Abstract
The paper is devoted to exponential estimates of eigenfunctions of the discrete spectrum of matrix Schrödinger operators with variable potentials, and Dirac operators for nonhomogeneous media with variable light speed and variable electric and magnetic potentials, For the study of exponential estimates we apply methods developed in our recent paper [25].
This work was partially supported by the German Research Foundation (DFG).
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References
S. Agmon, Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 4 (1975), 151–218.
S. Agmon, Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations. Princeton University Press, Princeton, 1982.
M.S. Agranovich, Elliptic Operators on Smooth Manifolds. In: Itogi Nauki i Tekhniki, Sovremennie Problemi Matematiki, Fundamentalnie Napravlenia, V. 63, Partial Differential Equations-6 (Russian), 5 130.
E. Buzano, Super-exponential decay of solutions to differential equations inℝr. in: Modern Trends in Pseudo-Differential Operators, Operator Theory: Adv. Appl. 172, Birkhäuser, Basel 2007, 117–133.
M. Cappiello, T. Gramchev, L. Rodino, Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients. J. Functional Anal. 237 (2006), 2, 634–654.
M. Cappiello, T. Gramchev, L. Rodino, Gelfand-Shilov spaces, pseudo-differential operators and localization operators. in: Modern Trends in Pseudo-Differential Operators, Operator Theory: Adv. Appl. 172, Birkhäuser, Basel 2007, 297–312.
H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin, Heidelberg, New York 1987.
P.A.M. Dirac, The Principles of Quantum Mechanics. Oxford: Clarendon Press, 1958.
G. Esposito, Dirac Operators and Spectral Geometry, Cambridge Lecture Notes in Physics, Cambridge, University Press, 1998.
V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer, Berlin Heidelberg New York, 1998.
V.B. Berestecskiy, E.M. Lifshitz, L.P. Pitaevsky, Quantum Electrodynamics, 3 Edition, Nauka, Moskow, 1989 (in Russian).
R. Froese, I. Herbst, Exponential bound and absence of positive eigenvalue for N-body Schrödinger operators. Comm. Math. Phys. 87 (1982), 429–447.
R. Froese, I. Herbst, M. Hoffman-Ostenhof, T. Hoffman-Ostenhof, L 2-exponential lower bound of the solutions of the Schrödinger equation. Comm. Math. Phys. 87 (1982), 265–286.
J.M. Jauch, Foundation of Quantum Mechanics, Addison-Wesley, London, 1973
M. Klein, A. Martinez, R. Seiler, X.P. Wang, On the Born-Oppenheimer expansion for polyatomic molecules. Comm. Math. Phys. 143 (1992), 3, 607–639.
Ya.A. Luckiy, V.S. Rabinovich, Pseudodifferential operators on spaces of functions of exponential behavior at infinity. Funct. Anal. Prilozh. 4 (1977), 79–80.
M. Mântoiu, Weighted estimations from a conjugate operator. Letter Math. Phys. 51 (2000), 17–35.
A. Martinez, Eigenvalues and resonances of polyatomic molecules in the Born-Oppenheimer approximation, in: Schrödinger Operators. The Quantum Mechanical Many-Body Problem, Lecture Notes in Physics 403, Springer-Verlag, Berlin, Heidelberg 1992, 145–152.
A. Martinez, Microlocal exponential estimates and application to tunnelling. In: Microlocal Analysis and Spectral Theory, L. Rodino (Editor), NATO ASI Series, Series C: Mathematical and Physical Sciences 490, 1996, 349–376.
A. Martinez, An Introduction to Semiclassical and Microlocal Analysis. Springer, New York 2002.
S. Nakamura, Agmon-type exponential decay estimates for pseudodifferential operators. J. Math. Sci. Univ. Tokyo 5 (1998), 693–712.
L. Nedelec, Resonances for matrix Schrödinger operators. Duke Math. J. 106 (2001), 2, 209–236.
V.S. Rabinovich, Pseudodifferential operators with analytic symbols and some of its applications. In: Linear Topological Spaces and Complex Analysis 2, Metu-Tübitak, Ankara 1995, 79–98.
V. Rabinovich, Pseudodifferential operators with analytic symbols and estimates for eigenfunctions of Schrödinger operators. Z. Anal. Anwend. (J. Anal. Appl.) 21 (2002), 2, 351–370.
V.S. Rabinovich, S. Roch, Essential spectrum and exponential decay estimates of solutions of elliptic systems of partial differential equations. Applications to Schrödinger and Dirac operators. Georgian Math. J. 15 (2008), 2, 333–351.
M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, Heidelberg, New York 2001.
B. Thaller, The Dirac Equation, Springer Verlag, Berlin, Heidelberg, New York 1992.
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Dedicated to Prof. N. Vasilevsky on the occasion of his 60th birthday.
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Rabinovich, V., Roch, S. (2010). Exponential Estimates of Eigenfunctions of Matrix Schrödinger and Dirac Operators. In: Duduchava, R., Gohberg, I., Grudsky, S.M., Rabinovich, V. (eds) Recent Trends in Toeplitz and Pseudodifferential Operators. Operator Theory: Advances and Applications, vol 210. Springer, Basel. https://doi.org/10.1007/978-3-0346-0548-9_12
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