• Previous Article
    Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case
  • DCDS Home
  • This Issue
  • Next Article
    Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms
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 and Continuous Dynamical Systems, 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.

[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. 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.

[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.

[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. 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.

[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. 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.

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.

[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. 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.

[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.

[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. 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.

[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. 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.

[1]

Günter Leugering, Gisèle Mophou, Maryse Moutamal, Mahamadi Warma. Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022015

[2]

Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405

[3]

N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050

[4]

Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030

[5]

Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185

[6]

Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Controllability for Schrödinger type system with mixed dispersion on compact star graphs. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022019

[7]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems and Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004

[8]

Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3399-3434. doi: 10.3934/dcdsb.2021190

[9]

Guglielmo Feltrin. Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities. Communications on Pure and Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052

[10]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko, Xiao-Chuan Xu. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems and Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068

[11]

Elimhan N. Mahmudov. Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints. Journal of Industrial and Management Optimization, 2020, 16 (1) : 169-187. doi: 10.3934/jimo.2018145

[12]

Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066

[13]

Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2417-2434. doi: 10.3934/dcdss.2020171

[14]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[15]

Fengshuang Gao, Yuxia Guo. Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1181-1204. doi: 10.3934/cpaa.2020055

[16]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337

[17]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024

[18]

Nobu Kishimoto. A remark on norm inflation for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1375-1402. doi: 10.3934/cpaa.2019067

[19]

Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359

[20]

Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (214)
  • HTML views (86)
  • Cited by (1)

Other articles
by authors

[Back to Top]