arXiv subjects
Publications and source records attributed to Milivoje Lukic.
We study decaying half-line Schrödinger operators and the local eigenvalue spacing of their Dirichlet restrictions. While absolutely continuous spectrum is strongly associated with bulk universality and clock behavior, singular spectral measures can correspond to varied local behaviors. In this work, the rate of decay of the potential is shown to give upper bounds for the spacing of Dirichlet eigenvalues on finite intervals.
We consider Jacobi matrices with eventually increasing sequences of diagonal and off-diagonal Jacobi parameters. We describe the asymptotic behavior of the subordinate solution at the top of the essential spectrum, and the asymptotic behavior of the spectral density at the top of the essential spectrum. In particular, allowing on both diagonal and off-diagonal Jacobi parameters perturbations of the free case of the form $- \sum_{j=1}^J c_j n^{-τ_j} + o(n^{-τ_1-1})$ with $0 < τ_1 < τ_2 < \dots < τ_J$ and $c_1>0$, we find the asymptotic behavior of the $\log$ of spectral density to order $O(\log(2-x))$ as $x$ approaches $2$. Apart from its intrinsic interest, the above results also allow us to describe the asymptotics of the spectral density for orthogonal polynomials on the unit circle with real-valued Verblunsky coefficients of the same form.
We study an initial value problem for the Toda lattice with almost periodic initial data. We consider initial data for which the associated Jacobi operator is absolutely continuous and has a spectrum satisfying a Craig-type condition, and show the boundedness and almost periodicity in time and space of solutions.
We disprove a conjecture of Breuer-Last-Simon concerning the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an $\ell^2$ bounded variation condition with step $q$. We prove existence of a.c. spectrum on a smaller set than that specified by the conjecture and prove that our result is optimal.
We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of $\mathrm{GL}(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.
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 $|d(m)| \le \sqrt{2 \varepsilon} \exp(-\frac{κ_0}{2} |m|)$. Our derivation relies on the study of the approximation via Hill operators with potentials $\tilde V (x) = \sum_{m \in \mathbb{Z}^ν} c(m) \exp (2 πi m \tilde ωx)$, where $\tilde ω$ is a rational approximation of $ω$. It turns out that the multi-scale analysis method of \cite{DG} applies to these Hill operators. Namely, in \cite{DGL} we developed the multi-scale analysis for the operators dual to the Hill operators in question. The main estimates obtained in \cite{DGL} allow us here to establish the estimates for the gap lengths and the Fourier coefficients in a form which is considerably stronger than the estimates known in the theory of Hill operators with analytic potentials in the general setting. Due to these estimates, the approximation procedure for the quasi-periodic potentials is effective, despite the fact that the rate of approximation $|ω- \tilde ω| \thicksim \tilde T^{-δ}$, $0 < δ< 1/2$ is slow, on the scale of the period $\tilde T$ of the Hill operator.
We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in $x-vt$ is replaced by exponential decay in $x-vt^α$ with $0<α<1$. In fact, we can characterize the values of $α$ for which such a bound holds as those exceeding $α_u^+$, the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of \cite{HSS11}, we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan-Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in \cite{DT07, DT08, D05, DGY} to our purposes. We also explain why our method does not extend to yield anomalous Lieb-Robinson bounds of power-law type for the random dimer model.
We consider Schrödinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show that for a dense set of limit-periodic potentials, the spectrum of the associated Schrödinger operator has Hausdorff dimension zero. In both results one can introduce a coupling constant $λ\in (0,\infty)$, and the respective statement then holds simultaneously for all values of the coupling constant.
We prove the following higher-order Szego theorems: if a measure on the unit circle has absolutely continuous part $w(θ)$ and Verblunsky coefficients $α$ with square-summable variation, then for any positive integer $m$, $\int (1-\cos θ)^m \log w(θ) dθ$ is finite if and only if $α\in \ell^{2m+2}$. This is the first known equivalence result of this kind in the regime of very slow decay, i.e. with $\ell^p$ conditions with arbitrarily large $p$. The usual difficulty of controlling higher-order sum rules is avoided by a new test sequence approach.
We study the Cauchy problem for the KdV equation $\partial_t u - 6 u \partial_x u + \partial_x^3 u = 0$ with almost periodic initial data $u(x,0)=V(x)$. We consider initial data $V$, for which the associated Schrödinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Percy Deift for this class of initial data. The result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.
We prove that the spectrum of a limit-periodic Schrödinger operator is homogeneous in the sense of Carleson whenever the potential obeys the Pastur--Tkachenko condition. This implies that a dense set of limit-periodic Schrödinger operators have purely absolutely continuous spectrum supported on a homogeneous Cantor set. When combined with work of Gesztesy--Yuditskii, this also implies that the spectrum of a Pastur--Tkachenko potential has infinite gap length whenever the potential fails to be uniformly almost periodic.
