Discrete spectrum of many body Schrödinger operators with non-constant magnetic fields I

Discrete spectrum of many body Schrödinger operators with non-constant magnetic fields II

“…We now prove the main theorem by using Rayleigh-Ritz method. Proof of Theorem 1. by the same method as in [10]. This completes the proof of Theorem 1.1.…”

“…In this case the finiteness or the infiniteness of the discrete spectrum of H N z (b) depends not only on N and Z but also on magnetic fields. In fact we have the following result in [10]. Our main result of this paper is the following theorem, which is contrastive to the above.…”

“…In this case the finiteness or the infiniteness of the discrete spectrum of H N z (b) depends not only on N and Z but also on magnetic fields. In fact we have the following result in [10]. THEOREM 0.3 For any positive number ε there exists a vector potential b ε €= C (R ) , which is independent of N and Z, such that the number of the discrete spectrum of H NZ (b) is finite for Z > ε and N > 1.…”

“…general scalar potentials (see [10] [10]). Then the next lemma follows from the proof of Lemma 2.5 in [10].…”

“…This paper is continuation from [10], in which we studied the discrete spectrum of atomic Hamiltonians with non-constant magnetic fields and, more precisely, we showed that any atomic system has only finitely many bound states, corresponding to the discrete energy levels, in a suitable magnetic field. In this paper we show another phenomenon in non-constant magnetic fields that any atomic system has infinitely many bound states in a suitable magnetic field.…”