The Eigenvalue Problem Of The Schrodinger Operator With Electromagnetic Field

This paper is concerned with the stability of the normal state of the superconductors in the presence of an electric current and an applied magnetic filed. The model studied in this paper is the time-dependent Ginzburg-Landau model:The steady state solutions of this system satisfy the following (time-independent) Ginzburg-Landausystem:This paper is organized as follows. In section 2 we study some mathematical problems about (2), including the discussions on the normal state solutions. In section 3 we reduce the stability problem to the eigenvalue problem on a bounded domain. Then we consider the asymptotic analysis of the eigenvalue problem in the case with strong electronic currentsand strong applied magnetic fields. This leads to the study of the limit equation on the entire plane or on the half plane. The eigenvalues of the limit equation in the entire plane is studied in section 4. Section 5 is devoted to the study of the asymptotic behavior of the eigenvalue problem with a weak applied magnetic filed or a weak electric current.The main results are the following.Theorem 1 LetΩbe a simply-connected domain with C² boundary. (0,A_e,φ_e) is the stable solution of (2), that is (A_e,φ_e) satisfies the following equation:Assumeκ² is not an eigenvalue of the operator (?) + iφ, nor of (?) - iφ, then the equation (2) has no non-trivial solution in a small domain of (0, A_e,φ_e).Theorem 2 Consider the operator P = (?) + iφin R², where A = (-x₂,0),φ= Jx₁ + Jx₂, then the eigenvalues of the operator P are the following:Theorem 3 Supposeψ∈W_loc^1,2(R²Ã—R₊)∩L²(R²Ã—R₊) is a solution of the equation:Then, (i)ifκ²ï¼œÎ±(A), we have ||ψ||₂ (?) 0, where ||ψ||₂² = (ψ,ψ) = (?)(ii)Let curlA = 1, = Jx₁ + Jx₂, then ifκ²ï¼ž1 + (?),ψ= 0 is unstable.Theorem 4 Letφ_ε=εφ, whereφis a smooth functionε＞0. let us consider theequation:Whenε→0, there exists an asymptotical solution:where...