Local Spectral Asymptotics for the Schrödinger Operator with Strong Magnetic Field
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 μ ≤ h −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 (μ ≤ h δ−1) and in section 6.4 we derive non-Weylian formulas in the case of a strong magnetic field (μ ≥ h δ−1); these formulas permit us to justify the (quasi-)Weylian formulas under weaker restrictions than those of section 6.3.
KeywordsIntegral Formula Total Contribution Extra Term Principal Symbol Magnetic Line
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