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Weakly coupled bound states of Pauli operators

Sergey Morosov
(UCL, London)
Analysis Seminar
Tuesday, 9 November, 2010, at 15:15, in Aud. G1 (1532-116)

We will consider two-dimensional Pauli operator $P$ with finite normalized flux $F$ of magnetic field. It turns out that for any non-negative, compactly supported function $V$ and $\alpha > 0$ the perturbed operator $P - \alpha V$ has a finite number of negative eigenvalues. The amount of these eigenvalues has a finite limit as $\alpha$ approaches zero, which differs by one or two from the dimension $N$ of the kernel of $P$. We are interested in the asymptotic behavior of these eigenvalues in the limit of small positive $\alpha$.

We will see that $N$ of the eigenvalues behave like $-C\alpha$ with some appropriate $C$, but the "additional" one or two eigenvalues have quite different asymptotics. For radially symmetric operators we will prove that:

  1. If the magnetic flux $F$ is non-integer, the single "extra" eigenvalue $\lambda_{N+1}$ is, to the first order, equal to $-C\alpha^{1/\mu}$, with $\mu$ being the fractional part of $F$;
  2. If $F$ is integer, then $\lambda_{N+1}$ is, to the first order, proportional to $\alpha/|\log(\alpha)|$, while $|\log(\lambda_{N+2})|$ is proportional to $-1/\alpha$.

The proof is based on purely variational ideas.

Contact person: Søren Fournais