Resolvent Operator of the Matrix Schrödinger Equation on the Half-Line with Quasi-selfadjoint Potential

Year 2021, Volume: 8 Issue: 2, 197 - 207, 28.06.2021

Gökhan Mutlu

Abstract

We obtain the resolvent operator of the matrix Schrödinger equation on the half-line with a quasi-selfadjoint matrix potential Q. We also assume each entry of Q is Lebesgue measurable on (0,∞) and Q has a finite first moment. We impose the general boundary condition at x=0. This boundary value problem is not selfadjoint which makes it valuable and difficult in terms of the spectral analysis. Moreover, considering the most general boundary conditions generalizes many studies in the literature. We introduce the Jost matrix of this boundary value problem. We examine asymptotical and analytical properties of the Jost matrix in order to derive the resolvent operator and point spectrum. We use the quasi-selfadjointness of the matrix potential Q to obtain these properties. We show that the resolvent set consists of squares of the non-singular points of the Jost matrix in the upper complex plane. Moreover, we obtain the Green’s function of this boundary value problem with the help of the Jost matrix. In the light of this main result, we show that the continuous spectrum is [0,∞) and the point spectrum consist of squares of the singular points of the Jost matrix in the upper complex plane. We also show that the set of spectral singularities is empty.

Keywords

Matrix Schrödinger Equation, Jost Matrix, Resolvent Operator, Continuous Spectrum, Eigenvalues, Spectral Singularities

References

• Agranovic, Z. S., & Marchenko, V. A. (1965). The Inverse Problem of Scattering Theory, Gordon and Breach.
• Aktosun, T., Klaus, M., & Weder, R. (2011). Small-energy analysis for the self-adjoint matrix Schrödinger equation on the half line. Journal of Mathematical Physics, 52, 102101. doi:10.1063/1.3640029
• Aktosun, T., & Weder, R. (2013). High-energy analysis and Levinson's theorem for the selfadjoint matrix Schrödinger operator on the half line. Journal of Mathematical Physics, 54, 012108. doi:10.1063/1.4773904
• Aktosun, T., & Weder, R. (2018). Inverse scattering on the half line for the matrix Schrödinger equation, arXiv:1806.01644.
• Aktosun, T., & Weder, R. (2020). Direct and Inverse Scattering for the Matrix Schrödinger Equation, Applied Mathematical Sciences, 203, Cham: Springer.
• Arpat, E. K. & Mutlu, G. (2015). Spectral properties of Sturm-Liouville system with eigenvalue-dependent boundary conditions. International Journal of Mathematics, 26(10), 1550080-1550088. doi:10.1142/S0129167X15500809
• Bagarello, F., Gazeau, J. P., Szafraniec, F. H., & Znojil, M. (2015). Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects, John Wiley & Sons.
• Birman, M. S., & Solomjak, M. Z. (1987). Spectral Theory of Self-Adjoint Operators in Hilbert Space, Mathematics and its Applications, vol 5, Netherlands: Springer.
• Bairamov, E., Arpat, E. K., & Mutlu, G. (2017). Spectral properties of non-selfadjoint Sturm-Liouville operator with operator coefficient. Journal of Mathematical Analysis and Applications 456(1), 293-306. doi:10.1016/j.jmaa.2017.07.001
• Bender, C. M., Brody, D. C., & Jones, H. F. (2002). Complex extension of quantum mechanics. Physical Review Letters, 89, 270401. doi:10.1103/PhysRevLett.89.270401
• Bender, C. M., Brody, D. C., & Jones, H. F. (2003). Must a Hamiltonian be Hermitian? American Journal of Physics, 71, 1095. doi:10.1119/1.1574043
• Bender, C. M. (2007) Making sense of non-Hermitian Hamiltonians. Reports on Progress in Physics, 70, 947–1018. doi:10.1088/0034-4885/70/6/R03
• Berkolaiko, G., & Kuchment, P. (2013). Introduction to Quantum Graphs (Mathematical Surveys and Monographs vol 186), Rhode Island: American Mathematical Society.
• Gasymov, M. G., Zikov, V. V., & Levitan, B. M. (1967). Conditions for the negative spectrum of the Schrödinger equation operator to be discrete and finite. Mathematical notes of the Academy of Sciences of the USSR, 2, 813–817. doi:10.1007/BF01093944
• Kottos, T., & Smilansky, U. (1997). Quantum chaos on graphs. Physical Review Letters, 79, 4794-4797. doi:10.1103/PhysRevLett.79.4794
• Levitan, B. M., & Sargsyan, I. S. (1975). Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, American Mathematical Society.
• Mostafazadeh, A. (2010). Pseudo-Hermitian representation of quantum mechanics. International Journal of Geometric Methods in Modern Physics, 7(7), 1191-1306. doi:10.1142/S0219887810004816
• Mutlu, G., & Kır Arpat, E. (2020). Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis. Hacettepe Journal of Mathematics and Statistics, 49(5), 1686-1694. doi:10.15672/hujms.577991
• Naimark, M. A. (1968). Linear Differential Operators, II, New York: Ungar.
• Olgun, M., & Coskun, C. (2010). Non-selfadjoint matrix Sturm-Liouville operators with spectral singularities. Applied Mathematics and Computation, 216, 2271-2275. doi:10.1016/j.amc.2010.03.062
• Scholtz, F. G., Geyer, H. B. & Hahne, F. J. W. (1992). Quasi-Hermitian operators in quantum mechanics and the variational principle. Annals of Physics, 21, 74-101. doi:10.1016/0003-4916(92)90284-S
• Schmüdgen, K. (2012). Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics, vol 265, Netherlands: Springer.
• Sjöstrand, J. (2019). Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations, Pseudo-Differential Operators, vol 14, Basel: Birkhäuser.
• Weder, R. (2017). The number of eigenvalues of the matrix Schrödinger operator on the half line with general boundary conditions. Journal of Mathematical Physics, 58, 102107. doi:10.1063/1.5008655
• Yokus, N., & Coskun, N. (2018). The spectrum of quadratic eigenparameter-dependent non-selfadjoint matrix sturm-liouville operators. Advances in Differential Equations and Control Processes, 19(2), 139-152. doi:10.17654/DE019020139
• Yokus, N., & Coskun, N. (2019). A note on the matrix Sturm‐Liouville operators with principal functions. Mathematical Methods in Applied Sciences, 42, 5362-5370. doi:10.1002/mma.5383