论文标题
具有单数势和方形可集成数据的非平滑schrödinger方程的溶解度
Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data
论文作者
论文摘要
我们为一阶运算符$ db $开发一个全体形态功能计算，以解决Schrödinger方程的边界价值问题$ - \ Mathrm {div} \，A \ nabla U+a v u = 0 $ in the Uphass Space $ \ Mathbb {R}这取决于$ db $的二次估计，这些估计值是针对系数$ a，a，v $独立于横向到边界的横向方向的，并且由一个复杂的纤维化对$（a，a，a，a，a）组成，这些$（a，a，a）是有界和可测量的，并且是$ l^{n/2 $ l^{n/2}（n/2）（n/2} $ v $） $ b^{q}（\ mathbb {r}^n）$带有$ q \ geq \ geq \ max \ {\ tfrac {n} {2} {2}，2 \} $。在后一种情况下，平方函数边界还显示等于非界限最大函数边界。这使我们能够证明（dirichlet）规律性和neumann边界价值问题与$ l^2（\ mathbb {r}^n）$ - 数据在且仅当功能计算定义的某些边界跟踪运算符时，数据是很好的。当主系数矩阵$ A $具有遗传学或区块结构时，我们证明了这一属性。更一般而言，所有复杂系数的集合被证明是开放的。
We develop a holomorphic functional calculus for first-order operators $DB$ to solve boundary value problems for Schrödinger equations $-\mathrm{div}\, A \nabla u + a V u = 0$ in the upper half-space $\mathbb{R}^{n+1}_+$ with $n\in\mathbb{N}$. This relies on quadratic estimates for $DB$, which are proved for coefficients $A,a,V$ that are independent of the transversal direction to the boundary, and comprised of a complex-elliptic pair $(A,a)$ that are bounded and measurable, and a singular potential $V$ in either $L^{n/2}(\mathbb{R}^n)$ or the reverse Hölder class $B^{q}(\mathbb{R}^n)$ with $q\geq\max\{\tfrac{n}{2},2\}$. In the latter case, square function bounds are also shown to be equivalent to non-tangential maximal function bounds. This allows us to prove that the (Dirichlet) Regularity and Neumann boundary value problems with $L^2(\mathbb{R}^n)$-data are well-posed if and only if certain boundary trace operators defined by the functional calculus are isomorphisms. We prove this property when the principal coefficient matrix $A$ has either a Hermitian or block structure. More generally, the set of all complex coefficients for which the boundary value problems are well-posed is shown to be open.
