Abstract

In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let $P\\colon L^2(\\mathbb{R})\\to L^2(\\mathbb{R})$ have the form $$ P:=-\\frac{d^2}{dx^2}+W, $$ 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, $\\mathbf{1}{(-\\infty,\\lambda^2]}(P)$ 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, the class of potentials includes certain potentials with \_dense pure point spectrum. The proof combines the gauge transform methods of Parnovski–Shterenberg and Sobolev with Melrose's scattering calculus.

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