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 \({{\Im{V}}}\).     

Dispersive Estimates for Schrödinger Operators in Dimension Two

We prove L¹(²)L(²) for the two-dimensional Schrödinger operator –+V with the decay rate t^–1. We assume that zero energy is neither an eigenvalue nor a resonance. This condition is formulated as in the recent paper by Jensen and Nenciu on threshold expansions for the two-dimensional resolvent.The author was partially supported by the NSF grant DMS-0300081 and a Sloan Fellowship     

Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

We consider L¹L estimates for the time evolution of Hamiltonians H=–+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is (1+|x|)|V(x)|dx<, whereas in d=3 it is |V(x)|C(1+|x|)^–3–.Supported by the NSF grant DMS-0070538 and a Sloan fellowship.     

Structural Resolvent Estimates and Derivative Nonlinear Schrödinger Equations

A refinement of a uniform resolvent estimate is given and several smoothing estimates for Schrödinger equations in the critical case are induced from it. The relation between this resolvent estimate and a radiation condition is discussed. As an application of critical smoothing estimates, we show a global existence result for derivative nonlinear Schrödinger equations.     

A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions

In dimension n > 3 we show the existence of a compactly supported potential in the differentiability class , for which the solutions to the linear Schrödinger equation in,

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.     

Tunneling Estimates for Magnetic Schrödinger Operators

We study the behavior of eigenfunctions in the semiclassical limit for Schr?dinger operators with a simple well potential and a (non-zero) constant magnetic field. We prove an exponential decay estimate on the low-lying eigenfunctions, where the exponent depends explicitly on the magnetic field strength. Received: 30 March 1998 / Accepted: 1 May 1998     

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.     

Schr?dinger Dispersive Estimates for a Scaling-Critical Class of Potentials

We prove a dispersive estimate for the evolution of Schr?dinger operators H = ??? + V(x) in ${{\mathbb R}^3}$ . The potential should belong to the closure of ${C^c_b(\mathbb{R}^3)}$ with respect to the global Kato norm. Some additional spectral conditions are imposed, namely that no resonances or eigenfunctions of H exist anywhere within the interval [0, ??). The proof is an application of a new version of Wiener??s L ¹-inversion theorem.     

Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose–Einstein condensates in a one-dimensional periodic lattice is also discussed.     

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     

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.     

Anderson Localization for Time Quasi-Periodic Random Schrödinger and Wave Equations

We prove that at large disorder, with large probability and for a corresponding set of Diophantine frequencies of large measure, Anderson localization in ^d is stable under localized time quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Fröhlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. This paper paves the way for the construction of time quasi-periodic KAM type of solutions of non linear random Schrödinger equations in [BW].Wei-Min Wang thanks A. Soffer and T. Spencer for many useful conversations and for initiations to the subject. She also thanks M. Combescure and J. Sjöstrand for helpful discussions on the quasi-energy operator formulation of time dependent Schrödinger equations. The support of NSF grant DMS 9729992 is gratefully acknowleged.     

Nonlinear Schrödinger Equations and the Separation Property

Abstract

We investigate hierarchies of nonlinear Schrödinger equations for multiparticle systems satisfying the separation property, i.e., where product wave functions evolve by the separate evolution of each factor. Such a hierarchy defines a nonlinear derivation on tensor products of the single-particle wave-function space, and satisfies a certain homogeneity property characterized by two new universal physical constants. A canonical construction of hierarchies is derived that allows the introduction, at any particular “threshold” number of particles, of truly new physical effects absent in systems having fewer particles. In particular, if single quantum particles satisfy the usual (linear) Schrödinger equation, a system of two particles can evolve by means of a fairly simple nonlinear Schrödinger equation without violating the separation property. Examples of Galileian-invariant hierarchies are given.     

A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Let λ _i (Ω,V) be the i ^th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^nLet λ _i (Ω,V) be the i ^th eigenvalue of the Schr?dinger operator with Dirichlet boundary conditions on a bounded domain and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V _*, we prove that λ₂(Ω,V) ≤ λ₂(S ₁,V _*). Here S ₁ denotes the ball, centered at the origin, that satisfies the condition λ₁(Ω,V)=λ₁(S ₁,V _*).Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ₂(B _R , V) / λ₁(B _R , V) decreases when the radius R of the ball B _R increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.R.B. was supported by FONDECYT project # 102-0844.H.L. gratefully acknowledges financial support from DIPUC of the Pontifí cia Universidad Católica de Chile and from CONICYT.