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Local Spectral Asymptotics for the Schrödinger Operator with Strong Magnetic Field

  • Victor Ivrii
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we analyze two- and three-dimensional Schrödinger operators with strong magnetic field; now we have not only a small parameter h but a large parameter μ (a coupling constant with the magnetic field or simply a magnetic parameter); moreover, a natural condition μh −1 arises. In section 6.1, which has a preliminary character, we obtain a formula for e(x,x,τ), for operators in ℝ d with constant g jk ,V and F jk (which is the tensor intensity of the magnetic field) and we also prove some preliminary assertions. In section 6.2 we obtain microlocal canonical forms for d = 2, 3 which are the basis for the advanced analysis. In section 6.3 we derive (quasi-)Weylian and related formulas for d = 2,3 in the case of a weak magnetic field (μh δ−1) and in section 6.4 we derive non-Weylian formulas in the case of a strong magnetic field (μh δ−1); these formulas permit us to justify the (quasi-)Weylian formulas under weaker restrictions than those of section 6.3.

Keywords

Integral Formula Total Contribution Extra Term Principal Symbol Magnetic Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Victor Ivrii
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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