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The spectrum of a hydrogen atom in an intense magnetic field. (English) Zbl 0814.35107

Rev. Math. Phys. 6, No. 5, 699-832 (1994); errata ibid. 8, No. 5, 761-762 (1996).
The authors consider a hydrogen atom in a constant magnetic field \(B\), \(3 \cdot 10^ 9G \ll B \ll 4 \cdot 10^{13}G\), pointed to \(x_ 3\) direction. If \(L^ 2 (R^ 3)\) is decomposed into a direct sum of eigenspaces of \(i \partial/ \partial \theta\), \(\theta = \arctan (x_ 2/x_ 1)\), the Hamiltonian \(H\) of this system splits into \(H = m_ e e^ 4/ \hbar^ 2 \oplus^ \infty_{m = - \infty} H^{(m)}\), \[ H^{(m)} = \lambda^{-2} (- \partial^ 2/ \partial r^ 2 - 1/r \cdot \partial/ \partial r + m^ 2/r^ 2 + r^ 2 + 2m) - \partial^ 2/ \partial z^ 2 - 1/ \sqrt {\lambda^ 2r^ 2 + z^ 2}, \] where \(\lambda = (2m^ 2_ e e^ 3c/(\hbar^ 3 B))^{1/2}\). They expect that relativistic effect is small and \(\rho_{\text{mag}}\) is smaller than the Bohr radius.
Results: There is some \(\lambda_ 0\) with \(0 < \lambda_ 0 < 1\) such that if \(\lambda \leq \lambda_ 0\), the followings are true: There is a unique eigenvalue \(\delta_ 0 \leq - 1\) of \(H^{(m)} - \lambda^{-2} E_ 0^{(m)}\) by \(E_ 0^{(m)} = (m + | m | + 1)\) satisfying \(\ln (1/ \lambda) = \sqrt {-\delta_ 0} + \ln (\sqrt {- \delta_ 0}) + 0(1)\). The eigenfunction \(\varphi_ 0 (r,z)\) corresponding to \(\delta_ 0\) satisfies \[ \biggl \| \varphi_ 0 (r,z) - \sqrt {4 \sqrt {- \delta_ 0}/ | m |!} r^{| m |} e^{-r^ 2/2} e^{- \sqrt {-\delta_ 0} | z |} \biggr \| \leq C \ln \bigl( \ln (1/ \lambda) \bigr)/ \ln (1/ \lambda). \] The asymptotics for the eigenvalues \(\delta_ k^{(\pm)}\) and the eigenfunction \(\varphi_ k^{(\pm)} (r,z)\), \(k = \{1,2,3, \dots\}\) of \(H^{(m)} - \lambda^{-2} E_ 0^{(m)}\) and rigorous error bounds dependent on \(\lambda\) are also given.
Reviewer: H.Yamagata (Osaka)

MSC:

35Q40 PDEs in connection with quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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