We consider a two-dimensional electron with an anomalous magnetic moment, <i>g</i>&gt;2, interacting with a nonzero magnetic field <i>B</i> perpendicular to the plane which gives rise to a flux <i>F</i>. Recent results about the discrete spectrum of the Pauli operator are extended to fields with the \mathcal{O}\left( {r^{ - 2 - \delta } } ight) decay at infinity: we show that if \mathcal{O}\left( {r^{ - 2 - \delta } } ight) exceeds an integer <i>N</i>, there is at least \mathcal{O}\left( {r^{ - 2 - \delta } } ight) bound states. Furthermore, we prove that weakly coupled bound states exist under mild regularity assumptions also in the zero flux case.

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We analyse two-dimensional Schrdinger operators with the potential| xy| p (x 2+ y 2) p/(p+ 2) where p 1 and 0. We show that there is a critical value of such that the spectrum for &lt; crit is bounded below and purely discrete, while for &gt; crit it is unbounded from below. In the subcritical case, we prove upper and lower bounds for the eigenvalue sums.

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