Asymptotic distribution of eigenvalues for Schrödinger operators with homogeneous magnetic fields. II (Q1813918)

[For part I, see ibid. 25, 633-647 (1988, Zbl 0731.35073).] Considered is the selfadjoint realization in \(L^ 2(\mathbb{R}^{3N})\) of the Hamiltonian \[ H=\sum^ N_{j=1}\{{1\over 2}\mu_ jT^ 2_ j+V_{0j}(x^ j)\}+\sum_{1\leq i<j<n}V_{ij}(x^ j-x^ i) \] for an \(N\)-particle system where \(T_ j=-i\nabla_ j+A_ j(x^ j)\), \(A_ j(e_ j/2)B\times x^ j\). Here \(\mu_ j\) is the mass and \(e_ j\) the charge of the particle with coordinates \(x^ j\) \((1\leq j\leq N\)), the \(V_{ij}\) are potentials, and \(B=(0,0,b)\) with \(b\neq0\) is a homogeneous magnetic field. Let \(\Sigma(H)\) be the infimum of the essential spectrum of \(H\). For \(\lambda>0\), \(N(\lambda)\) denotes the number of the eigenvalues of \(H\) not exceeding \(\Sigma(H)-\lambda\), counted according to their multiplicity. Under certain assumptions, an asymptotic formula for \(N(\lambda)\) as \(\lambda\to0\) is proved. In particular, \(N(\lambda)\to\infty\) as \(\lambda\to0\). This is not always true if \(B=0\).