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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
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References:
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