Let ${γ_q(n)}_{n \in \mathbb{N}}$ be the lengths of spectral gaps in a continuous spectrum of the Hill-Schrödinger operators S(q)u=-u''+q(x)u,\quad x\in \mathbb{R}, with 1-periodic real-valued potentials $q \in L^{2}(\mathbb{T})$. Let weight function $ω:\;[1,\infty)\to (0,\infty)$. We prove that under the condition \exists s\in [0,\infty):\quad k^{s}\llω(k)\ll k^{s+1},\; k\in \mathbb{N}, the map $γ:\, q \mapsto \{γ_{q}(n)}_{n \in \mathbb{N}}$ satisfies the equalities: \verb"i")\quad γ(H^ω) = h_{+}^ω, \verb"ii")\quad γ^{-1}(h_{+}^ω) = H^ω, where the real function space H^ω & ={f=\sum_{k\in \mathbb{Z}}\hat{f}\,(k)e^{i k2πx}\in L^{2}(\mathbb{T})| \sum_{k\in \mathbb{N}} ω^{2}(k)|\hat{f}(k)|^{2}<\infty,\; \hat{f}(k)=\bar{\hat{f}(-k)},\;k\in \mathbb{Z}.}, and h^ω = {a=\{a(k)\}_{k\in \mathbb{N}}|\sum_{k\in \mathbb{N}}ω^{2}(k)|a(k)|^{2}<\infty.}, h_{+}^ω = {a=\{a(k)\}_{k\in \mathbb{N}}\in h^ω| a(k)\geq 0.}. If the weight $ω$ is such that \exists a>1,c>1:\qquad c^{-1}\leq \frac{ω(λt)}{ω(t)}\leq c\quad\forall t\geq 1,\;λ\in [1,a] then the function class $H^ω$ is a real Hörmander space $H_{2}^ω(\mathbb{T},\mathbb{R})$ with the weight $ω(\sqrt{1+ξ^{2}})$.
The Hill operators Ly=-y''+v(x)y, considered with singular complex valued π-periodic potentials v of the form v=Q' with Q in L^2([0,π]), and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large n, the disc {z: |z-n^2|<n} contains two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λ_n^-, λ_n^+ and one Neumann eigenvalue ν_n. We show that rate of decay of the sequence |λ_n^+-λ_n^-|+|λ_n^+ - ν_n| determines the potential smoothness, and there is a basis consisting of periodic (or antiperiodic) root functions if and only …
The paper studies the Hill--Schrödinger operators with potentials in the space $H^ω\subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences arising as the lengths of spectral gaps of these operators. The space $H^ω$ coincides with the Hörmander space $H^ω_2\left(\mathbb{T}, \mathbb{R}\right)$ with the weight function $ω(\sqrt{1+ξ^{2}})$ if $ω$ belongs to Avakumovich's class $\mathrm{OR}$. In particular, if the functions $ω$ are power, then these spaces coincide with the Sobolev spaces. The functions $ω$ may be nonmonotonic.
The Hill operator admits a band gap structure. As a special case, like the Mathieu operator, one has only open gaps, however, the instability intervals of the Whittaker-Hill operator may be open or closed. In 2007, P. Djakov and B. Mityagin gave the asymptotics of band gaps for a special Whittaker-Hill operator [P. Djakov and B. Mityagin, J. Funct. Anal., 242, 157-194 (2007).]. In this paper, a more general Whittaker-Hill operator is considered and the asymptotics of the instability intervals are studied.
