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On the large order asymptotics of general states in semiclassical quantum mechanics. (English) Zbl 0791.35113
Summary: We consider the limit $$\hslash\to 0$$ of the solution $$\Phi(t,x,\hslash)$$ of the Schrödinger equation: $i\hslash {{\partial\Phi(t,x,\hslash)} \over {\partial t}}=- {{\hslash^ 2} \over {2m}} {{d^ 2 \Phi(t,x,\hslash)} \over {dx^ 2}} +V(x) \Phi(t,x,\hslash).$ We prove that, for any integer $$l\geq 2$$ and any initial condition $$\Phi(0,x,\hslash)$$ that belongs to the Schwartz-class, a solution $$\Phi^*(t,x,\hslash)$$ of the semiclassical equation approximates $$\Phi(t,x,\hslash)$$ such as $\|\Phi^*(t,\cdot,\hslash)- \Phi(t,\cdot,\hslash)\|_{L^ 2} \leq C\hslash^{1/2} \qquad (\hslash\to 0)$ .

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
##### Keywords:
Schrödinger equation; semiclassical equation
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##### References:
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