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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
##### Keywords:
weak-type inequalities; reverse Hölder inequality
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