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.

Abstract. This article is concerned with the dynamical properties of solutions of the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity. It is shown that the TDGL equations define a dynamical process when the applied magnetic field varies with time, and a dynamical system when the applied magnetic field is stationary. The dynamical system describes the large-time asymptotic behavior: Every solution of the TDGL equations is attracted to a set of stationary solutions, which are .divergence free. These results are obtained in the ''4 = -w(V A)" gauge, which reduces to the standard %qj = -V -A" gauge if w = 1 and to the zero-electric potential gauge if w = 0; the treatment captures both in a unified framework. This gauge forces the London gauge, V -A = 0, for any stationary solution of the TDGL equations.

Abstract. This article is concerned with the dynamical properties of solutions of the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity. It is shown that the TDGL equations define a dynamical process when the applied magnetic field varies with time, and a dynamical system when the applied magnetic field is stationary. The dynamical system describes the large-time asymptotic behavior: Every solution of the TDGL equations is attracted to a set of stationary solutions, which are .divergence free. These results are obtained in the ''4 = -w(V A)" gauge, which reduces to the standard %qj = -V -A" gauge if w = 1 and to the zero-electric potential gauge if w = 0; the treatment captures both in a unified framework. This gauge forces the London gauge, V -A = 0, for any stationary solution of the TDGL equations.

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Abstract. In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain iî C R" with fractal boundary 3Q . We construct an open set S for which we can effectively compute the second term of the asymptotics of the "counting function" N(X, S), the number of eigenvalues less than A . In this example, contrary to the M. V. Berry conjecture, the second asymptotic term is proportional to a periodic function of In X , not to a constant. We also establish some properties of the f-function of this problem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set t? derived from @ . Analogous periodic functions still appear in our inequalities. These results have been announced in [FV].