The Maslov and Morse indices for Sturm-Liouville systems on the half-line

Peter Howard; Alim Sukhtayev

doi:10.3934/dcds.2020068

Article Contents

2020, Volume 40, Issue 2: 983-1012. Doi: 10.3934/dcds.2020068

This issue Previous Article Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms Next Article Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case

The Maslov and Morse indices for Sturm-Liouville systems on the half-line

Peter Howard^1, , and
Alim Sukhtayev^2,

1.
Department of Mathematics, Texas A & M University, College Station, TX 77843, USA
2.
Department of Mathematics, Miami University, Oxford, OH 45056, USA

^* Corresponding author: Peter Howard
^* Corresponding author: Peter Howard

Received: March 2019

Revised: August 2019

Published: November 2019

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Abstract

We show that for Sturm-Liouville Systems on the half-line $ [0, \infty) $, the Morse index can be expressed in terms of the Maslov index and an additional term associated with the boundary conditions at $ x = 0 $. Relations are given both for the case in which the target Lagrangian subspace is associated with the space of $ L^2 ((0, \infty), \mathbb{C}^{n}) $ solutions to the Sturm-Liouville System, and the case in which the target Lagrangian subspace is associated with the space of solutions satisfying the boundary conditions at $ x = 0 $. In the former case, a formula of Hörmander's is used to show that the target space can be replaced with the Dirichlet space, along with additional explicit terms. We illustrate our theory by applying it to an eigenvalue problem that arises when the nonlinear Schrödinger equation on a star graph is linearized about a half-soliton solution.

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Mathematics Subject Classification: Primary: 34B24, 34B45; Secondary: 34L15.

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References

[1]	A. Ben-Artzi, I. Gohberg and M. A. Kaashoek, Invertibility and dichotomy of differential operators on a half-line, J. Dynamics and Differential Equations, 5 (1993), 1-36. doi: 10.1007/BF01063733.
[2]	G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, , Mathematical Surveys and Monographs, 186, AMS, Providence, RI, 2013.
[3]	W.-J. Beyn and J. Lorenz, Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals, Num. Functional Anal. and Optim., 20 (1999), 201-244. doi: 10.1080/01630569908816889.
[4]	W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, MA, 1965.
[5]	J. J. Duistermaat, On the Morse index in variational calculus, Adv. Math., 21 (1976), 173-195. doi: 10.1016/0001-8708(76)90074-8.
[6]	K. Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, Journal of Geometry and Physics, 51 (2004), 269-331. doi: 10.1016/j.geomphys.2004.04.001.
[7]	F. Gesztesy and M. Zinchenko, Renormalized oscillation theory for Hamiltonian systems, Adv. Math., 311 (2017), 569-597. doi: 10.1016/j.aim.2017.03.005.
[8]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840, Springer-Verlag, Berlin-New York, 1981.
[9]	D. B. Hinton and J. K. Shaw, On the spectrum of a singular Hamiltonian system, Quaes. Math., 5 (1982), 29-81. doi: 10.1080/16073606.1982.9631879.
[10]	P. Howard, Y. Latushkin and A. Sukhtayev, The Maslov index for Lagrangian pairs on $\mathbb{R}^{2n}$, Journal of Mathematical Analysis and Applications, 451 (2017), 794-821. doi: 10.1016/j.jmaa.2017.02.022.
[11]	P. Howard, Y. Latushkin and A. Sukhtayev, The Maslov and Morse indices for system Schrödinger operators on $\mathbb{R}$, Indiana J. Mathematics, 67 (2018), 1765-1815. doi: 10.1512/iumj.2018.67.7462.
[12]	L. Hörmander, Fourier integral operators I, Acta Math., 127 (1971), 79-183. doi: 10.1007/BF02392052.
[13]	P. Howard and A. Sukhtayev, The Maslov and Morse indices for Schrödinger operators on [0, 1], J. Differential Equations, 260 (2016), 4499-4549. doi: 10.1016/j.jde.2015.11.020.
[14]	P. Howard and A. Sukhtayev, Renormalized oscillation theory for linear Hamiltonian systems on $[0, 1]$ via the Maslov index, preprint, arXiv: 1808.08264.
[15]	T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer, New York, 2013. doi: 10.1007/978-1-4614-6995-7.
[16]	A. Kairzhan and D. Pelinovsky, Nonlinear instability of half-solitons on star graphs, J. Differential Equations, 264 (2018), 7357-7383. doi: 10.1016/j.jde.2018.02.020.
[17]	A. M. Krall, Hilbert Space, Boundary Value Problems and Orthogonal Polynomials, Birkhäuser Verlag, Basel, Boston, Berlin 2002. doi: 10.1007/978-3-0348-8155-5.
[18]	V. Kostrykin and R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A: Math. Gen., 32 (1999), 595-630. doi: 10.1088/0305-4470/32/4/006.
[19]	P. Kuchment, Quantum graphs: I. Some basic structures, Waves in Random Media, 14 (2004), S107–S128. doi: 10.1088/0959-7174/14/1/014.
[20]	Y. Latushkin and S. Sukhtaiev, An index theorem for Schrödinger operators on metric graphs, preprint, arXiv: 1809.09344v2.
[21]	J. Phillips, Self-adjoint Fredholm operators and spectral flow, Canad. Math. Bull., 39 (1996), 460-467. doi: 10.4153/CMB-1996-054-4.
[22]	B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D, 145 (2000), 233-277. doi: 10.1016/S0167-2789(00)00114-7.
[23]	B. Simon, G. Teschl and F. Gesztesy, Zeros of the Wronskian and renormalized oscillation theory, American J. Math., 118 (1996), 571-594. doi: 10.1353/ajm.1996.0024.
[24]	G. Teschl, Oscillation theory and renormalized oscillation theory for Jacobi operators, J. Differential Equations, 129 (1996), 532-558. doi: 10.1006/jdeq.1996.0126.
[25]	G. Teschl, Renormalized oscillation theory for Dirac operators, Proceedings of the AMS, 126 (1998), 1685-1695. doi: 10.1090/S0002-9939-98-04310-X.
[26]	J. Weidmann, Spectral Theory of Ordinary Differential Operators, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987. doi: 10.1007/BFb0077960.
[27]	K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana U. Math. J., 47 (1998), 741-871. doi: 10.1512/iumj.1998.47.1604.
[28]	Y. Zhou, Li Wu and C. Zhu, Hörmander index in finite-dimensional case, Front. Math. China, 13 (2018), 725-761. doi: 10.1007/s11464-018-0702-3.