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20
Magnetic Bottles in Connection With Superconductivity
, 2001
"... Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semiclassical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here ..."
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Cited by 41 (13 self)
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Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semiclassical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here we would like to mention the works by BernoffSternberg, LuPan and Del PinoFelmerSternberg. This recovers partially questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significatively the lower bounds and as a byproduct we solve a conjecture proposed by BernoffSternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields.
Compactness in the ∂Neumann problem, Magnetic Schrödinger operators, and the AharonovBohm effect, Adv
"... Abstract. Compactness of the Neumann operator in the dbar Neumann problem is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth example is given in which condition (P) fails to hold, yet the Neumann operator is compact. The main result, in contrast, is that for s ..."
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Cited by 20 (2 self)
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Abstract. Compactness of the Neumann operator in the dbar Neumann problem is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth example is given in which condition (P) fails to hold, yet the Neumann operator is compact. The main result, in contrast, is that for smoothly bounded Hartogs domains, condition (P) of Catlin and Sibony is equivalent to compactness. The analyses of both compactness and condition (P) boil down to properties of the lowest eigenvalues of certain sequences of Schrodinger operators, with and without magnetic fields, parametrized by a Fourier variable resulting from the Hartogs symmetry. The nonsmooth counterexample is based on the AharonovBohm phenomenon of quantum mechanics. For smooth domains, we prove that there always exists an exceptional sequence of Fourier variables for which the AharonovBohm effect is weak. This sequence can be very sparse, so that the failure of compactness is due to a rather subtle effect. 1.
Nodal Sets, Multiplicity and Superconductivity in Non Simply Connected Domains
, 2001
"... This is a survey on [HHOO] and further developments of the theory [He4]. We explain in detail the origin of the problem in superconductivity as first presented in [BeRu], recall the results of [HHOO] and explain the extension to the Dirichlet case. As illustration of the theory, we detail some semi ..."
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Cited by 5 (5 self)
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This is a survey on [HHOO] and further developments of the theory [He4]. We explain in detail the origin of the problem in superconductivity as first presented in [BeRu], recall the results of [HHOO] and explain the extension to the Dirichlet case. As illustration of the theory, we detail some semiclassical aspects and give examples where our estimates are sharp.
Spectral Theory For The Dihedral Group.
"... Let H = \Gamma\Delta + V be a twodimensional Schrodinger operator defined on a bounded with Dirichlet boundary conditions on @ Suppose that H commutes with the actions of the dihedral group D 2n , the group of the regular ngone. We analyze completely the multiplicity of the groundstate eige ..."
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Cited by 4 (4 self)
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Let H = \Gamma\Delta + V be a twodimensional Schrodinger operator defined on a bounded with Dirichlet boundary conditions on @ Suppose that H commutes with the actions of the dihedral group D 2n , the group of the regular ngone. We analyze completely the multiplicity of the groundstate eigenvalues associated to the different symmetry subspaces related to the irreducible representations of D 2n . In particular we find that the multiplicities of these groundstate eigenvalues equal the degree of the corresponding irreducible representation. We also obtain an ordering of these eigenvalues. In addition we analyze the qualitative properties of the nodal sets of the corresponding ground state eigenfunctions.
Zero modes in a system of AharonovBohm fluxes
 Rev. Math. Phys
"... We study zero modes of twodimensional Pauli operators with Aharonov–Bohm fluxes in the case when the solenoids are arranged in periodic structures like chains or lattices. We also consider perturbations to such periodic systems which may be infinite and irregular but they are always supposed to be ..."
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Cited by 4 (0 self)
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We study zero modes of twodimensional Pauli operators with Aharonov–Bohm fluxes in the case when the solenoids are arranged in periodic structures like chains or lattices. We also consider perturbations to such periodic systems which may be infinite and irregular but they are always supposed to be sufficiently scarce. 1.
PERIODIC SCHRÖDINGER OPERATORS AND AHARONOV–BOHM HAMILTONIANS
"... Abstract. Let H = − ∆ + V be a twodimensional Schrödinger operator defined on a domain Ω ⊂ R 2 with Dirichlet boundary conditions. Suppose that H and Ω are invariant with respect to translations in the x1 direction, so that V (x1, x2) = V (x1 +1, x2); suppose in addition that V (x1, x2) = V (−x1, ..."
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Cited by 3 (2 self)
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Abstract. Let H = − ∆ + V be a twodimensional Schrödinger operator defined on a domain Ω ⊂ R 2 with Dirichlet boundary conditions. Suppose that H and Ω are invariant with respect to translations in the x1 direction, so that V (x1, x2) = V (x1 +1, x2); suppose in addition that V (x1, x2) = V (−x1, x2) and that (x1, x2) ∈ Ω implies (x1 + 1, x2) ∈ Ω and (−x1, x2) ∈ Ω. We investigate the associated Floquet operator H (q) , 0 ≤ q < 1. In particular, we show that the lowest eigenvalue λq is simple for q = 1/2 and strictly increasing in q for 0 < q < 1/2 and that the associated complexvalued eigenfunction uq has empty zero set. For the Dirichlet realization of the Aharonov–Bohm Hamiltonian in an annuluslike domain with an axis of symmetry, HA,V = (i∂x1 + A1) 2 + (i∂x2 + A2) 2 + V, we obtain similar results, where the parameter q is replaced by the 1 2πflux through the hole, under the assumption that the magnetic field curl A vanishes identically.
Stability estimate in an inverse problem for non autonomous magnetic Schrödinger equations
, 2012
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Numerical Analysis of Nodal Sets for Eigenvalues of Aharonov–Bohm Hamiltonians on the Square and Application to Minimal Partitions
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