Abstract
We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rn, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman & Hall/CRC, Boca Raton, FL, 2001.
S. Albeverio and K. Makarov, Attractors in a model related to the three body quantum problem, Acta Appl. Math. 48 (1997) 113–184.
V. I. Arnold, Characteristic classes entering in quantization conditions, Funct. Anal. Appl. 1 (1967), 1–14.
V. I. Arnold, Sturm theorems and symplectic geometry, Funct. Anal. Appl. 19 (1985), 1–10.
J. Behrndt and J. Rohleder, An inverse problem of Calderón type with partial data, Comm. Partial Differential Equations, 37 (2012), 1141–1159.
B. Booss-Bavnbek and K. Furutani, The Maslov index: a functional analytical definition and the spectral flow formula, Tokyo J. Math. 21 (1998), 1–34.
B. Boos-Bavnbek and K. Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston, MA, 1993.
R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171–206.
S. Cappell, R. Lee, and E. Miller, On the Maslov index, Comm. Pure Appl. Math. 47 (1994), 121–186.
F. Chardard and T. J. Bridges, Transversality of homoclinic orbits, the Maslov index, and the symplectic Evans function, Nonlinearity 28 (2015) 77–102.
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), 14545–14557.
F. Chardard, F. Dias, and T. J. Bridges, Computing the Maslov index of solitary waves. I. Hamiltonian systems on a four-dimensional phase space, Phys. D 238 (2009), 1841–1867; II. Phase space with dimension greater than four, Phys. D 240 (2011), 1334–1344.
C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207–253.
G. Cox, C. K. R. T. Jones, Y. Latushkin, and A. Sukhtayev, The Morse and Maslov indices for multidimentional Schrödinger operators with matrix valued potential, Trans. Amer. Math. Soc. 368 (2016), 8145–8207.
G. Cox, C. K. R. T. Jones, and J. Marzuola, A Morse index theorem for elliptic operators on bounded domains, Comm. Partial Differential Equations 40 (2015), 1467–1497.
G. Cox, C. K. R. T. Jones, and J. Marzuola, Manifold decompositions and indices of Schrödinger operators, Indiana U. Math. J. 66 (2017), 1573–1602.
R. Cushman and J. J. Duistermaat, The behavior of the index of a periodic linear Hamiltonian system under iteration, Adv. Math. 23 (1977) 1–21.
F. Dalbono and A. Portaluri, Morse–Smale index theorems for elliptic boundary deformation problems, J. Differential Equations 253 (2012), 463–480.
Ju. Daleckii and M. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1974.
J. Deng and C. Jones, Multi-dimensional Morse index theorems and a symplectic view of elliptic boundary value problems, Trans. Amer. Math. Soc. 363 (2011), 1487–1508.
J. J. Duistermaat, On the Morse index in variational calculus, Adv. Math. 21 (1976), 173–195.
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1989.
K. Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004), 269–331.
F. Gesztesy and M. Mitrea, A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas in nonsmooth domains, J. Anal. Math. 113 (2011), 53–172.
F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, in Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Amer. Math. Soc., Providence, RI, 2008, pp. 105–173.
M. A. de Gosson, The Principles of Newtonian and Quantum Mechanics, Imperial College Press, London, 2001.
P. Howard and A. Sukhtayev, The Maslov and Morse indices for Schrödinger operators on [0, 1], J. Differential Equations 260 (2016), 4499–4549.
C. K. R. T. Jones, Y. Latushkin, and R. Marangel, The Morse and Maslov indices for matrix Hill’s equations, Proc. Symp. Pure Math. 87 (2013), 205–233.
C. K. R. T. Jones, Y. Latushkin, and S. Sukhtaiev, Counting spectrum via the Maslov index for one dimensional θ-periodic Schrödinger operators, Proc. Amer. Math. Soc. 145 (2016), 363–377.
Y. Karpeshina, Perturbation Theory for the Schrödinger Operator with a Periodic Potential, Springer-Verlag, Berlin, 1997.
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980.
D. McDuff and D. Salamon Introduction to Symplectic Topology, second edition, Clarendon Press, Oxford, 1998.
J. Milnor, Morse Theory, Princeton Univ. Press, Princeton, NJ, 1963.
B. S. Pavlov and M. D. Faddeev, Scattering on a hollow resonator with a small opening, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 126 (1983), 159–169.
A. Portaluri and N. Waterstraat, A Morse-Smale index theorem for indefinite elliptic systems and bifurcation, J. Differential Equations 258, 1715–1748.
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Volume 1, Academic Press, London, 1980.
J. Robbin and D. Salamon, The Maslov index for paths, Topology 32 (1993), 827–844.
J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), 1–33.
D. Salamon and K. Wehrheim, Instanton Floer homology with Lagrangian boundary conditions, Geom. Topol. 12 (2008), 747–918.
S. Smale, On the Morse index theorem, J. Math. Mech. 14 (1965), 1049–1055; The Collected Papers by Stephen Smale, V. 2, City University of Hong Kong, 2000, pp. 535–543.
R. C. Swanson, Fredholm intersection theory and elliptic boundary deformation problems, J. Differential Equations 28 (1978), I, 189–201, II, 202–219.
M. E. Taylor, Partial Differential Equations I. Basic Theory, Springer-Verlag, 2011.
K. Uhlenbeck, The Morse Index Theorem in Hilbert spaces, J. Differential Geom. 8 (1973), 555–564.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the NSF grants DMS-1067929 and DMS-1710989, by the Simons Foundation, and by the Research Council and the Research Board of the University of Missouri.
Rights and permissions
About this article
Cite this article
Latushkin, Y., Sukhtaiev, S. & Sukhtayev, A. The Morse and Maslov indices for Schrödinger operators. JAMA 135, 345–387 (2018). https://doi.org/10.1007/s11854-018-0043-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-018-0043-x