Differential Analysis on Complex Manifolds

Wells, Raymond O.

doi:10.1007/978-0-387-73892-5

Cited by 210 publications

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“…see [Wel80]. In [Shi91] it was shown that the Dirac and Pauli operators can be defined in terms of W and operators acting on differential forms in a very nice way.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Zero Modes for the Magnetic Pauli Operator in Even-Dimensional Euclidean Space

Persson

2008

Lett Math Phys

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Abstract. We study the ground state of the Pauli Hamiltonian with a magnetic field in R 2d . We consider the case where a scalar potential W is present and the magnetic field B is given by B = 2i ∂∂ W . The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993. Introduction and main resultThe Pauli operator P in R n describes a charged spin-1 2 particle in a magnetic field. Along with the Dirac operator, it lies in the base of numerous models in quantum physics. The problem about zero modes, the bound states with zero energy, is one of many questions to be asked about the spectral properties of these operators.Zero modes were discovered in [AC79] in dimension n = 2. Unlike the purely electric interaction, a compactly supported magnetic field can generate zero modes, as soon as the total flux of the field is sufficiently large. Quantitatively, this is expressed by the famous Aharonov-Casher formula. The two-dimensional case is by now quite well studied; the AC formula is extended to rather singular magnetic field, moreover, if the total flux is infinite (and the field has constant sign), there are infinitely many zero modes.On the other hand, in the three-dimensional case the presence of zero modes is a rather exceptional feature, and the conditions for them to appear are not yet found, see the discussion in [MR03] and references therein.Even less clear is the situation in the higher dimensions. In [Shi91], for even n some sufficient conditions for the infiniteness of the number of zero modes were found, requiring, in particular, that the field decays rather slowly (more slowly than r −2 ) at infinity. On the other hand, in [Ogu93], again for even n, the case where a finite number of zero modes should appear was considered. Under the assumption of a rather regular behavior of the scalar potential of the magnetic field at infinity the number of zero modes was calculated. In particular, for a field with compact support or decaying faster than quadratically at infinity the formula in [Ogu93] implies the absence of zero modes, thus making a difference with the two-dimensional situation.2000 Mathematics Subject Classification. 81Q05 (Primary); 35Q40, 81Q10 (Secondary).

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“…see [Wel80]. In [Shi91] it was shown that the Dirac and Pauli operators can be defined in terms of W and operators acting on differential forms in a very nice way.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We know from [Wel80] that solutions W exist, the essential part of this Lemma is that there exist a bounded solution to (1.5).…”

Section: Proof Ofmentioning

confidence: 99%

Zero Modes for the Magnetic Pauli Operator in Even-Dimensional Euclidean Space

Persson

2008

Lett Math Phys

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“…It is well known [1,2] that elliptic differential and pseudodifferential operators on a closed infinitely smooth manifold are Fredholm between appropriate Sobolev spaces. This fundamental property is used in the theory of elliptic differential equations and elliptic boundary-value problems.…”

Section: Introductionmentioning

confidence: 99%

“…However, elliptic operators on vector bundles have not been covered by this theory. These operators have important applications to elliptic boundary problems on vector bundles, elliptic complexes, spectral theory of elliptic differential operators, and others (see, e.g., [1,2,38]). …”

Section: Introductionmentioning

confidence: 99%

Elliptic operators on refined Sobolev scales on vector bundles

Zinchenko

2017

Open Mathematics

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: We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.

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“…In §2 we present a very brief sketch of sheaf theory for the benefit of readers less acquainted with the subject. This is intended to provide the bare minimum of information necessary for following the ensuing discussion; for further details on the sheaf theory, see Bredon (1997), Kultze (1970), or Wells (2008). In §3 we develop the basic mathematical machinery used in later sections.…”

Section: Introductionmentioning

confidence: 99%