Some remarks on the linearized operator about the radial solution for the Ginzburg--Landau equation. (Q1400280)

For a small positive parameter \(\varepsilon \) and each positive integer \(d\), the Ginzburg-Landau equation \[ -\Delta u=\frac1{\varepsilon ^2}u(1-| u| ^2) \tag{1} \] admits a radial solution \[ u_{d,\varepsilon }(r,\theta)=e^{id\theta }f_d\left(\frac r\varepsilon \right), \] where \(f_d\) is the unique solution to \(f''-\frac{f'}r+d^2\frac f{r^2}=f(1-f^2)\) in \([0,\infty )\) satisfying \(f\geq0\), \(f(0)=0\), \(\lim\limits_{r\to\infty }f(r)=1\). Linearising (1) about a solution \(u\), one arrives at the operator \[ Lw=\Delta w+\frac w{\varepsilon ^2}(1-| u| ^2) -\frac2{\varepsilon ^2}u(\overline{u}w+\overline{w}u). \] For \(u=u_{d,\varepsilon }\), this operator is denoted by \(L_{d,\varepsilon }\), and, finally, the operator \(\mathcal L_{d,\varepsilon }=e^{-id\theta }L_{d,\varepsilon }e^{id\theta }\) is considered on \((H_0^1\cap H^2)(B(0,1), \mathbb{C})\). The behavior of the eigenvalues of this operator family is investigated. Let \(V_n=\{a(r)e^{-in\theta }+b(r)e^{in\theta }\}\cap (H_0^1\cap H^2)(B(0,1),\mathbb{C})\) and let \(\delta \) be the dimension of the real vector space of the solutions \(w\in V_n\) to \(\mathcal L_{d,1}w=0\) that are bounded in \(\mathbb{R}^2\). It is shown that \(\delta \in \{0,2,4\}\), that \(\delta =0\) for \(n\geq 2d-1\), and that the number of eigenvalues of \(\mathcal L_{d,\varepsilon }| _{V_n}\) tending to \(0\) as \(\varepsilon \to 0\) is \(\delta /2\). Estimates on these eigenvalues and corresponding eigenfunctions are given. Furthermore, it is shown that there is \(R_0>0\) such that all eigenvalues of \(\mathcal L_{d,\varepsilon }\) restricted to \((H_0^1\cap H^2)(B(0,1)\setminus B(0,\varepsilon R), \mathbb{C})\) are bounded away from \(0\) for all \(R\in (0,R_0)\).