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Microlocal WKB expansions. (English) Zbl 0941.35136
Under some analyticity assumptions on the symbol \(p\) of a semiclassical pseudodifferential operator \(P=p(x,hD_x)\) (\(h\) being the semiclassical parameter, and the action of \(P\) on Schwartz functions being defined by \[ Pu(x)={{1}\over{(2\pi h)^n}}\iint e^{i\langle x-y,\xi\rangle/h}p\left({{x+y}\over{2}},\xi\right)u(y) dy d\xi), \] the authors establish the existence of a microlocal WKB expansion of the form \[ h^{-{m_0}}e^{-\Phi(x,\xi)/h}\sum_{j\geq 0}h^j a_j(x,\xi), \] as \(h\to 0+,\) of the FBI transform of the first eigenfunction of \(P\) near a point \((x_0,\xi_0)\in{\mathbb R}^{2n}\), which is a non-degenerate minimum for \(p(x,\xi).\) Here \(\Phi\) and the \(a_j\) are smooth (complex) functions, and such an expansion is valid on a domain that is characterized in terms of deformation properties. Such a result can be applied, for instance, to electromagnetic Schrödinger operators \[ P_A(x,hD_x)=\sum_{j=1}^n(hD_{x_j}-A_j(x))^2+V(x), \] where the electric potential \(V\) has a non-degenerate minimum at some point \(x_0.\)

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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