Magnetic Schrödinger operators and the \(\overline{\partial}\)-equation
The author characterizes compactness of the canonical solution of \(\widetilde\partial\) on weighted \(L^2\) spaces on \(\mathbb{C}\). Precisely if \(\varphi\) is a weight function, \(L^2_\varphi = \{f: \mathbb{C} \to \mathbb{C} \mid \|f\|^2_{\varphi^2}= \int_{\mathbb{C}} |f(x)|^2 e^{-2\varphi(z)} \,d\lambda (z) < \infty\}\) it is classical that there is a continuous solution operator \(\widetilde S : L^2_\varphi \to L^2_\varphi\) such that \(\overline\partial \widetilde S(f) = f\) [\textit{L. Hörmander}, An introduction to complex analysis in several variables. Princeton, N.J. et al.f: van Nostrand (1966; Zbl 0138.06203), \textit{M. Christ}, J. Geom. Anal. 1, No. 3, 193--230 (1991; Zbl 0737.35011)]. The solution \(S = (I - P_\varphi)\widetilde S\) is the canonical solution operator of \(\overline\partial\), such that \(S (f) \perp A^2_\varphi\) where \(A_\varphi\) is the space of entire functions belonging to \(L^2_\varphi\) and \(P_\varphi\) denotes the Bergmann projection: \(L^2_\varphi \to A^2_\varphi\). The author proves that the solution operator \(S\) to \(\overline\partial\) is compact iff there is a real valued continuous function \(\mu\) on \(\mathbb{C}\), s.t. \(\mu(z) \to \infty\) as \(|z| \to \infty\), and \(\int_{\mathbb{C}} S \varphi \overline\varphi \,d\lambda \geq \int \mu |\varphi|^2 \,d\lambda \) for all \(\varphi \in C_0^\infty (\mathbb{C})\) where \(S\) is explicitly given on \(\mathbb{C}\) in terms of the weight function. In particular, if this weight function is \(|z|^2\) then the canonical solution is not compact. On the other side, if the weight function \(\varphi\) supposed to be subharmonic and s.t. \(\Delta \varphi\) belong to the space of doubling measures (\(\mu\) is doubling if there is a constant \(C\) s.t. \(\nu (B(z, 2r)) \leq C \nu (B(z, r))\) for all \(z \in \mathbb{C}\), \(r \in \mathbb{R}^+\)) and if \(\Delta \varphi(z) \to \infty\) as \(|z| \to \infty\), then the canonical solution operator \(S \cdot L^2_\varphi \to L^2_\varphi\) is compact. Finally, the author tries to generalize the method for the case of several complex variables. If \(v \in L^2(\mathbb{C}^n)\) and \(c_j = \sum g_j |z_j|)\) a \((0, 1)\) form with coefficients in \(L^2(\mathbb{C}^n, \varphi)\) (\(\varphi\) a \(\mathcal{C}^2\)-weight function), then \(\overline D v\) is defined to be \[ \overline D v = \sum \left(\frac{\partial v}{\partial \overline z_n} + \frac{\partial \varphi}{\partial \overline z_n v}\right) d\overline z_k\;\text{ and }\;\overline D^* g = \sum\left( \frac{\partial \varphi}{\partial z_j} g_j - \frac{\partial g_j}{\partial z_j}\right). \] If \(\varphi\) is of the form \(\varphi_1(z_1) + \ldots + \varphi_n(z_n)\) the operator \(\overline D{} \overline D^* g = h\) splits into \(n\) equations, which can be represented as Schrödinger operators \(S_k\) with magnetic fields. The effective potentials [\textit{V. Kondratiev} and \textit{M. Shubin}, Commun. Partial Differ. Equations 27, No. 3--4, 477--525 (2002; Zbl 1090.35066)] are causing obstruction for the Schrödinger operators to have compact resolvents (for example in the case \(\varphi(z) = \sum |z_j|^2\)).
scientific article; zbMATH DE number 5128986