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February  2020, 40(2): 983-1012. doi: 10.3934/dcds.2020068

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

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

Received  March 2019 Revised  August 2019 Published  November 2019

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.

Citation: Peter Howard, Alim Sukhtayev. The Maslov and Morse indices for Sturm-Liouville systems on the half-line. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 983-1012. doi: 10.3934/dcds.2020068
References:
[1]

A. Ben-ArtziI. 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.  Google Scholar

[2]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, , Mathematical Surveys and Monographs, 186, AMS, Providence, RI, 2013.  Google Scholar

[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.  Google Scholar

[4]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, MA, 1965.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[10]

P. HowardY. 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.  Google Scholar

[11]

P. HowardY. 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.  Google Scholar

[12]

L. Hörmander, Fourier integral operators I, Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.  Google Scholar

[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.  Google Scholar

[14]

P. Howard and A. Sukhtayev, Renormalized oscillation theory for linear Hamiltonian systems on $[0, 1]$ via the Maslov index, preprint, arXiv: 1808.08264. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. Latushkin and S. Sukhtaiev, An index theorem for Schrödinger operators on metric graphs, preprint, arXiv: 1809.09344v2. Google Scholar

[21]

J. Phillips, Self-adjoint Fredholm operators and spectral flow, Canad. Math. Bull., 39 (1996), 460-467.  doi: 10.4153/CMB-1996-054-4.  Google Scholar

[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.  Google Scholar

[23]

B. SimonG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Weidmann, Spectral Theory of Ordinary Differential Operators, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987. doi: 10.1007/BFb0077960.  Google Scholar

[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.  Google Scholar

[28]

Y. ZhouLi Wu and C. Zhu, Hörmander index in finite-dimensional case, Front. Math. China, 13 (2018), 725-761.  doi: 10.1007/s11464-018-0702-3.  Google Scholar

show all references

References:
[1]

A. Ben-ArtziI. 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.  Google Scholar

[2]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, , Mathematical Surveys and Monographs, 186, AMS, Providence, RI, 2013.  Google Scholar

[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.  Google Scholar

[4]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, MA, 1965.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[10]

P. HowardY. 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.  Google Scholar

[11]

P. HowardY. 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.  Google Scholar

[12]

L. Hörmander, Fourier integral operators I, Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.  Google Scholar

[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.  Google Scholar

[14]

P. Howard and A. Sukhtayev, Renormalized oscillation theory for linear Hamiltonian systems on $[0, 1]$ via the Maslov index, preprint, arXiv: 1808.08264. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. Latushkin and S. Sukhtaiev, An index theorem for Schrödinger operators on metric graphs, preprint, arXiv: 1809.09344v2. Google Scholar

[21]

J. Phillips, Self-adjoint Fredholm operators and spectral flow, Canad. Math. Bull., 39 (1996), 460-467.  doi: 10.4153/CMB-1996-054-4.  Google Scholar

[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.  Google Scholar

[23]

B. SimonG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Weidmann, Spectral Theory of Ordinary Differential Operators, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987. doi: 10.1007/BFb0077960.  Google Scholar

[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.  Google Scholar

[28]

Y. ZhouLi Wu and C. Zhu, Hörmander index in finite-dimensional case, Front. Math. China, 13 (2018), 725-761.  doi: 10.1007/s11464-018-0702-3.  Google Scholar

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