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Localization and Multiplicity of a Self-Adjoint Elliptic 2×2 Positive NCHO in \(\mathbb{R}^n\)

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1992)

Abstract

Using the machinery developed in the past chapters, we can now describe the beautiful connection between the spectrum of a self-adjoint elliptic 2×2 positive NCHO in n variables, and the periods of the bicharacteristics of the principal symbol (i.e., in the case of systems, the periods of the bicharacteristic curves associated with the eigenvalues of the principal symbol). This connection is a very deep result and its history, in the scalar case, is based on fundamental papers by Chazarain, Coline de Verdière, Duistermaat and Guillemin, Ivrii, Helffer and Robert (see also Cardoso and Mendoza [4]), and many others (for such results in the semiclassical setting, see Chazarain [5], Dimassi-Sjöstrand [7], Dozias [8], Helffer-Robert [18, 20], Ivrii [34], Robert [65]). However, very few are the results for systems (see Ivrii [34], Parmeggiani [52, 55]). Our results here (see Section 12.3) for elliptic NCHOs in Rn, whose symbols have distinct eigenvalues, are based on the approach followed in Parmeggiani [55]. In Section 12.4 we shall specialize these results to NCHOs of the kind Qw (α,β)(x,D) (with α,β > 0 and αβ > 1), and in the final Section 12.5 we shall give a slightly more general result (namely, when the subprincipal part of the diagonalization has constant average on periodic bicharacteristics).

Keywords

  • Large Eigenvalue
  • Energy Interval
  • Principal Symbol
  • Maslov Index
  • Ferential Operator

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Correspondence to Alberto Parmeggiani .

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Parmeggiani, A. (2010). Localization and Multiplicity of a Self-Adjoint Elliptic 2×2 Positive NCHO in \(\mathbb{R}^n\) . In: Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction. Lecture Notes in Mathematics(), vol 1992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11922-4_12

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