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Energy estimation of density operator to von Neumann equation

Consider the Schrödinger equation on $\mathbb R_+\times\mathbb R^n$ as follows: $$i\partial_t\varphi(t,x)=-\frac12\Delta_x\varphi(t,x),\quad \varphi(0,x)=\varphi_0(x).$$ Denote by $\varphi$ its soluti …
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