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Register a new userComplete 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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We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists an almost periodic magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} eq 0$, then generically $operatorname{dim} operatorname{Ker} H = infty$. If $b_{0} = 0$, then for each $m in {mathbb N} cup { infty }$, we construct almost periodic $b$ such that $operatorname{dim} operatorname{Ker} H = m$. This construction depends strongly on results concerning the asymptotic behavior of Dirichlet series, also obtained in the present article.
Let $H_0 = -Delta + V_0(x)$ be a Schroedinger operator on $L_2(mathbb{R}^ u),$ $ u=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $mathbb{R}^ u.$ Let $V$ be an operator of multiplication by a bounded integrable real-valued function $V(x)$ and put $H_r = H_0+rV$ for real $r.$ We show that the associated spectral shift function (SSF) $xi$ admits a natural decomposition into the sum of absolutely continuous $xi^{(a)}$ and singular $xi^{(s)}$ SSFs. This is a special case of an analogous result for resolvent comparable pairs of self-adjoint operators, which generalises the known case of a trace class perturbation while also simplifying its proof. We present two proofs -- one short and one long -- which we consider to have value of their own. The long proof along the way reframes some classical results from the perturbation theory of self-adjoint operators, including the existence and completeness of the wave operators and the Birman-Krein formula relating the scattering matrix and the SSF. The two proofs demonstrate the equality of the singular SSF with two a priori different but intrinsically integer-valued functions: the total resonance index and the singular $mu$-invariant.
We consider the 3D Schrodinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $Omega_{rm in} subset {mathbb R}^3$. We introduce the Krein spectral shift functions $xi(E;H_pm,H_0)$, $E geq 0$, for the operator pairs $(H_pm,H_0)$, and study their singularities at the Landau levels $Lambda_q : = b(2q+1)$, $q in {mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. We show that $xi(E;H_+,H_0)$ remains bounded as $E uparrow Lambda_q$, $q in {mathbb Z}_+$ being fixed, and obtain three asymptotic terms of $xi(E;H_-,H_0)$ as $E uparrow Lambda_q$, and of $xi(E;H_pm,H_0)$ as $E downarrow Lambda_q$. The first two terms are independent of the perturbation while the third one involves the {em logarithmic capacity} of the projection of $Omega_{rm in}$ onto the plane perpendicular to $B$.
We study perturbations of the self-adjoint periodic Sturm--Liouville operator [ A_0 = frac{1}{r_0}left(-frac{mathrm d}{mathrm dx} p_0 frac{mathrm d}{mathrm dx} + q_0right) ] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.
We show that for a one-dimensional Schrodinger operator with a potential whose (j+1)th moment is integrable the jth derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrodinger equation in the resonant case.
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