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Renormalized oscillation theory for regular linear non-Hamiltonian systems
Department of Mathematics, Texas A&M University, College Station, TX 77843 USA
Received: May 2022
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In recent work, Baird et al. have generalized the definition of the Maslov index to paths of Grassmannian subspaces that are not necessarily contained in the Lagrangian Grassmannian [2]. Such an extension opens up the possibility of applications to non-Hamiltonian systems of ODE, and Baird and his collaborators have taken advantage of this observation to establish oscillation-type results for obtaining lower bounds on eigenvalue counts in this generalized setting. In the current analysis, the author shows that renormalized oscillation theory, appropriately defined in this generalized setting, can be applied in a natural way, and that it has the advantage, as in the traditional setting of linear Hamiltonian systems, of ensuring monotonicity of crossing points as the independent variable increases for a wide range of system/boundary-condition combinations. This seems to mark the first effort to extend the renormalized oscillation approach to the non-Hamiltonian setting.
Renormalized oscillation theory,
generalized Maslov index,
boundary value problems,
eigenvalues,
spectral counts.
Mathematics Subject Classification: Primary: 34C10, 34L15; Secondary: 34L16.
Figure 1. Local contributions to $ \mathfrak{m} $
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Figure 2. The Maslov Box
Figure 3. Full Maslov Box and spectral curve for (6.15)
Figure 4. Full Maslov Box and spectral curves for (6.18)–(6.30)
Figure 5. Spectral curves for the system (6.18)–(6.31)
[1] V. I. Arnol'd, The Sturm theorems and symplectic geometry, Fund. Anal. Appl., 19 (1985), 251-259.
[2] T. J. Baird, P. Cornwell, G. Cox, C. Jones and R. Marangell, Generalized Maslov indices for non-Hamiltonian systems, SIAM J. Math. Anal., 54 (2022), 1623-1668. doi: 10.1137/20M1381319.
[3] M. Bohner, W. Kratz and R. Šimon Hilscher, Oscillation and spectral theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter, Nath. Nachr., 285 (2012), 1343-1356. doi: 10.1002/mana.201100172.
[4] V. Barutello, D. Offin, A. Portaluri and L. Wu, Sturm theory with applications in geometry and classical mechanics, Math. Zeit., 299 (2021), 257-297. doi: 10.1007/s00209-020-02686-3.
[5] F. Chardard and T. J. Bridges, Transversality of homoclinic orbits, the Maslov index, and the symplectic Evans function, Nonlinearity, 28 (2015), 77-102. doi: 10.1088/0951-7715/28/1/77.
[6] G. Cox, C. K. R. T. Jones, Y. Latushkiun and A. Sukhtayev, The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials, Trans. Amer. Math. Soc., 368 (2016), 8145-8207. doi: 10.1090/tran/6801.
[7] M. De Gosson, S. De Gosson and P. Piccione, On a product formula for the Conley-Zehnder index of symplectic paths and its applications, Ann. Glob. Anal. Geom., 34 (2008), 167-183. doi: 10.1007/s10455-008-9106-z.
[8] J. Deng and C. K. R. T. Jones, Multi-dimensional Morse Index Theorems and a symplectic view of elliptic boundary value problems, Trans. Amer. Math. Soc., 363 (2011), 1487-1508. doi: 10.1090/S0002-9947-2010-05129-3.
[9] H. M. Edwards, A generalized Sturm theorem, Ann. Math., 80 (1964) 22-57. doi: 10.2307/1970490.
[10] J. Elyseeva, Renormalized oscillation theory for symplectic eigenvalue problems with nonlinear dependence on the spectral parameter, J. Differ. Equ. Appl., 26 (2020), 458-487. doi: 10.1080/10236198.2020.1748020.
[11] K. Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, J. Geomet. Phys., 51 (2004), 269-331. doi: 10.1016/j.geomphys.2004.04.001.
[12] F. Gesztesy, B. Simon and G. Teschl, Zeros of the Wronskian and renormalized oscillation theory, Amer. J. Math., 118 (1996), 571-594.
[13] 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.
[14] P. Howard, Asymptotic behavior near transition fronts for equations of generalized Cahn-Hilliard form, Commun. Math. Phys., 269 (2007), 765-808. doi: 10.1007/s00220-006-0102-5.
[15] P. Howard, Hörmander's index and oscillation theory, J. Math. Anal. Appl., 500 (2021), 1-38. doi: 10.1016/j.jmaa.2021.125076.
[16] P. Howard and A. Sukhtayev, The Maslov and Morse indices for Schrödinger operators on [0, 1], J. Differ. Equ., 260 (2016), 4499-4549. doi: 10.1016/j.jde.2015.11.020.
[17] P. Howard and A. Sukhtayev, Renormalized oscillation theory for linear Hamiltonian systems on $[0, 1]$ via the Maslov index, J. Dynam. Differ. Equ., (2022), 41 pp. doi: 10.1007/s10884-021-10121-2.
[18] P. Howard and A. Sukhtayev, Renormalized oscillation theory for singular linear Hamiltonian systems, J. Funct. Anal., 283, (2022), 74 pp. doi: 10.1016/j.jfa.2022.109525.
[19] P. Howard and K. Zumbrun, Pointwise estimates and stability for dispersive–diffusive shock waves, Arch. Rational Mech. Anal., 155 (2000), 85-169. doi: 10.1007/s002050000110.
[20] R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. Lond. A, 340 (1992), 47-94. doi: 10.1098/rsta.1992.0055.
[21] H. Schulz-Baldes, Sturm intersection theory for periodic Jacobi matrices and linear Hamiltonian systems, Linear Algebra Appl., 436 (2012), 498-515. doi: 10.1016/j.laa.2011.06.052.
[22] B. Simon, Sturm oscillation and comparison theorems, in Sturm-Liouville Theory: Past and Present, Birkhäuser Verlag 2005, W. O. Amrein, A. M. Hinz, and D. B. Pearson, Eds. doi: 10.1007/3-7643-7359-8_2.
[23] G. Teschl, Oscillation theory and renormalized oscillation theory for Jacobi operators, J. Differ. Equ., 129 (1996), 532-558. doi: 10.1006/jdeq.1996.0126.
[24] G. Teschl, Renormalized oscillation theory for Dirac operators, Proceedings of the AMS, 126 (1998), 1685-1695. doi: 10.1090/S0002-9939-98-04310-X.
[25] 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.2002.51.2410.
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Local contributions to $ \mathfrak{m} $
The Maslov Box
Full Maslov Box and spectral curve for (6.15)
Full Maslov Box and spectral curves for (6.18)–(6.30)
Spectral curves for the system (6.18)–(6.31)