In this article we obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the self-adjoint operator generated by a system of Sturm-Liouville equations with summable coefficients and the quasiperiodic boundary conditions. Then using these asymptotic formulas, we find the conditions on the potential for which the number of gaps in the spectrum of the Hill's operator with matrix potential is finite.
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.
We find conditions on the potential of the non-self-adjoint Mathieu-Hill operator such that the all eigenvalues of the periodic, antiperiodic, Dirichlet and Neumann boundary value problems are simple.
First we study the spectral singularity at infinity and investigate the connections of the spectral singularities and the spectrality of the Hill operator. Then we consider the spectral expansion when there is not the spectral singularity at infinity.
In this paper we prove that the spectrum of the Mathieu-Hill Operators with potentials ae^{-i2πx}+be^{i2πx} and ce^{-i2πx}+de^{i2πx} are the same if and only if ab=cd, where a,b,c and d are complex numbers. This result implies some corollaries about the extension of Harrell-Avron-Simon formula. Moreover, we find explicit formulas for the eigenvalues and eigenfunctions of the t-periodic boundary value problem for the Hill operator with Gasymov's potential.
We obtain uniform, with respect to t asymptotic formulas for the eigenvalues of the operators generated in (0,1) by the Mathieu-Hill equation with a complex-valued potential and by the t-periodic boundary conditions. Then using it we investigate the non-self-adjoint Mathieu-Hill operator H generated in the allreal line by the same equation and establish the necessary and sufficient conditions for the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator.
A periodic one-dimensional Schroedinger operator is called semifinite-gap if every second gap in its spectrum is eventually closed. We construct explicit examples of semifinite-gap Schroedinger operators in trigonometric functions by applying Darboux transformations to the Whittaker-Hill equation. We give a criterion of the regularity of the corresponding potentials and investigate the spectral properties of the new operators.
In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential spectral singularities and singular quasimomenta.
We obtain the uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L_{t}(q) with a potential q\inL_{1}[0,1] and with t-periodic boundary conditions, t\in(-π,π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L_{2}(-\infty,\infty) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies the sufficient conditions.
In this paper we investigate the non-self-adjoint operator H generated in all real line by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator H by investigating the essential spectral singularities.
Let $l_{n}$ be the length of the $n$-th instability interval of the Hill operator $Ly=-y^{\prime\prime}+q(x)y$. We obtain that if $l_{n}=o(n^{-2})$ then $c_{n}=o(n^{-2})$, where $c_{n}$ are the Fourier coefficients of $q$. Using this inverse result, we prove: Let $l_{n}=o(n^{-2})$. If $\{(nπ)^{2}: \textrm{$n$ even and $n>n_{0}$}\}$ is a subset of the periodic spectrum of Hill operator then $q=0$ a.e., where $n_{0}$ is a positive large number such that $l_{n}<\varepsilon n^{-2}$ for all $n>n_{0}(\varepsilon)$ with some $\varepsilon>0$. A similar result holds for the anti-periodic case.
It is shown that the deviations $P_n -P_n^0$ of Riesz projections $$ P_n = \frac{1}{2πi} \int_{C_n} (z-L)^{-1} dz, \quad C_n=\{|z-n^2|= n\}, $$ of Hill operators $L y = - y^{\prime \prime} + v(x) y, x \in [0,π],$ with zero and $H^{-1}$ periodic potentials go to zero as $n \to \infty $ even if we consider $P_n -P_n^0$ as operators from $L^1$ to $L^\infty. $ This implies that all $L^p$-norms are uniformly equivalent on the Riesz subspaces $Ran P_n. $
The Hill operators $L y = - y^{\prime \prime} + v(x) y, x \in [0,π],$ with $H^{-1}$ periodic potentials, considered with periodic, antiperiodic or Dirichlet boundary conditions, have discrete spectrum, and therefore, for sufficiently large $N,$ the Riesz projections $$ P_n = \frac{1}{2πi} \int_{C_n} (z-L)^{-1} dz, \quad C_n=\{z: |z-n^2|= n\} $$ are well defined. It is proved that $$\sum_{n>N} \|P_n - P_n^0\|^2_{HS} < \infty, $$ where $P_n^0$ are the Riesz projection of the free operator and $\|\cdot\|_{HS}$ is the Hilbert--Schmidt norm.
