Semiclassical resolvent estimates for two and three–body schrödinger operators

We prove uniform resolvent estimates for semiclassical three–body Schrödinger operators under a non–trapping condition for the classical flow of all subsystems. We also prove resolvent estimates for two–body Schrödinger operators with positive potentials when the energy level and the Planck constant tend both to zero.     

Renormalized two-body low-energy scattering

For a class of long-range potentials, including ultra-strong perturbations of the attractive Coulomb potential in dimension d ≥ 3, we introduce a stationary scattering theory for Schrödinger operators which is regular at zero energy. In particular, it is well-defined at this energy, and we use it to establish a characterization there of the set of generalized eigenfunctions in an appropriately adapted Besov space, generalizing parts of [DS1]. Principal tools include global solutions of the eikonal equation and strong radiation condition bounds.     

Local energy decay of solutions to the wave equation for nontrapping metrics

We prove uniform local energy decay estimates of solutions to the wave equation on unbounded Riemannian manifolds with nontrapping metrics. These estimates are derived from the properties of the resolvent at high frequency. Applications to a class of asymptotically Euclidean manifolds as well as to perturbations by non-negative long-range potentials are given.     

Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting

We show that the singularities of a matrix-valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization. The proof uses a new notion of noncommutative backward shifts. As an application, we establish the commutative counterpart of the singularities theorem: the singularities of a matrix-valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its Fornasini-Marchesini realizations with the minimal possible state space dimension. The singularities results imply the absence of zero-pole cancellations in a minimal factorization, both in the noncommutative and in the commutative setting.     

Zero energy asymptotics of the resolvent for a class of slowly decaying potentials

We prove a limiting absorption principle at zero energy for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter tends to zero in cones which are adjacent to the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates.in final form: 14 November 2003S. Fournais was supported by a grant from the Carlsberg Foundation (before 31.12.02) and by a Marie Curie Fellowship of the European Community Programme Improving the Human Research Potential and the Socio-Economic Knowledge Base under contract number HPMF-CT-2002-01822 (from 01.01.03).E. Skibsted is (partially) supported by MaPhySto – A Network in Mathematical Physics and Stochastics funded by The Danish National Research Foundation.     

Continuum Schrödinger Operators Associated With Aperiodic Subshifts

We study Schrödinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik–Lenz and Klassert–Lenz–Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue measure if the subshift satisfies the Boshernitzan condition and the potentials are aperiodic and irreducible. We then study the case of the Fibonacci subshift in detail and prove results for the local Hausdorff dimension of the spectrum at a given energy in terms of the value of the associated Fricke–Vogt invariant. These results are elucidated for some simple choices of the local potential pieces, such as piecewise constant ones and local point interactions. In the latter special case, our results explain the occurrence of so-called pseudo bands, which have been pointed out in the physics literature.     

The semiclassical resolvent on conformally compact manifolds with variable curvature at infinity

We construct a semiclassical parametrix for the resolvent of the Laplacian acting on functions on nontrapping conformally compact manifolds with variable sectional curvature at infinity. We apply this parametrix to analyze the Schwartz kernel of the semiclassical resolvent and Poisson operator and to show that the semiclassical scattering matrix is a semiclassical Fourier Integral Operator of appropriate class that quantizes the scattering relation. We also obtain high energy estimates for the resolvent and show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to obtain semiclassical resolvent estimates on certain conformally compact manifolds with hyperbolic trapping.     

A finite‐time blow‐up result for a class of solutions with positive initial energy for coupled system of heat equations with memories

In this article, we are interested by a system of heat equations with initial condition and zero Dirichlet boundary conditions. We prove a finite‐time blow‐up result for a large class of solutions with positive initial energy.     

On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations

Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented. Received August 29, 1994 / Revised version received September 19, 1995     

Global Well-Posedness for a Nonlocal Gross–Pitaevskii Equation with Non-Zero Condition at Infinity

We study the Gross–Pitaevskii equation involving a nonlocal interaction potential. Our aim is to give sufficient conditions that cover a variety of nonlocal interactions such that the associated Cauchy problem is globally well-posed with non-zero boundary condition at infinity, in any dimension. We focus on even potentials that are positive definite or positive tempered distributions.     

Solution set splitting at low energy levels in Schrödinger equations with periodic and symmetric potential

The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to explicitly verify the condition, and we carry out the calculation in dimension one for a specific class of potentials.     

Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps

Summary. We consider a -dimensional Euclidean domain whose boundary is Lipschitz continuous but admits locally finite number of outward or inward H?lder cusp points. Using a method of Stampacchia and Moser for PDE, we first construct a conservative diffusion process on the Euclidean closure of possessing a strong Feller resolvent and associated with a second order uniformly elliptic differential operator of divergence form with measurable coefficients . The sample path of the constructed diffusion can be uniquely decomposed as a sum of a martingale additive functional and an additive functional locally of zero energy. The second additive functional will be proved to be of bounded variation with a Skorohod type expression whenever is weakly differentiable and the H?lder exponent at each outward cusp boundary point is greater than regardless the dimension . Received: 4 October 1995     

Spectral Asymptotics for Schrödinger Operators with Periodic Point Interactions

Spectrum of the second-order differential operator with periodic point interactions in L₂(R) is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely. It is proven that the spectrum of the operators with anomalous spectral asymptotics (when the ratio between the lengths of the bands and gaps tends to zero at infinity) can be approximated by standard periodic “weighted” operators with step-wise density functions. It is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal (generalized) resolvent of the periodic “Schrödinger operator” with certain energy-dependent boundary conditions. The operator acting in an extended Hilbert space such that its resolvent restricted to L₂(R) coincides with the formal resolvent is constructed explicitly.     

Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces H p σ and B p σ

In a bounded Lipschitz domain, we consider a strongly elliptic second-order equation with spectral parameter without assuming that the principal part is Hermitian. For the Dirichlet and Neumann problems in a weak setting, we prove the optimal resolvent estimates in the spaces of Bessel potentials and the Besov spaces. We do not use surface potentials. In these spaces, we derive a representation of the resolvent as a ratio of entire analytic functions with sharp estimates of their growth and prove theorems on the completeness of the root functions and on the summability of Fourier series with respect to them by the Abel-Lidskii method. Preliminarily, such questions for abstract operators in Banach spaces are discussed. For the Steklov problem with spectral parameter in the boundary condition, we obtain similar results. We indicate applications of the resolvent estimates to parabolic problems in a Lipschitz cylinder. We also indicate generalizations to systems of equations. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 4, pp. 2–23, 2008 Original Russian Text Copyright ? by M. S. Agranovich To dear Israel Moiseevich Gelfand in connection with his 95th birthday Supported by RFBR grant no. 07-01-00287.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Hausdorff dimension of perturbed Cantor sets without some boundedness condition

A perturbed Cantor set (without the uniform boundedness condition away from zero of contraction ratios) whose upper Cantor dimension and lower Cantor dimension coincide has its Hausdorff dimension of the same value of Cantor dimensions. We will show this using an energy theory instead of Frostman's density lemma which was used for the case of the perturbed Cantor set with the uniform boundedness condition. At the end, we will give a nontrivial example of such a perturbed Cantor set. This revised version was published online in August 2006 with corrections to the Cover Date.     

Noncoercive boundary value problems for the Laplace equation with a spectral parameter

In this paper we find conditions that guarantee that irregular boundary value problems for elliptic differential-operator equations of the second order in an interval are coercive with a defect and fredholm; compactness of a resolvent and estimations by spectral parameter; completeness of root functions. We apply this result to find some algebraic conditions that guarantee that irregular boundary value problems for elliptic partial differential equations of the second order in cylindrical domains have the same properties. Apparently this is the first paper where the regularity of an elliptic boundary value problem is not satisfied on a manifold of the dimension equal to the dimension of the boundary. Nevertheless, the problem is fredholm and the resolvent is compact. It is interesting to note that the considered boundary value problems for elliptic equations in a cylinder being with separating variables are noncoercive. I wish to thank the referee whose comments helped me improve the style of the paper. Supported in part by the Israel Ministry of Science and Technology and the Israel-France Rashi Foundation.     

Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators

We prove explicit coercivity estimates for the linearized Boltzmann and Landau operators, for a general class of interactions including any inverse-power law interactions, and hard spheres. The functional spaces of these coercivity estimates depend on the collision kernel of these operators. They cover the spectral gap estimates for the linearized Boltzmann operator with Maxwell molecules, improve these estimates for hard potentials, and are the first explicit coercivity estimates for soft potentials (including in particular the case of Coulombian interactions). We also prove a regularity property for the linearized Boltzmann operator with non locally integrable collision kernels, and we deduce from it a new proof of the compactness of its resolvent for hard potentials without angular cutoff.     

Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents

We investigate the spectral properties of discrete one-dimensional Schrödinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dimension and all spectral measures are purely singular continuous. The associated Lyapunov exponent is a continuous strictly positive function of the energy. It is possible to include a coupling constant in the model and these results then hold for every non-zero value of the coupling constant.     

On decay of the Schrödinger resolvent

We strengthen the known Agmon-Jensen-Kato decay of the resolvent for a special case of the Schrödinger equation in arbitrary dimension n ≥ 1. The decay is of crucial importance in applications to linear and nonlinear hyperbolic PDEs.