Semicontinuous Banach spaces for Schrödinger's Eq. with Dirac-$δ'$ potential
Bradly K Button
备注：
63 pages, 1 figure, additional references, revised abstract, Section 3.1 has been revised to better motivate the sheaf property in Def. 3.1 along with a slight revision of Def. 3.1, added citation in support of Def. 3.1 regarding recent work on diffeomorphism invariant Colombeau algebra
Schródinger's equation with distributional $δ$, or $δ'$ potentials has been well studied in the past. There are challenges in simultaneously addressing some of the inherent issues of the system: The functional operator cannot exist entirely within the standard $L^2$ Hilbert spaces. On differentiable manifolds, the domain of the free kinetic energy operator is in the space of harmonic forms. Locally, by the Hodge decomposition theorem and the standard distributional calculus, the space of functionals of a $δ$ or $δ'$ potential must be orthogonal to the free kinetic energy operator. Restricting to semicontinuous topologies presents opportunities to address these, and other issues. We develop, in great detail, a formalism of Banach spaces with semicontinuous topologies, and their properties are extensively defined and studied. For $C(\overline{\mathbb{R}})$ functions, the spaces are indistinguishable. The semicontinuous analogs of the $L^p$ spaces, are nontrivial and result in a dense topologically continuous embedding of the semicontinuous $L^p$ spaces into the semicontinuous $C(\overline{\mathbb{R}})$ spaces. Here, certain classes of distributions may be inverted in terms of their primitive functions. Also many operators are inherently self adjoint. We define equivalence relations between the cohomology classes of distributions and derivatives of their associated primitives on local sections of $\overline{\mathbb{R}}$. Here Hamilton's equations are canonical, and define a connection on the fibers of the base space. Semicontinuity provides a resolution to the above domain and interaction problems, and easily integrable Feynman functional. We arrive at a compatible domain which is Krein ($\mathfrak{H}$) over disjoint components of $\overline{\mathbb{R}}$. The subspaces of $\mathfrak{H}$ are isomorphic to the semicontinuous Hilbert spaces of the Hamiltonian.
On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers
CNRS IRMAR - UMR 6625（法国国家科学研究中心 IRMAR）；Univ. Lille, CNRS, Inria UMR 8524 - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，Inria UMR 8524 - 保罗·庞加莱实验室）；Department of Mathematics, McMaster University（麦克马斯特大学数学系）
On unique continuation for solutions of the Schr{ö}dinger equation on trees
我们证明了如果齐次树上的含时薛定谔方程的解在两个不同的时刻快速衰减，则该解是平凡的.对于自由薛定谔算子，我们利用Laplacian谱理论和复分析，得到了导致在任意时刻急剧衰减的初始条件的一个特征.然后，我们利用Colin de Verdi和Turc的紧支势的Schr dinger算子的谱分解，推广了我们在紧支势存在下的结果.最后，我们使用实变量的方法首先介绍了Escauriaza，Kenig，Ponce和Vega建立一个一般的尖锐的结果的情况下，有界的潜力。