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.
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 derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrödinger operator) $H=-d^2/dx^2+V$ to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential $V$. In the course of our analysis we also establish a functional model for periodic Schrödinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion. The problem of deciding which Hil…
We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq π, $$ subject to periodic or antiperiodic boundary conditions, with potentials $v$ which are trigonometric polynomials with nonzero coefficients, of the form (i) $ ae^{-2ix} +be^{2ix}; $ (ii) $ ae^{-2ix} +Be^{4ix}; $ (iii) $ ae^{-2ix} +Ae^{-4ix} + be^{2ix} +Be^{4ix}. $ Then the system of eigenfunctions and (at most finitely many) associated functions is complete but it is not a basis in $L^2 ([0,π], \mathbb{C})$ if $|a| \neq |b| $ in the case (i), if $|A| \neq |B| $ and neither $-b^2/4B$ nor $-a^2/4A$ is an …
Using the relation established by Johnson--Zumbrun between Hill's method of aproximating spectra of periodic-coefficient ordinary differential operators and a generalized periodic Evans function given by the $2$-modified characteristic Fredholm determinant of an associated Birman--Schwinger system, together with a Volterra integral computation introduced by Gesztesy--Makarov, we give an explicit connection between the generalized Birman--Schwinger-type periodic Evans function and the standard Jost function-type periodic Evans function defined by Gardner in terms of the fundamental solution of …
We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator H=-d^2/dx^2+V in L^2(R) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V.
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 …
We formulate several open questions related to enumerative combinatorics, which arise in the spectral analysis of Hill operators with trigonometric polynomial potentials.
The paper deals with nonlocal differential operators possessing a term with frozen (fixed) argument appearing, in particular, in modelling various physical systems with feedback. The presence of a feedback means that the external affect on the system depends on its current state. If this state is taken into account only at some fixed physical point, then mathematically this corresponds to an operator with frozen argument. In the present paper, we consider the operator $Ly\equiv-y''(x)+q(x)y(a),$ $y^{(ν)}(0)=γy^{(ν)}(1),$ $ν=0,1,$ where $γ\in{\mathbb C}\setminus\{0\}.$ The operator $L$ is a non…
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 …
Consider the Hill operator $Ty=-y''+q'(t)y$ in $L^2(\R)$, where $q\in L^2(0,1)$ is a 1-periodic real potential. The spectrum of $T$ is is absolutely continuous and consists of bands separated by gaps $\g_n,n\ge 1$ with length $|\g_n|\ge 0$. We obtain a priori estimates of the gap lengths, effective masses, action variables for the KDV. For example, if $\m_n^\pm$ are the effective masses associated with the gap $\g_n=(ł_n^-,ł_n^+)$, then $|\m_n^-+\m_n^+|\le C|\g_n|^2n^{-4}$ for some constant $C=C(q)$ and any $n\ge 1$. In order prove these results we use the analysis of a conformal mapping corre…
For Hill's equations with matrix valued periodic potential, we discuss relations between the Morse index, counting the number of unstable eigenvalues, and the Maslov index, counting the number of signed intersections of a path in the space of Lagrangian planes with a fixed plane. We adapt to the one dimensional periodic setting the strategy of a recent paper by J. Deng and C. Jones relating the Morse and Maslov indices for multidimensional elliptic eigenvalue problems.
