Ground state of a spin 1/2 charged particle in an even-dimensional magnetic field

Ogurisu, Osamu

doi:10.1007/bf00750959

Cited by 4 publications

(15 citation statements)

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“…This is the reason for us to return to the question on zero modes in the even higher-dimensional case. We try to revive the results in [Ogu93] and succeed partially.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…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.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…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.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The mistake in [Ogu93] occurs in calculating the L 2 norm of the form one gets after applying the∂ operator. We present the detailed analysis of this miscalculation in Section 3.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The strategy of our treatment of zero modes differs from the one in [AC79] and other previous papers including [Ogu93]. Usually, when studying zero modes, one shows first that they, after having been multiplied by some known factor, are holomorphic function in the whole space; then one easily counts the number of such functions.…”

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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“…This is the reason for us to return to the question on zero modes in the even higher-dimensional case. We try to revive the results in [Ogu93] and succeed partially.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The mistake in [Ogu93] occurs in calculating the L 2 norm of the form one gets after applying the∂ operator. We present the detailed analysis of this miscalculation in Section 3.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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