Finite Gap Potentials and WKB Asymptotics¶for One-Dimensional Schrödinger Operators

Consider the Schr?dinger operator H=−d ²/dx ²+V(x) with power-decaying potential V(x)=O(x ^−α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range of validity. The construction relies on pointwise bounds on finite gap potentials. These bounds are obtained by an analysis of the Jacobi inversion problem on hyperelliptic Riemann surfaces. Received: 14 March 2001 / Accepted: 27 June 2001     

On the Absolutely Continuous Spectrum¶of One-Dimensional Schrödinger Operators¶with Square Summable Potentials

For continuous and discrete one-dimensional Schrödinger operators with square summable potentials, the absolutely continuous part of the spectrum is essentially supported by [0,X) and [ф,2] respectively. This fact is proved by considering a priori estimates for the transmission coefficient.     

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.     

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 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.     

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.     

A Class of Schrödinger Operators with Decaying Oscillatory Potentials

We discuss Schrödinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous spectrum and give bounds on the Hausdorff dimension of the singular part of the spectral measure. We also discuss the analogs for orthogonal polynomials on the real line and unit circle.     

Schrödinger Operators with Random Sparse Potentials. Existence of Wave Operators

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) (No. 15340005) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject classifications (2000). primary, 34M35, secondary, 33E20.This revised version was published online in March 2005 with corrections to the cover date.     

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 \({{\Im{V}}}\).     

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     

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     

Inverse Spectral Problems for Schrödinger Operators

In this article we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrödinger operator. As an application of these new formulas for the wave invariants, we improve the inverse spectral results proved by Guillemin and Uribe in [GU]. They proved that under some symmetry assumptions on the potential V(x), the Taylor expansion of V(x) near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in \({\mathbb R^n}\) . We prove similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V(x).     

Spectral Theory for Periodic Schrödinger Operators with Reflection Symmetries

Let H=–+V be defined on ^d with smooth potential V, such that In addition we assume that where This is a periodic Schrödinger operator with additional reflection symmetries. We investigate the associated Floquet operators H ^q , q[0,1] ^d . In particular we show that the associated lowest eigenvalues _q are simple if q=(q ₁ ,q ₂ ,,q _d ) satisfies q _j 1/2 for each j=1,2,,d. Supported by Ministerium für Bildung, Wissenschaft und Kunst der Republik ÖsterreichSupported by the European Science Foundation Programme Spectral Theory and Partial Differential Equations (SPECT)     

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

On Minimal Eigenvalues¶of Schrödinger Operators on Manifolds

We consider the problem of minimizing the eigenvalues of the Schr?dinger operator H=−Δ+αF(κ) (α>0) on a compact n-manifold subject to the restriction that κ has a given fixed average κ₀. In the one-dimensional case our results imply in particular that for F(κ)=κ² the constant potential fails to minimize the principal eigenvalue for α>α_c=μ₁/(4κ₀ ²), where μ₁ is the first nonzero eigenvalue of −Δ. This complements a result by Exner, Harrell and Loss, showing that the critical value where the constant potential stops being a minimizer for a class of Schr?dinger operators penalized by curvature is given by α _c . Furthermore, we show that the value of μ₁/4 remains the infimum for all α >α _c . Using these results, we obtain a sharp lower bound for the principal eigenvalue for a general potential. In higher dimensions we prove a (weak) local version of these results for a general class of potentials F(κ), and then show that globally the infimum for the first and also for higher eigenvalues is actually given by the corresponding eigenvalues of the Laplace–Beltrami operator and is never attained. Received: 17 July 2000 / Accepted: 11 October 2000     

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.     

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).     

Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields

We consider Schrödinger operators H^h=(ihd+A)*(ihd+A) with the periodic magnetic field B=dA on covering spaces of compact manifolds. Using methods of a paper by Kordyukov, Mathai and Shubin [14], we prove that, under some assumptions on B, there are in arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of the strong magnetic field h0.Acknowledgement I am very thankful to Bernard Helffer for bringing these problems to my attention and useful discussions and to Mikhail Shubin for his comments.     

Level Statistics for One-Dimensional Schrödinger Operators and Gaussian Beta Ensemble

We study the level statistics for two classes of 1-dimensional random Schrödinger operators: (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random potential. As a byproduct of (2) with our previous result (Kotani and Nakano in Festschrift Masatoshi Fukushima, 2013) imply the coincidence of the limits of circular and Gaussian beta ensembles.