Communications in Mathematical Analysis

Fredholm property and essential spectrum of pseudodifferential operators with operator-valued symbols

Gennadiy Burlak , Yuri Karlovich , and Vladimir Rabinovich

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We consider the Fredholm property and essential spectrum of pseudodifferential operators with symbols having values in the space of bounded operators acting from a separable Hilbert space ${\mathcal{H}}_{1}$ into a separable Hilbert space ${\mathcal{H}}_{2}$ and satisfying additional estimates. There are many problems in mathematical physics which are reduced to the study of associated pseudodifferential operators with operator-valued symbols. In particular, this happens for problems of wave propagation in acoustic, electromagnetic and quantum waveguides. We consider so-called locally Fredholm pseudodifferential operators with operator-valued symbols for which the study of their Fredholmness is reduced to the problem of local invertibility at infinity. For investigation of the local invertibility at infinity we use the method of limit operators. We give applications of general results to the investigation of the essential spectra of Schrödinger operators with the Robin condition at the boundary in a cylindrical domain $\mathcal{D}=\mathbb{R}^{n}\times\Omega ,$ where $\Omega $ is a bounded domain in $\mathbb{R}^{m}$ with a smooth boundary ( quantum waveguides). We obtain the exact descriptions of the essential spectra of such operators in terms of the spectra of limit operators. In the case of slowly oscillating at infinity electric potentials the description of the essential spectra has an explicit form.

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Commun. Math. Anal., Conference 3 (2011), 50-65.

First available in Project Euclid: 25 February 2011

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Burlak , Gennadiy; Karlovich , Yuri; Rabinovich , Vladimir. Fredholm property and essential spectrum of pseudodifferential operators with operator-valued symbols. Commun. Math. Anal. (2011), no. 3, 50--65.

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  • R. Beals, Characterization of pseudodifferential operators and applications. Duke Math. J. 44 (1977), pp 45-57.
  • V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskiy, Operator separation of variables for adiabatic problems in quantum and wave mechanics. J. Engineering Math. 55 (2006), No. 1-4, pp 183-237.
  • H. L. Cycon, R. G. Froese, W. Kirsh, and B. Simon, Schrö dinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin, Heidelberg, New York 1987.
  • J. Dittrich and J. Kříž, Bound states in straight quantum waveguides with combined boundary conditions. J. Math. Phys. 43 (2002), pp 3892-3915.
  • J. Dittrich and J. Kříž, Curved planar quantum wires with Dirichlet and Neumann boundary conditions. J. Phys. A: Math. Gen. 35 (2002), pp L269-L275.
  • P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995), pp 73-102.
  • P. Duclos, P. Exner, and B. Meller, Open quantum dots: Resonances from perturbed symmetry and bound states in strong magnetic fields. Rep. Math. Phys. 47 (2001), pp 253-267.
  • P. Duclos, P. Exner, and B. Meller, Exponential bounds on curvature induced resonances in a two-dimensional Dirichlet tube. Helv. Phys. Acta. 71 (1998), pp 477-492.
  • P. Duclos, P. Exner, and P. Stov\i cek, Curvature induced resonances in a two-dimensional Dirichlet tube. Ann. Inst. Henri Poincare 62 (1995), No. 1, pp 81-101.
  • T. Ekholm and H. Kovar\i k, Stability of the magnetic Schrödinger operator in a waveguide. Comm. Part. Diff. Eq. 30 (2005), pp 539-565.
  • L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo- Differential Operators. Springer, New York, Berlin, Heidelberg 2007.
  • Y. Last, and B. Simon, The essential spectrum of Schrö dinger, Jacobi, and CMV operators. Journal D'Analysis Math. 98 (2006), pp 183-220.
  • S. Levendorskii, Asymptotic Distribution of Eigenvalues of Differential Operators. Kluwer, Dordrecht 1990.
  • S. Levendorskii, Degenerate Elliptic Equations. Kluwer, Dordrecht 1994.
  • M. Mantoiu, R. Purice, and S. Richard, Spectral and propagation results for magnetic Schrödinger operators; A C*-algebraic framework, Journal of Functional Analysis. 250 (2007), pp 42-67.
  • V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory. Birkhäuser Verlag, Basel, Boston, Berlin 2004.
  • V. Rabinovich, Essential spectrum of perturbed pseudodifferential operators. Applications to the Schrödinger, Klein-Gordon, and Dirac operators. Russian Journal of Math. Physics. 12 (2005), No.1, pp 62-80.
  • V. S. Rabinovich, An Introductionary Course on Pseudodifferential Operators. Instituto Superior Técnico. Textos de Matemática. Centro de Matemática Aplicada 1, Lisbon 1998, 79 p.
  • V. S. Rabinovich and S. Roch, Essential Spectra of Pseudodifferential Operators and Exponential Decay of their Solutions. Applications to Schrödinger Operators. Operator Algebras, Operator Theory and Applications, Ser. Oper. Theory: Adv. Appl. 181 (2008), pp 335-384.
  • M. A. Shubin, Pseudodifferential Operators and Spectral Theory, second ed., Springer 2001.
  • M. E. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey 1981.