Local Spectral Asymptotics for the Schrödinger Operator with Strong Magnetic Field

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 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.