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تسجيل مستخدم جديدComplete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrodinger operators in dimension one
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Jeffrey Galkowski
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In this article we consider asymptotics for the spectral function of Schrodinger operators on the real line. Let $P:L^2(mathbb{R})to L^2(mathbb{R})$ have the form $$ P:=-tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $mathbb{1}_{(-infty,lambda^2]}(P)$ has a full asymptotic expansion in powers of $lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, it includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melroses scattering calculus.
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