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Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the ...

Consider the following lemma: Let $g \in H^s_{x,y}(S)$ where $S = \mathbb{R}^2$ or $S = \mathbb{T}^2$, and $\eta \in C^\infty(\mathbb{R})$, $\operatorname{Supp}(\eta) \subset [-2,2]$, and $\eta \equiv ...

Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that $$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$ and ...

Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...

I have the following nonlinear dispersive PDEs $$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$ where $f$ is some nice complex-valued function. I am trying to use the ansatz $u(t,x) = e^{i \...