Parabolic Schrödinger operators

We characterize the domain of the parabolic Schrödinger operator t_∂−Δ+V in Lp(Rn+1), 1<p<∞, where the potential V is nonnegative and belongs to the Parabolic Reverse Hölder class p(PB).     

Generalized difference sequence spaces and their dual spaces

The definition of the pα-, pβ- and pγ-duals of a sequence space was defined by Et [Internat. J. Math. Math. Sci. 24 (2000) 785-791]. In this paper we compute pα- and N-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀, and compute β- and γ-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀.     

The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials

We study the integrated density of states of random Anderson-type additive and multiplicative perturbations of deterministic background operators for which the single-site potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L^p-theory of the spectral shift function for p?1 (Comm. Math. Phys.218 (2001), 113-130), and the vector field methods of Klopp (Comm. Math. Phys.167 (1995), 553-569). We discuss the application of this result to Schrödinger operators with random magnetic fields and to band-edge localization.     

Anderson localization and Lifshits tails for random surface potentials

We consider Schrödinger operators on L²(R^d) with a random potential concentrated near the surface R^d₁×{0}⊂R^d. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87-97] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Our proof of Lifshits tails relies on spectral properties of Schrödinger operators with partially periodic potentials. In particular, we show that the lowest energy band of such operators is parabolic.     

Spectral analysis of nonselfadjoint Schrödinger operators with a matrix potential

Dissipative Schrödinger operators with a matrix potential are studied in L₂((0,∞);E)(dimE=n<∞) which are extension of a minimal symmetric operator L₀ with defect index (n,n). A selfadjoint dilation of a dissipative operator is constructed, using the Lax-Phillips scattering theory, the spectral analysis of a dilation is carried out, and the scattering matrix of a dilation is founded. A functional model of the dissipative operator is constructed and its characteristic function's analytic properties are determined, theorems on the completeness of eigenvectors and associated vectors of a dissipative Schrödinger operator are proved.     

Existence of solutions for non-periodic superlinear Schrödinger equations without (AR) condition

The existence of solutions is obtained for a class of the non-periodic Schrödinger equation −Δu + V(x)u = f(x, u), x ∈ R^N, by the generalized mountain pass theorem, where V is large at infinity and f is superlinear as |u| → ∞.     

On some new generalized sequence spaces

In this paper we define the sequence sets ?_∞(u,Δ²,p), c(u,Δ²,p) and c₀(u,Δ²,p), and give α- and β-duals of these sets. Further we investigate matrix transformations in the spaces and give a characterization of the class (?_∞(u,Δ²,p),?_∞).     

Schrödinger operators and associated hyperbolic pencils

For a large class of Schrödinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are significantly easier to study. Then, we give some applications to the original Schrödinger operators including one-dimensional Schrödinger operators with L²-operator-valued potentials, multidimensional Schrödinger operators with slowly decaying potentials.     

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.     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).     

Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles

Mellin's transform is used to establish a functional calculus of a class of pseudodifferential-operators depending on a small parameter h > 0. We apply for exeample this result to prove the semi-classical behaviour of the discrete spectrum of Schrödinger operators ?h² · Δ + V, and of Dirac operators h ∑_{j = 1}³α_jD_j + α₄ ? V.     

The cubic fourth-order Schrödinger equation

Fourth-order Schrödinger equations have been introduced by Karpman and Shagalov to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. In this paper we investigate the cubic defocusing fourth-order Schrödinger equation

it_∂u+Δ²u+²|u|u=0     

Nonpositive curvature in p-Schatten class

We study the geometry of the set Δp, with 1<p<∞, which consists of perturbations of the identity operator by p-Schatten class operators, which are positive and invertible as elements of B(H). These manifolds have natural and invariant Finsler structures. In [C. Conde, Geometric interpolation in p-Schatten class, J. Math. Anal. Appl. 340 (2008) 920-931], we introduced the metric dp and exposed several results about this metric space. The aim of this work is to prove that the space (Δp,dp) behaves in many senses like a nonpositive curvature metric space.     

Computing Schrödinger propagators on Type-2 Turing machines

We study Turing computability of the solution operators of the initial-value problems for the linear Schrödinger equation u_t=iΔu+φ and the nonlinear Schrödinger equation of the form iu_t=-Δu+mu+|u|²u. We prove that the solution operators are computable if the initial data are Sobolev functions but noncomputable in the linear case if the initial data are L^p-functions and p≠2. The computations are performed on Type-2 Turing machines.     

Asymptotics of odd solutions for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for a cubic nonlinear Schrödinger equation in the case of an odd initial data from H²∩H0,2. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.     

Asymptotics of odd solutions of quadratic nonlinear Schrödinger equations

We consider the Cauchy problem for a quadratic nonlinear Schrödinger equation in the case of odd initial data from H²∩H0,2. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.     

Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation

We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V(x)=±²|x|.     

Schrödinger operators and unique continuation. Towards an optimal result

In this article we prove the property of unique continuation (also known for C^∞ functions as quasianalyticity) for solutions of the differential inequality |Δu|?|Vu| for V from a wide class of potentials (including class) and u in a space of solutions YV containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds?     

The focusing energy-critical Hartree equation

We consider the focusing energy-critical nonlinear Hartree equation iut+Δu=−(⁻⁴|x|∗²|u|)u. We proved that if a maximal-lifespan solution u:I×Rd→C satisfies supt∈I‖∇u(t)_‖2<‖∇W_‖2, where W is the static solution of the equation, then the maximal-lifespan I=R, moreover, the solution scatters in both time directions. For spherically symmetric initial data, similar result has been obtained in [C. Miao, G. Xu, L. Zhao, Global wellposedness, scattering and blowup for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., in press]. The argument is an adaptation of the recent work of R. Killip and M. Visan [R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint] on energy-critical nonlinear Schrödinger equations.     

Global existence of solutions for semilinear damped wave equation in 2-D exterior domain

We consider a mixed problem of a damped wave equation u_tt−Δu+u_t=|u|^p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term |u|^p satisfies p∗=2<p<+∞. For this purpose we shall deal with a radially symmetric solution in the exterior domain. A new device developed in Ikehata-Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.