The Aharonov-Bohm Solenoids in a Constant Magnetic Field

Abstract

We study the spectral properties of a two-dimensional magnetic Schrödinger operatorThe magnetic field is given by rot . . . , N) and the points {z j } N j=1 are uniformly separated. We give an upper bound for the number of eigenvalues of H N between two Landau levels or below the lowest Landau level, when N is finite. We prove the spectral localization of H N near the spectrum of the single solenoid operator, when {z j } N j=1 are far from each other, all the values {α j } N j=1 are the same, and the boundary conditions at z j are uniform. We determine the deficiency indices of the minimal operator and give a characterization of self-adjoint extensions of the minimal operator.

The spectrum of Schrödinger operators with random \delta magnetic fields

“…by Geȋler [16], Avishai-Redheffer-Band [6], Avishai-Redheffer [5], AvishaiAzbel-Gredeskul [4] and Dorlas-Macris-Pulé [14], or in the case of δ magnetic fields [19], [20], [32], [27]. It may be interesting to compare the above results with those in the case of regular potentials (see Zak [34], DinaburgSinai-Soshnikov [13]); in that case, it is widely believed that the Landau levels are broadened and there exist some extended states corresponding to the center of the Landau level.…”

“…The case α γ = 0 is (essentially) given in [2,Chapter I.5], and the case α = 1 is reduced to the case α = 0 by the unitary transform U u = e −iθγ u. The case 0 < α γ < 1 is given in [15] or [27,Lemma 5.15]. …”

“…Especially, GeylerGrishanov [19] and Geyler-Šťovíček [20] studied the infinite degeneracy of the zero modes of the 2-dimensional Pauli operator with δ magnetic fields; Rozenblum-Shirokov [32] also studied the same subject in the case the magnetic field is a signed Borel measure. One of the authors [27] studied the structure of the whole spectrum of the Schrödinger operators with a constant magnetic field plus δ magnetic fields, in the case the number of δ fields is finite, or in the case δ fields are well-separated; the authors [28] also studied the same subject in the case δ fields vary periodically. In the present paper, we consider the case there is some randomness in the positions of δ magnetic fields or in their intensities, and study some fundamental spectral properties of the Schrödinger operators with random δ magnetic fields.…”

The heat kernel for two Aharonov–Bohm solenoids in a uniform magnetic field

“…A thorough and detailed analysis of the spectral properties of H Λ due to T. Mine can be found in [19]. The spectrum spec(H Λ ) is known to be positive and pure point.…”

“…This quantum model has been approached also with some completely different techniques like asymptotic methods for largely separated AB solenoids, semiclassical analysis and a complex scaling method [11,28,3,4]. One may also mention more complex models comprising, apart of AB magnetic fluxes, also additional potentials or magnetic fields [16,19], or models with an arbitrary finite number of AB solenoids or even with countably many solenoids arranged in a lattice [26,17,18]. On the other hand, the method stemming from the original ideas of Schulman and Sunada turned out to be fruitful also in analysis of other interesting models like Brownian random walk on the twice punctured plane [10,7].…”