Dynamical Analysis of Schrödinger Operators with Growing Sparse Potentials

We consider discrete half-line Schrödinger operators H with potentials of the form V(n)=S(n)+Q(n). Here Q is any compactly supported real function, if n=L_N and S(n)=0 otherwise, where (0,1) and L_N is a very fast growing sequence. We study in a rather detailed manner the time-averaged dynamics exp(–itH) for various initial states . In particular, for some we calculate explicitly the intermittency function ^–(p) which turns out to be nonconstant. The dynamical results obtained imply that the spectral measure of H has exact Hausdorff dimension for all boundary conditions, improving the result of Jitomirskaya and Last.Acknowledgement I would like to thank F. Germinet for useful discussions.     

On the AC Spectrum of One-dimensional Random Schrödinger Operators with Matrix-valued Potentials

We consider discrete one-dimensional random Schrödinger operators with decaying matrix-valued, independent potentials. We show that if the ?²-norm of this potential has finite expectation value with respect to the product measure then almost surely the Schrödinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schrödinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon et al. (Ann. Inst. H. Poincaré Phys. Théor. 42(3):283–309, 1985).     

Estimates for Eigenvalues of Schrödinger Operators with Complex-Valued Potentials

New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrödinger operators are obtained in terms of L_p-norms of the potentials. The results cover and improve those known previously, in particular, due to Frank (Bull Lond Math Soc 43(4):745–750, 2011), Safronov (Proc Am Math Soc 138(6):2107–2112, 2010), Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009). We mention the estimations of the eigenvalues situated in the strip around the real axis (in particular, the essential spectrum). The method applied for this case involves the unitary group generated by the Laplacian. The results are extended to the more general case of polyharmonic operators. Schrödinger operators with slowly decaying potentials and belonging to weak Lebesgue’s classes are also considered.     

Eigenvalue Estimates for Schrödinger Operators with Complex Potentials

We discuss properties of eigenvalues of non-self-adjoint Schrödinger operators with complex-valued potential V. Among our results are estimates of the sum of powers of imaginary parts of eigenvalues by the L ^p -norm of \({{\Im{V}}}\).     

Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

Let H be a one-dimensional discrete Schrödinger operator. We prove that if _ess(H)[–2,2], then H–H₀ is compact and _ess(H)=[–2,2]. We also prove that if has at least one bound state, then the same is true for H₀+V. Further, if has infinitely many bound states, then so does H₀+V. Consequences include the fact that for decaying potential V with , H₀+V has infinitely many bound states; the signs of V are irrelevant. Higher-dimensional analogues are also discussed. Supported in part by NSF grant DMS-0227289On leave from Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801-2975, USASupported in part by NSF grant DMS-0140592     

Absolutely Continuous Spectrum of Schrödinger Operators with Slowly Decaying and Oscillating Potentials

The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by [0,). Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables.Acknowledgement A.L and O.S. are grateful for the partial support of the ESF European programme SPECT. S.N. would like to thank the Gustafsson Foundation which has allowed him to spend one month at the Royal Institute of Technology in Stockholm. This research was also partly supported by the KBN grant 5, PO3A/026/21. g1925l.     

Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential

We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models, we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.     

Eigenvalue Order Statistics for Random Schrödinger Operators with Doubly-Exponential Tails

We consider random Schrödinger operators of the form \({\Delta+\xi}\), where \({\Delta}\) is the lattice Laplacian on \({\mathbb{Z}^{d}}\) and \({\xi}\) is an i.i.d. random field, and study the extreme order statistics of the Dirichlet eigenvalues for this operator restricted to large but finite subsets of \({\mathbb{Z}^{d}}\). We show that, for \({\xi}\) with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class, and the corresponding eigenfunctions are exponentially localized in regions where \({\xi}\) takes large, and properly arranged, values. The picture we prove is thus closely connected with the phenomenon of Anderson localization at the spectral edge. Notwithstanding, our approach is largely independent of existing methods for proofs of Anderson localization and it is based on studying individual eigenvalue/eigenfunction pairs and characterizing the regions where the leading eigenfunctions put most of their mass.     

Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

In this paper, we study the loss of coherence of a wave propagating according to the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence phenomena is a properly rescaled Wigner transform of the solution of the random Schrödinger equation. We exhibit anomalous wave decoherence effects at different propagation scales.     

Continuity of the Measure of the Spectrum for Quasiperiodic Schrödinger Operators with Rough Potentials

We study discrete quasiperiodic Schrödinger operators on ${\ell^2(\mathbb{Z})}$ with potentials defined by γ-Hölder functions. We prove a general statement that for γ > 1/2 and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of convergence from above in the subadditive ergodic theorem for strictly ergodic cocycles.     

On the Absolutely Continuous Spectrum of Multi-Dimensional Schrödinger Operators with Slowly Decaying Potentials

We consider a multi-dimensional Schrödinger operator –+V in L²(R^d) and find conditions on the potential V which guarantee that the absolutely continuous spectrum of this operator is essentially supported by the positive real line. We prove some results which go beyond the case L¹+L^p with p<d.The author is grateful to Gunter Stolz for useful discussions. The work was supported by the grant of NSF DMS-0245210.     

Anderson Localization for Schrödinger Operators on ℤ with Strongly Mixing Potentials

In this paper we show that for a.e. x∈[ 0,2 π) the operators defined on as

Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators

We prove a unique continuation principle for spectral projections of Schrödinger operators. We consider a Schrödinger operator H = ?Δ + V on ${{\rm L}^2(\mathbb{R}^d)}$ L 2 ( R d ) , and let H _Λ denote its restriction to a finite box Λ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type χ _I (H _Λ ) W χ _I (H _Λ ) ≥ κ χ _I (H _Λ ) with κ > 0 for appropriate potentials W ≥ 0 and intervals I. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schrödinger operators with alloy-type random potentials (‘crooked’ Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.     

The Absolute Continuity of the Integrated Density¶of States for Magnetic Schrödinger Operators¶with Certain Unbounded Random Potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr?dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schr?dinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schr?dinger operators with Neumann boundary conditions. Received: 20 October 2000 / Accepted: 8 March 2001     

Perturbations of Magnetic Schrödinger Operators

We study Schrödinger operators H(a, V): = (P – a)² + V acting in L ²(³). We assume that the magnetic field B = rot a may be decomposed as B = B ⁰ + B, where B ⁰ is a very general field having constant direction. The perturbations B and V will be small in a certain sense in the direction of B ⁰, but in the orthogonal plane they may even grow for certain fields B ⁰. Commutator methods are used to derive spectral properties of H(a, V).     

WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

Consider a Schr?dinger operator on L ² of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a finite sum of terms, each of which has a derivative of some order in L ¹+L ^p for some exponent p<2, then an essential support of the the absolutely continuous spectrum equals ℝ⁺. Almost every generalized eigenfunction is bounded, and satisfies certain WKB-type asymptotics at infinity. If moreover these derivatives belong to L ^p with respect to a weight |x|^γ with γ >0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one. Received: 27 July 2000 / Accepted: 23 October 2000     

Schrödinger Operators with Few Bound States

We show that whole-line Schrödinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schrödinger operators, H, with bounded positive ground states: Given a potential V, if both H±V are bounded from below by the ground-state energy of H, then V≡0.D. D. was supported in part by NSF grant DMS–0227289.R. K. was supported in part by NSF grant DMS–0401277.B. S. was supported in part by NSF grant DMS–0140592.