolvent of a non-selfadjoint difference operator. Ukrainian Mathematical Journal, 20, 422-434. doi:10.1007/BF01085212
• Mamedov, K. (2010). On an inverse scattering problem for a discontinuous Sturm-Liouville equation with a spectral parameter in the boundary condition. Boundary Value Problems, 2010, 1-17. doi:10.1155/2010/171967
• Mamedov, K. R., & Cetinkaya, F. A. (2015). Boundary value problem for a Sturm-Liouville operator with piecewise continuous coefficient. Hacettepe Journal of Mathematics and Statistics, 44(4), 867-874.
• Marchenko, V. A. (1986). Sturm-Liouville Operators and Applications. Basel, Switzerland: Birkhauser Verlag.
• Mutlu, G., & Kir Arpat, E. (2020). Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis. Hacettepe Journal of Mathematics and Statistics, 49(5), 1-9. doi:10.15672/hujms.577991
• Nabiev, A. A., & Mamedov, Kh. R. (2015). On the Jost solutions for a class of Schrödinger equations with piecewise constant coefficients. Journal of Mathematical Physics, Analysis, Geometry, 11(3), 279-296. doi:10.15407/mag11.03.279
• Naimark, M. A. (1954). Investigation of the spectrum and the expansion in eigenfunctions of a nonselfadjoint operator of the second order on a semi-axis (in Russian). Trudy Moskovskogo Matematicheskogo Obshchestva, 3, 181-270.
• Naimark, M. A. (1968). Linear Differential Operators I, II. New York, USA: Ungar.
• Olgun, M., & Coskun, C. (2010). Non-selfadjoint matrix Sturm–Liouville operators with spectral singularities. Applied Mathematics and Computation, 216(8), 2271-2275. doi:10.1016/j.amc.2010.03.062
• Pavlov, B. S. (1962). On the spectral theory of non-selfadjoint differential operators. Doklady Akademii Nauk, 146(6), 1267-1270.
• Yokus, N., & Coskun, N. (2016). Jost solution and the spectrum of the discrete Sturm-Liouville equations with hyperbolic eigenparameter. Neural, Parallel, and Scientific Computations, 24, 419-430.
• Yokus, N., & Coskun, N. (2019). A note on the matrix Sturm-Liouville operators with principal functions. Mathematical Methods in the Applied Sciences, 42(16), 5362-5370. doi:10.1002/mma.5383