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 consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq π, $$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \quad a, b \neq 0, r,s \in \mathbb{N}, r\neq s. $$ It is shown that the system of root functions does not contain a basis in $L^2 ([0,π], \mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r, s$ are odd or $r=1$ and $s \geq 3. $
We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator $L= -d^2/dx^2 + v(x), $ $x \in [0,π], $ with $H_{per}^{-1} $-potential and the free operator $L^0=-d^2/dx^2, $ subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $$ \|S_N - S_N^0: L^a \to L^b \| \to 0 \quad \text{if} \;\; 1<a \leq b< \infty, \;\; 1/a - 1/b <1/2, $$ where $S_N$ and $S_N^0 $ are the $N$-th partial sums of the spectral decompositions of $L$ and $L^0.$ Moreover, if $v \in H^{-α} $ with $1/2 < α< 1$ and $\frac{1}{a}=(3/2)-α, $ then …
Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,π]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ 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 describe classes of complex potentials $v(x)= \sum_{2\mathbb{Z}} V(k) e^{ikx}$ in weighted spaces (defined in terms of the Fourier coefficients of $v$) such that the periodic (or antiperiodic) root function system of $L(v) $ contains a Riesz basis if and only …
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…
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 extend the well-known trace formula for Hill's equation to general one-dimensional Schrödinger operators. The new function $ξ$, which we introduce, is used to study absolutely continuous spectrum and inverse problems.
We study the quasi-periodic Schrödinger operator $$ -ψ"(x) + V(x) ψ(x) = E ψ(x), \qquad x \in \mathbb{R} $$ in the regime of "small" $V(x) = \sum_{m\in\mathbb{Z}^ν}c(m)\exp (2πi mωx)$, $ω= (ω_1, \dots, ω_ν) \in \mathbb{R}^ν$, $|c(m)| \le \varepsilon \exp(-κ_0|m|)$. We show that the set of reflectionless potentials isospectral with $V$ is homeomorphic to a torus. Moreover, we prove that any reflectionless potential $Q$ isospectral with $V$ has the form $Q (x) = \sum_{m \in \mathbb{Z}^ν} d(m) \exp (2πi mωx)$, with the same $ω$ and with $|d(m)| \le \sqrt{2 \varepsilon} \exp(-\frac{κ_0}{2} |m|)$. …
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 $$ \{…
Consider the discrete 1D Schrödinger operator on $\Z$ with an odd $2k$ periodic potential $q$. For small potentials we show that the mapping: $q\to $ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of $2^k$ distinct potentials. Finally, the asymptotics of the spectrum are determined as $q\to 0$.
In this paper, we build up Hill-type formula for linear Hamiltonian systems with Lagrangian boundary conditions, which include standard Neumann, Dirichlet boundary conditions. Such a kind of boundary conditions comes from the brake symmetry periodic orbits in $n$-body problem naturally. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Consequently, we derive the Krein-type trace formula and give nontrivial estimation for the eigenvalue problem. Combined …
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…
We consider Schrödinger operators with periodic potentials in the positive quadrant for dim $>1$ with Dirichlet boundary condition. We show that for any integer $N$ and any interval $I$ there exists a periodic potential such that the Schrödinger operator has $N$ eigenvalues counted with the multiplicity on this interval and there is no other spectrum on the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for other domains. The proof is based on the inverse spectral theory for Hill operators on…
Applying perturbation theory methods, the absence of the point spectrum for some nonselfadjoint integro-differential operators is investigated. The considered differential operators are of arbitrary order and act in either $\mathbf{L}_{p}(\mathbb{R}_{+})$ or $\mathbf{L}_{p}(\mathbb{R}) (1\leq p<\infty)$. As an application of general results, new spectral properties of the perturbed Hill operator are derived.
For any $N\ts N$ monodromy matrix we define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy. We show that the endpoints of each gap are periodic (anti-periodic) eigenvalues or resonances (real branch points). Moreover, the following results are obtained: 1) we…
Let $λ_j$ be the $j$-th eigenvalue of Sturm-Liouville systems with separated boundary conditions, we build up the Hill-type formula, which represent $\prod\limits_{j}(1-λ_j^{-1})$ as a determinant of finite matrix. This is the first attack on such a formula under non-periodic type boundary conditions. Consequently, we get the Krein-type trace formula based on the Hill-type formula, which express $\sum\limits_{j}{1\over λ_j^m}$ as trace of finite matrices. The trace formula can be used to estimate the conjugate point alone a geodesic in Riemannian manifold and to get some infinite sum identitie…
