Magnetic Schrödinger operators with compact resolvent

Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators

“…This means that a magnetic field can only improve the situation from our point of view. Papers by J. Avron, I. Herbst and B. Simon [1], Y. Colin de Verdière [4], A. Dufresnoy [6] and A. Iwatsuka [14] provide some quantitative results which show that even in case V = 0 the magnetic field can make the spectrum discrete. (This situation is called magnetic bottle.…”

“…Other, more effective sufficient conditions (which do not include µ 0 ) and related results (in particular, asymptotics of eigenvalues under appropriate conditions) can be found in [4,6,8,11,12,13,14,15,20,23,29,32,34].…”

Asymptotic Distribution of Eigenvalues for Schrödinger Operators with Magnetic Fields

“…The following the assumptions imply the compactness property of (H + ί)~\ so that H has only discrete spectrum: Here we should refer to the results of [4] and [6] on the compactness property of (H + ί)~\ Consider the condition The compactness of (H + i)" 1 has been proved by [4] under (A.2) and (B), with p = = 3/2 and by [6] under (A.2) and (B), with p = 2. Furthermore, it has been shown that the compactness property does not follow from (A.2) only.…”

“…Furthermore, it has been shown that the compactness property does not follow from (A.2) only. In particular, Iwatsuka [6] has obtained that p = 2 is the border line for the compactness, by constructing an example in which b(x) satisfies \Vb,(x)\ = O(b{xf) but (H + i)~l is not compact. ext we shall discuss the problem on the semi-classical asymptotic distribution of eigenvalues.…”

“…We assume (A.I), (A. ), H h has only discrete spectrum by the above result due to Iwatsuka [6]. THEOREM …”