Half-line eigenfunction estimates and singular continuous spectrum of zero Lebesgue measure
David Damanik, Daniel Lenz
备注：
15 pages; extended and corrected version of the paper "Half-line eigenfunction estimates and stability of singular continuous spectrum"; contains thorough discussion of pure point spectrum arising in the models in question under rank one perturbations
We consider discrete one-dimensional Schrödinger operators with strictly ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely singular continuous spectrum of zero Lebesgue measure is further elucidated. We provide a unified approach to both the study of the spectral type as well as the measure of the spectrum as a set. We apply this approach to Schrödinger operators with Sturmian potentials. Finally, in the appendix, we discuss the two different strictly ergodic dynamical systems associated to a circle map.
We prove Polterovich's conjecture concerning the growth of the number of nodal domains for eigenfunctions on a unit square domain, under the assumption that the eigenfunctions do not have any singular points.
We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a rescaled function that is constructed in terms of the $j$th eigenfunction of fourth or second order order effective opera…
In this study, singular diffusion operator with jump conditions is considered. Integral representations have been derived for solutions that satisfy boundary conditions and jump conditions. Some properties of eigenvalues and eigenfunctions are investigated. Asymtotic representation of eigenvalues and eigenfunction have been obtained. Reconstruction of the singular diffusion operator have been shown by the Weyl function.
In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterisa…
We consider the one-dimensional Schrödinger equation $-f"+q_κf = Ef$ on the positive half-axis with the potential $q_κ(r)=(κ^2-1/4)r^{-2}$. For each complex number $\vartheta$, we construct a solution $u^κ_\vartheta(E)$ of this equation that is analytic in $κ$ in a complex neighborhood of the interval $(-1,1)$ and, in particular, at the "singular" point $κ= 0$. For $-1<κ<1$ and real $\vartheta$, the solutions $u^κ_\vartheta(E)$ determine a unitary eigenfunction expansion operator $U_{κ,\vartheta}\colon L_2(0,\infty)\to L_2(\mathbb R,\mathcal V_{κ,\vartheta})$, where $\mathcal V_{κ,\vartheta}$ …
We introduce a dynamically defined class of unbounded, connected, equilateral metric graphs on which the Kirchhoff Laplacian has zero Lebesgue measure spectrum and a nontrivial singular continuous part. A new local Borg--Marchenko uniqueness result is obtained in order to utilize Kotani theory for aperiodic subshifts satisfying Boshernitzan's condition.
Let $M$ be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a compact Lie group $G$, and $P_0$ an invariant elliptic classical pseudodifferential operator on $M$. Using Fourier integral operator techniques, we prove a local Weyl law with remainder estimate for the equivariant (or reduced) spectral function of $P_0$ for each isotpyic component in the Peter-Weyl decomposition of $L^2(M)$, generalizing work of Avacumovič, Levitan, and Hörmander. From this we deduce a generalized Kuznecov sum formula for periods of G-orbits, and recover the local Weyl law f…
This paper is dedicated to $L^p$ bounds on eigenfunctions of a Schödinger-type operator $(-Δ_g)^{α/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-Δ_g)^{α/2}$ is defined spectrally in terms of the eigenfunctions of $-Δ_g$. We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation $(\partial_t^2+(-Δ_g)^{α/2}+V)u=0$. The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao and Shao-Yao play a key role in this paper. We construct a new reproducing opera…
We investigate the interior nodal sets $\mathcal{N}_λ$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $Cλ$. The singular sets $\mathcal{S}_λ$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_λ)\leq Cλ^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_λ)\leq Cλ^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.
We obtain quasimode, eigenfunction and spectral projection bounds for Schrödinger operators, $H_V=-Δ_g+V(x)$, on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$, which extend the results of the third author~\cite{sogge88} corresponding to the case where $V\equiv 0$. We are able to handle critically singular potentials and consequently assume that $V\in L^{\tfrac{n}2}(M)$ and/or $V\in {\mathcal K}(M)$ (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where $V\equiv 0$ that go back to the third author \cite{sogge88} as w…
Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $φ^\ell_m$ be the orthonormal basis of joint eigenfunctions of $Δ_g$ and $\partial_θ$, the generator of the rotation action. The main result is an explicit formula for the weak-* limit of the normalized empirical measures, $Σ_{m = -\ell}^\ell ||φ^\ell_m||^2_{L^2(H)} δ_{\frac{m}{\ell}}(c)$ on $[-1,1]$. The explicit formula shows that, asymptotically, the $L^2$ norms of restricted eigenfunctions are minimal for the zonal eigenfunction $m = 0$, maximal for Gaussian beams $m = \pm 1$…
