Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 345–387 | Cite as

The Morse and Maslov indices for Schrödinger operators



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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman & Hall/CRC, Boca Raton, FL, 2001.MATHGoogle Scholar
  2. [AM]
    S. Albeverio and K. Makarov, Attractors in a model related to the three body quantum problem, Acta Appl. Math. 48 (1997) 113–184.MathSciNetCrossRefMATHGoogle Scholar
  3. [A67]
    V. I. Arnold, Characteristic classes entering in quantization conditions, Funct. Anal. Appl. 1 (1967), 1–14.CrossRefGoogle Scholar
  4. [A85]
    V. I. Arnold, Sturm theorems and symplectic geometry, Funct. Anal. Appl. 19 (1985), 1–10.MathSciNetGoogle Scholar
  5. [BR]
    J. Behrndt and J. Rohleder, An inverse problem of Calderón type with partial data, Comm. Partial Differential Equations, 37 (2012), 1141–1159.MathSciNetCrossRefMATHGoogle Scholar
  6. [BF]
    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.MathSciNetCrossRefMATHGoogle Scholar
  7. [BW]
    B. Boos-Bavnbek and K. Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston, MA, 1993.CrossRefMATHGoogle Scholar
  8. [B]
    R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171–206.MathSciNetCrossRefMATHGoogle Scholar
  9. [CLM]
    S. Cappell, R. Lee, and E. Miller, On the Maslov index, Comm. Pure Appl. Math. 47 (1994), 121–186.MathSciNetCrossRefMATHGoogle Scholar
  10. [CB]
    F. Chardard and T. J. Bridges, Transversality of homoclinic orbits, the Maslov index, and the symplectic Evans function, Nonlinearity 28 (2015) 77–102.MathSciNetMATHGoogle Scholar
  11. [CDB6]
    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.MathSciNetCrossRefMATHGoogle Scholar
  12. [CDB9]
    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.MathSciNetCrossRefMATHGoogle Scholar
  13. [CZ]
    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.MathSciNetCrossRefMATHGoogle Scholar
  14. [CJLS]
    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.MathSciNetCrossRefMATHGoogle Scholar
  15. [CJM1]
    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.MathSciNetCrossRefMATHGoogle Scholar
  16. [CJM2]
    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.CrossRefMATHGoogle Scholar
  17. [CD]
    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.MathSciNetCrossRefMATHGoogle Scholar
  18. [DP]
    F. Dalbono and A. Portaluri, Morse–Smale index theorems for elliptic boundary deformation problems, J. Differential Equations 253 (2012), 463–480.MathSciNetCrossRefMATHGoogle Scholar
  19. [DK]
    Ju. Daleckii and M. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI, 1974.Google Scholar
  20. [DJ]
    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.MathSciNetCrossRefMATHGoogle Scholar
  21. [D]
    J. J. Duistermaat, On the Morse index in variational calculus, Adv. Math. 21 (1976), 173–195.MathSciNetCrossRefMATHGoogle Scholar
  22. [EE]
    D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1989.MATHGoogle Scholar
  23. [F]
    K. Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004), 269–331.MathSciNetCrossRefMATHGoogle Scholar
  24. [GM10]
    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.MathSciNetCrossRefMATHGoogle Scholar
  25. [GM08]
    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.CrossRefGoogle Scholar
  26. [G]
    M. A. de Gosson, The Principles of Newtonian and Quantum Mechanics, Imperial College Press, London, 2001.CrossRefMATHGoogle Scholar
  27. [HS]
    P. Howard and A. Sukhtayev, The Maslov and Morse indices for Schrödinger operators on [0, 1], J. Differential Equations 260 (2016), 4499–4549.MathSciNetCrossRefMATHGoogle Scholar
  28. [JLM]
    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.MathSciNetCrossRefMATHGoogle Scholar
  29. [JLS]
    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.CrossRefMATHGoogle Scholar
  30. [Ka]
    Y. Karpeshina, Perturbation Theory for the Schrödinger Operator with a Periodic Potential, Springer-Verlag, Berlin, 1997.CrossRefMATHGoogle Scholar
  31. [K]
    T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980.MATHGoogle Scholar
  32. [MS]
    D. McDuff and D. Salamon Introduction to Symplectic Topology, second edition, Clarendon Press, Oxford, 1998.MATHGoogle Scholar
  33. [M]
    J. Milnor, Morse Theory, Princeton Univ. Press, Princeton, NJ, 1963.MATHGoogle Scholar
  34. [PF]
    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.MathSciNetMATHGoogle Scholar
  35. [PW]
    A. Portaluri and N. Waterstraat, A Morse-Smale index theorem for indefinite elliptic systems and bifurcation, J. Differential Equations 258, 1715–1748.Google Scholar
  36. [RS]
    M. Reed and B. Simon, Methods of Modern Mathematical Physics. Volume 1, Academic Press, London, 1980.MATHGoogle Scholar
  37. [RS93]
    J. Robbin and D. Salamon, The Maslov index for paths, Topology 32 (1993), 827–844.MathSciNetCrossRefMATHGoogle Scholar
  38. [RS95]
    J. Robbin and D. Salamon, The spectral flow and the Maslov index, Bull. London Math. Soc. 27 (1995), 1–33.MathSciNetCrossRefMATHGoogle Scholar
  39. [SW]
    D. Salamon and K. Wehrheim, Instanton Floer homology with Lagrangian boundary conditions, Geom. Topol. 12 (2008), 747–918.MathSciNetCrossRefMATHGoogle Scholar
  40. [S]
    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.MathSciNetMATHGoogle Scholar
  41. [Sw]
    R. C. Swanson, Fredholm intersection theory and elliptic boundary deformation problems, J. Differential Equations 28 (1978), I, 189–201, II, 202–219.MathSciNetCrossRefGoogle Scholar
  42. [T]
    M. E. Taylor, Partial Differential Equations I. Basic Theory, Springer-Verlag, 2011.CrossRefMATHGoogle Scholar
  43. [U]
    K. Uhlenbeck, The Morse Index Theorem in Hilbert spaces, J. Differential Geom. 8 (1973), 555–564.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  • Yuri Latushkin
    • 1
  • Selim Sukhtaiev
    • 2
  • Alim Sukhtayev
    • 3
  1. 1.Department of MathematicsThe University of MissouriColumbiaUSA
  2. 2.Department of MathematicsRice UniversityHoustonUSA
  3. 3.Department of MathematicsMiami UniversityOxfordUSA

Personalised recommendations