Nodal Sets, Multiplicity and Superconductivity in Non-simply Connected Domains

Helffer, Bernard; Hoffmann‐Ostenhof, M.; Hoffmann‐Ostenhof, Thomas; Owen, Mark P.

doi:10.1007/3-540-44532-3_4

Cited by 17 publications

(38 citation statements)

References 10 publications

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“…Clearly, a nodal line emanates from a point z k if and only if z k ∈N( ). In particular, if is a ground state, some of our present results parallel the corresponding results for a bounded domain with holes obtained in References [2,3], although our mathematical tools are quite di erent from theirs. In our setting, we obtain …”

Section: Introductionsupporting

confidence: 78%

“…We recall that the ÿrst eigenvalue 1 (the ground state energy) has been deÿned in (3). A ground state is a minimizer for the Rayleigh quotient …”

Section: Proofmentioning

confidence: 99%

“…In contrast, certain superconductivity models considered in References [1][2][3] are set up in a bounded two-dimensional domain with a ÿnite number of holes k (k = 1; 2; : : : ; K), that is,…”

Section: Introductionmentioning

confidence: 98%

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Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in ℝ²

Alziary

Fleckinger-Pellé

Takáč

2003

Math Methods in App Sciences

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SUMMARYThe zero set {z ∈ R 2 : (z) = 0} of an eigenfunction of the Schr odinger operatorwith an Aharonov-Bohm-type magnetic potential is investigated. It is shown that, for the ÿrst eigenvalue 1 (the ground state energy), the following two statements are equivalent: (I) the magnetic ux through each singular point of the magnetic potential A is a half-integer; and (II) a suitable eigenfunction associated with 1 (a ground state) may be chosen in such a way that the zero set of is the union of a ÿnite number of nodal lines (curves of class C 2 ) which emanate from the singular points of the magnetic potential A and slit the two-dimensional plane R 2 . As an auxiliary result, a Hardy-type inequality near the singular points of A is proved. The C 2 di erentiability of nodal lines is obtained from an asymptotic analysis combined with the implicit function theorem.

show abstract

Section: Introductionsupporting

confidence: 78%

“…We recall that the ÿrst eigenvalue 1 (the ground state energy) has been deÿned in (3). A ground state is a minimizer for the Rayleigh quotient …”

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in ℝ²

Alziary

Fleckinger-Pellé

Takáč

2003

Math Methods in App Sciences

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show abstract

“…theory but, for the sake of completeness, we present it here. See [11] for a more general result. Suppose then that there exists a value so E [0, 27t] such that V,i(so) = 0.…”

Section: Propositionmentioning

confidence: 99%

A modified Ginzburg-Landau model for Josephson junctions in a ring

Hill¹,

Rubinstein²,

Sternberg³

2002

Quart. Appl. Math.

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Abstract.We consider the Ginzburg-Landau functional to analyze SNS junctions in a one-dimensional ring. We compare several canonical scalings. The linearized problem is solved to obtain the phase transition curves. We compute the V-limit of the functional in the different scalings. The interaction of several junctions is analyzed. We study the zero set of the order parameter for distinguished values of the flux. Finally, we compute the currents in the weakly nonlinear regime.

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“…We describe here briefly the "gauge-periodicity" of the operators with the vector potentialã; details can be found e.g. in [79,80,81,82]. First we shall assume that the considered solenoids carry equal fluxes of the value θ AB .…”

Section: Elimination Of Aharonov-bohm Solenoids With Integer Fluxesmentioning

confidence: 99%