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Interior Hölder estimates for solutions of Schrödinger equations and the regularity of nodal sets. (English) Zbl 0846.35036
The authors study qualitative properties of the solution of the Schrödinger equation (1) \(\Delta u= Vu\) in the distributional sense, where \(V\in L^1(\Omega)\) is a potential from the class \(K^{n, \delta}(\Omega)\) (for the definition see the reviewed paper) for \(n\geq 2\), \(\delta\in (0, 2)\) and \(\Omega\in \mathbb{R}^n\) and \(\Omega\subset \mathbb{R}^n\) is an open bounded set. The main results are presented in two theorems.
In the first theorem, the authors prove that the distributional solution of (1) can be approximated by harmonic homogeneous polynomials in the neighbourhood of the nodal point. The second theorem contains of the assertions on the nodal set as a local hypersurface from the class \(C^{1, \delta}\) or \(C^{\alpha, \delta- 1}\). The authors also consider the case of \(V\in C^{k, \alpha}(\Omega)\) for \(k\in N_0\) and \(\alpha\in (0, 1)\). These theorems generalize some previous results of the present authors.

MSC:
35J10 Schrödinger operator, Schrödinger equation
35B45 A priori estimates in context of PDEs
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