In the paper we study the behaviour of the lengths of spectral gaps $\{γ_{q}(n)\}_{n\in \mathbb{N}}$ in a continuous spectrum of the Hill-Schrödinger operators $$S(q)u=-u"+q(x)u,\quad x\in\mathbb{R},$$ with 1-periodic real-valued distribution potentials $$q(x)=\sum_{k\in \mathbb{Z}}\hat{q}(k) e^{i k 2πx}\in H^{-1}(\mathbb{T}),\quad\text{and}\quad\hat{q}(k)=\bar{\hat{q}(-k)}, k\in \mathbb{Z},$$ in dependence on the weight $ω$ of the Hörmander spaces $H^ω(\mathbb{T})\ni q$, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Let $h^ω(\mathbb{N})$ be a Hilbert space of weighted sequences. It is proved that $$ \{…
We describe a new, short proof of some facts relating the gap lengths of the spectrum of a potential of Hill's equation to its regularity. For example, a real potential is in a weighted Gevrey-Sobolev space if and only if its gap lengths belong to a similarly weighted sequence space. An extension of this result to complex potentials is proven as well. We also recover Trubowitz results about analytic potentials. The proof essentially employs the implicit function theorem.
The Hill operators $Ly=-y"+v(x)y$, considered with complex valued $π$-periodic potentials $v$ and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large $n,$ close to $n^2$ there are two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-$, $λ_n^+$ and one Neumann eigenvalue $ν_n$. We study the geometry of "the spectral triangle" with vertices ($λ_n^+$,$λ_n^-$,$ν_n$), and show that the rate of decay of triangle size characterizes the potential smoothness. Moreover, it is proved, for $v\in L^p ([0,π]), \; p>1,$ tha…
Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,π].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $|n|$ close to $n^2 $ in the Hill case, or close to $n, \; n\in \mathbb{Z}$ in the Dirac case, there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We give estimates for the asymptotics of the spectral gaps $γ_n = λ_n^+ - λ_n^-$ and deviations $ …
We study the system of root functions (SRF) of Hill operator $Ly = -y^{\prime \prime} +vy $ with a singular potential $v \in H^{-1}_{per}$ and SRF of 1D Dirac operator $ Ly = i {pmatrix} 1 & 0 0 & -1 {pmatrix} \frac{dy}{dx} + vy $ with matrix $L^2$-potential $v={pmatrix} 0 & P Q & 0 {pmatrix},$ subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property o…
We consider the Schrödinger operator $Hy=-y"+(p+q)y$ with a periodic potential $p$ plus a compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple eigenvalues below the spectrum and in each spectral gap $\g_n\ne \es, n\ge1$. We prove the following results: 1) the distribution of resonances in the disk with large radius is determined, 2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, 3) if $H$ has infinitely many open gaps in the continuous spectrum, then for any sequ…
We formulate the inverse spectral theory of infinite gap Hill's operators with bounded periodic potential as a Riemann--Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann--Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potential evolves according to the KdV equation, we use integrability to derive an associated Riemann--Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann--Hilbert problems yield …
We present an abstract multiscale analysis scheme for matrix functions $(H_{\varepsilon}(m,n))_{m,n\in \mathfrak{T}}$, where $\mathfrak{T}$ is an Abelian group equipped with a distance $|\cdot|$. This is an extension of the scheme developed by Damanik and Goldstein for the special case $\mathfrak{T} = \mathbb{Z}^ν$. Our main motivation for working out this extension comes from an application to matrix functions which are dual to certain Hill operators. These operators take the form $H_{\tildeω}=-\frac{d^2}{dx^2} + \varepsilon U(\tildeωx)$, where $U$ is a real smooth function on the torus $\mat…
For 1D Dirac operators Ly= i J y' + v y, where J is a diagonal 2x2 matrix with entrees 1,-1 and v(x) is an off-diagonal matrix with L^2 [0,π]-entrees P(x), Q(x) we characterize the class X of pi-periodic potentials v such that: (i) the smoothness of potentials v is determined only by the rate of decay of related spectral gaps gamma (n) = | λ(n,+) - λ(n,-)|, where λ(..) are the eigenvalues of L=L(v) considered on [0,π] with periodic (for even n) or antiperiodic (for odd n) boundary conditions (bc); (ii) there is a Riesz basis which consists of periodic (or antiperiodic) eigenfunctions and (at m…