Prüfer variables are a standard tool in spectral theory, developed originally for perturbations of the free Schrödinger operator. They were generalized by Kiselev, Remling, and Simon to perturbations of an arbitrary Schrödinger operator. We adapt these generalized Prufer variables to the setting of Jacobi and Szegő recursions. We present an application to random $L^2$ perturbations of Jacobi and CMV matrices, and an application to decaying oscillatory perturbations of periodic Jacobi and CMV matrices.
We investigate quantum dynamics with the underlying Hamiltonian being a Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms modulated by a periodic or a limit-periodic sequence. In particular, we investigate the transport exponents. In the periodic case we demonstrate ballistic transport, while in the limit-periodic case we discuss various phenomena such as quasi-ballistic transport and weak dynamical localization. We also present applications to some quantum many body problems. In particular, we establish for the anisotropic XY chain on $\mathbb{Z}$ with periodic parameters an explicit strictly positive lower bound for the Lieb-Robinson velocity.
We consider the quasi-periodic Schrödinger operator $$ [H ψ](x) = -ψ"(x) + V(x) ψ(x) $$ in $L^2(\mathbb{R})$, where the potential is given by $$ V(x) = \sum_{m \in \mathbb{Z}^ν\setminus \{ 0 \}} c(m)\exp (2πi m ωx) $$ with a Diophantine frequency vector $ω= (ω_1, \dots, ω_ν) \in \mathbb{R}^ν$ and exponentially decaying Fourier coefficients $|c(m)| \le \varepsilon \exp(-κ_0|m|)$. In the regime of small $\varepsilon > 0$ we show that the spectrum of the operator $H$ is homogeneous in the sense of Carleson.
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 $\mathbb{T}^ν$, $\tildeω\in \mathbb{R}^ν$ is a vector with rational components, and $\varepsilon$ is a small parameter. The group in this particular case is the quotient $\mathfrak{T} = \mathbb{Z}^ν/\{m\in\mathbb{Z}^ν:m\tildeω=0\}$. We show that the general theory indeed applies to this special case, provided that the rational frequency vector $\tildeω$ obeys a suitable Diophantine condition in a large box of modes. Despite the fact that in this setting the orbits $k + mω$, $k \in \mathbb{R}$, $m\in\mathbb{Z}^ν$ are not dense, the dual eigenfunctions are exponentially localized and the eigenvalues of the operators can be described as $E(k+mω)$ with $E(k)$ being a "nice" monotonic function of the impulse $k \ge 0$. This enables us to derive a description of the Floquet solutions and the band-gap structure of the spectrum, which we will use in a companion paper to develop a complete inverse spectral theory for the Sturm-Liouville equation with small quasi-periodic potential via periodic approximation of the frequency. The analysis of the gaps in the range of the function $E(k)$ plays a crucial role in this approach.
We consider products of the matrices associated with the Szegő recursion from the theory of orthogonal polynomials on the unit circle and show that under suitable assumptions, their norms grow exponentially in the number of factors. In the language of dynamical systems, this result expresses a uniform hyperbolicity statement. We present two applications of this result. On the one hand, we identify explicitly the almost sure spectrum of extended CMV matrices with non-negative random Verblunsky coefficients. On the other hand, we show that no Ising model in one dimension exhibits a phase transition. Also, in the case of dynamically generated interaction couplings, we describe a gap labeling theorem for the Lee-Yang zeros in the thermodynamic limit.
We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a quasi-periodic transversal magnetic field. By "anomalous", we mean that the usual effective light cone defined by $|x|\leq v|t|$ is replaced by the region $|x|\leq v|t|^α$ for some $0<α<1$. In fact, we can characterize exactly the values of $α$ for which this holds as those exceeding the upper transport exponent $α_u^+$ of an appropriate one-body discrete Schrödinger operator. Previous study has produced a good amount of quantitative information on $α_u^+$. The result is obtained by mapping to free fermions, obtaining good dynamical bounds on the one-body level by adapting techniques developed by Damanik, Gorodetski, Tcheremchantsev, and Yessen and then "pulling back" these bounds through the non-local Jordan-Wigner transformation, following an idea of Hamza, Sims, and Stolz. To our knowledge, this is the first rigorous derivation of anomalous many-body transport. We also explain why our method does not extend to yield anomalous LR bounds of power-law type if one replaces the quasi-periodic field by a random dimer field.
Recent results of Denisov and Kaluzhny-Shamis describe the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an l^2 bounded variation condition with step p and are asymptotically periodic. We extend these results to orthogonal polynomials on the unit circle. We also replace the asymptotic periodicity condition by the weaker condition of convergence to an isospectral torus and, for p=1 and p=2, we remove even that condition.