Abstract
We consider the Schur complement S(λ) with real spectral parameter λ corresponding to a certain 3 × 3 block operator matrix. In our case the essential spectrum of S(λ) can have gaps. We obtain formulas for the number and multiplicities of eigenvalues belonging to an arbitrary interval outside the essential spectrum of S(λ).
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Hübner M. and Spohn H., “Spectral properties of the spin-boson Hamiltonian,” Ann. Inst. H. PoincarÉ, 62, No. 3, 289–323 (1995).
Minlos R. A. and Spohn H., “The three-body problem in radioactive decay: the case of one atom and at most two photons,” in: Topics in Statistical and Theoretical Physics, Amer. Math. Soc., Providence, 1996, pp. 159–193 (Amer. Math. Soc. Transl., Ser. 2; V. 177).
Mogilner A. I., “Hamiltonians in solid state physics as multiparticle discrete Schr¨odinger operators: problems and results,” Adv. Sov. Math., 5, 139–194 (1991).
Friedrichs K. O., Perturbation of Spectra in Hilbert Space, Amer. Math. Soc., Providence (1965).
Malyshev V. A. and Minlos R. A., Linear Infinite-Particle Operators, Amer. Math. Soc., Providence (1995) (Transl. Math. Monogr.; V. 143).
Lifschitz A. E., Magnetohydrodynamic and Spectral Theory, Kluwer Academic Publishers, Dordrecht (1989) (Dev. Electromagn. Theory Appl.; V. 4).
Thaller B., The Dirac Equation, Springer-Verlag, Berlin, Heidelberg, and New York (1992.).
Feynman R. P., Statistical Mechanics: a Set of Lectures (2nd ed.), Addison-Wesley, Reading (1998).
Tretter C., Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London (2008).
Schur I., “über potenzreihen, die im innern des einheitskreises beschr¨ankt sint,” J. Reine Angew. Math., 147, 205–232 (1917).
Haynsworth E. V., “Determination of the inertia of a partitioned Hermitian matrix,” Linear Algebra Appl., 1, No. 1, 73–81 (1968).
Kreĭn M. G., “The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I,” Mat. Sb., 20, 365–404 (1947).
Zhang F., The Schur Complement and Its Applications, Springer-Verlag, New York (2005) (Numer. Methods Algorithms; V. 4).
Bart H., Gohberg I. C., Kaashoek M. A., and Ran A. C. V., “Schur complements and state space realizations,” Linear Algebra Appl., 399, 203–224 (2005).
Nagel R., “Well-posedness and positivity for systems of linear evolution equations,” Conf. Sem. Mat. Univ. Bari, 203, 1–29 (1985).
Nagel R., “The spectrum of unbounded operator matrices with non-diagonal domain,” J. Func. Anal., 89, No. 2, 291–302 (1990).
Atkinson F. V., Langer H., Menniken R., and Shkalikov A. A., “The essential spectrum of some matrix operators,” Math. Nachr., 167, 5–20 (1994).
Lakaev S. N., “Some spectral properties of the generalized Friedrichs model,” J. Soviet Math., 45, No. 6, 1540–1554 (1989).
Boldrighini C., Minlos R. A., and Pellegrinotti A., “Random walks in a random (fluctuating) environment,” Russian Math. Surveys, 62, No. 4, 663–712 (2007).
Lakshtanov E. L. and Minlos R. A., “The spectrum of two-particle bound states for the transfer matrices of Gibbs fields (an isolated bound state),” Funct. Anal. Appl., 38, No. 3, 202–216 (2004).
Akchurin E. R., “Spectral properties of the generalized Friedrichs model,” Theoret. Math. Phys., 163, No. 1, 414–428 (2010).
Motovilov A. K., Sandhas W., and Belyaev Y. B., “Perturbation of a lattice spectral band by a nearby resonance,” J. Math. Phys., 42, 2490–2506 (2001).
Lakaev S. N. and Rasulov T. Kh., “A model in the theory of perturbations of the essential spectrum of multiparticle operators,” Math. Notes, 73, No. 4, 521–528 (2003).
Reed M. and Simon B., Methods of Modern Mathematical Physics. Vol. 4: Analysis of Operators, Academic Press, New York (1978).
Muminov M. È., “Expression for the number of eigenvalues of a Friedrichs model,” Math. Notes, 82, No. 1, 67–74 (2007).
Sobolev A. V., “The Efimov effect. Discrete spectrum asymptotics,” Comm. Math. Phys., 156, No. 1, 101–126 (1993).
Birman M. Sh. and Solomyak M. Z., Spectral Theory of Selfadjoint Operators in Hilbert Space, D. Reidel Publ. Co., Dordrecht (1987).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2015 Muminov M. É. and Rasulov T.Kh.
The authors were supported by the Program of the Einstein Fund of the International Mathematical Society. The second author is grateful to the Berlin Mathematical School and the Weierstrass Institute for Applied Analysis and Stochastics for invitation, support, and hospitality.
Skudai; Bukhara. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 878–895, July–August, 2015; DOI: 10.17377/smzh.2015.56.412.
Rights and permissions
About this article
Cite this article
Muminov, M.É., Rasulov, T.K. An eigenvalue multiplicity formula for the Schur complement of a 3 × 3 block operator matrix. Sib Math J 56, 699–713 (2015). https://doi.org/10.1134/S0037446615040126
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446615040126