Counting spectrum via the Maslov index for one dimensional 𝜃-periodic Schrödinger operators

Jones, Christopher; Latushkin, Yuri; Sukhtaiev, Selim

doi:10.1090/proc/13192

Counting spectrum via the Maslov index for one dimensional $\theta -$periodic Schrödinger operators
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by Christopher K. R. T. Jones, Yuri Latushkin and Selim Sukhtaiev

Proc. Amer. Math. Soc. 145 (2017), 363-377

DOI: https://doi.org/10.1090/proc/13192

Published electronically: July 6, 2016

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Abstract:

We study the spectrum of the Schrödinger operators with $n\times n$ matrix valued potentials on a finite interval subject to $\theta -$periodic boundary conditions. For two such operators, corresponding to different values of $\theta$, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to $\theta$ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schrödinger operator for a fixed $\theta$.

References

V. I. Arnol′d, Sturm theorems and symplectic geometry, Funktsional. Anal. i Prilozhen. 19 (1985), no. 4, 1–10, 95 (Russian). MR 820079
F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
Margaret Beck and Simon J. A. Malham, Computing the Maslov index for large systems, Proc. Amer. Math. Soc. 143 (2015), no. 5, 2159–2173. MR 3314123, DOI 10.1090/S0002-9939-2014-12575-5
Bernhelm Booss-Bavnbek and Kenro Furutani, The Maslov index: a functional analytical definition and the spectral flow formula, Tokyo J. Math. 21 (1998), no. 1, 1–34. MR 1630119, DOI 10.3836/tjm/1270041982
Raoul Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171–206. MR 90730, DOI 10.1002/cpa.3160090204
Sylvain E. Cappell, Ronnie Lee, and Edward Y. Miller, On the Maslov index, Comm. Pure Appl. Math. 47 (1994), no. 2, 121–186. MR 1263126, DOI 10.1002/cpa.3160470202
F. Chardard, F. Dias, and T. J. Bridges, Fast computation of the Maslov index for hyperbolic linear systems with periodic coefficients, J. Phys. A 39 (2006), no. 47, 14545–14557. MR 2277067, DOI 10.1088/0305-4470/39/47/002
Frédéric Chardard, Frédéric Dias, and Thomas J. Bridges, Computing the Maslov index of solitary waves. I. Hamiltonian systems on a four-dimensional phase space, Phys. D 238 (2009), no. 18, 1841–1867. MR 2598511, DOI 10.1016/j.physd.2009.05.008
Charles Conley and Eduard Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), no. 2, 207–253. MR 733717, DOI 10.1002/cpa.3160370204
G. Cox, C. Jones, Y. Latushkin, and A. Sukhtayev, The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials, Trans. Amer. Math. Soc., to appear.
Graham Cox, Christopher K. R. T. Jones, and Jeremy L. Marzuola, A Morse index theorem for elliptic operators on bounded domains, Comm. Partial Differential Equations 40 (2015), no. 8, 1467–1497. MR 3355500, DOI 10.1080/03605302.2015.1025979
G. Cox, C. K. R. T. Jones, and J. Marzuola, Manifold decompositions and indices of Schrödinger operators, Preprint, http://arxiv.org/abs/1506.07431.
Francesca Dalbono and Alessandro Portaluri, Morse-Smale index theorems for elliptic boundary deformation problems, J. Differential Equations 253 (2012), no. 2, 463–480. MR 2921202, DOI 10.1016/j.jde.2012.04.008
Jian Deng and Christopher Jones, Multi-dimensional Morse index theorems and a symplectic view of elliptic boundary value problems, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1487–1508. MR 2737274, DOI 10.1090/S0002-9947-2010-05129-3
J. J. Duistermaat, On the Morse index in variational calculus, Advances in Math. 21 (1976), no. 2, 173–195. MR 649277, DOI 10.1016/0001-8708(76)90074-8
H. I. Dwyer and A. Zettl, Computing eigenvalues of regular Sturm-Liouville problems, Electron. J. Differential Equations (1994), No. 06, approx. 10 pp.}, review= MR 1292109,
Kenro Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004), no. 3, 269–331. MR 2079414, DOI 10.1016/j.geomphys.2004.04.001
C. Grudziena and C. K. R. T. Jones, Geometric phase in the Hopf bundle and the stability of non-linear waves, preprint (2015).
P. Howard and A. Sukhtayev, The Maslov and Morse indices for Schrödinger operators on $[0,1]$, J. Differential Equations 260 (2016), no. 5, 4499–4549. MR 3437596, DOI 10.1016/j.jde.2015.11.020
Christopher K. R. T. Jones, Yuri Latushkin, and Robert Marangell, The Morse and Maslov indices for matrix Hill’s equations, Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fritz Gesztesy’s 60th birthday, Proc. Sympos. Pure Math., vol. 87, Amer. Math. Soc., Providence, RI, 2013, pp. 205–233. MR 3087908, DOI 10.1090/pspum/087/01436
Y. Latushkin, S. Sukhtaiev, and A. Sukhtayev, The Morse and Maslov indices for Schrödinger operators, J. D’Analyse Math., to appear, http://arxiv.org/abs/1411.1656.
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
Alessandro Portaluri and Nils Waterstraat, A Morse-Smale index theorem for indefinite elliptic systems and bifurcation, J. Differential Equations 258 (2015), no. 5, 1715–1748. MR 3295598, DOI 10.1016/j.jde.2014.11.010
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
Joel Robbin and Dietmar Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), no. 1, 1–33. MR 1331677, DOI 10.1112/blms/27.1.1
S. Smale, On the Morse index theorem, J. Math. Mech. 14 (1965), 1049–1055. MR 0182027, DOI 10.1111/j.1467-9876.1965.tb00656.x