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Estimates in \(L^ p\) for magnetic Schrödinger operators. (English) Zbl 0880.35034
Let \(H({\mathbf a},V)\) be a magnetic Schrödinger operator \[ \sum_{j=1}^n\Biggl({1 \over i} {\partial \over \partial x_j } - a_j(x)\Biggr)^2 + V(x) \] in \({\mathbb{R}}^n, n \geq 3\) and \(L_j = {1 \over i} {\partial \over \partial x_j } - a_j\). Under certain conditions on \(\text{curl}{\mathbf a}\) and \(V\), given in terms of the reverse Hölder inequality (in particular, they are valid for polynomial coefficients), the estimates of the following type \[ \sum_{1\leq j,k \leq n}|L_jL_k f|\leq C_p|H({\mathbf a},V)f|, \qquad 1<p<\infty \] are proved. There are also similar weak-type (1,1) inequalities.

MSC:
35J10 Schrödinger operator, Schrödinger equation
35B45 A priori estimates in context of PDEs
35Q40 PDEs in connection with quantum mechanics
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