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According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $...

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 ...

Suppose I have the following equality $$\hat{u}_\epsilon(t,k) = \alpha(t,k) + \int_0^t\int_{\mathbb{R}^n} e^{ik\cdot(x +\epsilon \phi(s,x))}u_\epsilon(s,x)dxds$$ Where $\alpha(t,k) \geq 0$ and $\alpha(...

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\...