Wave front set for solutions to Schrödinger equations

We consider solutions to Schrödinger equation on Rd with variable coefficients. Let H be the Schrödinger operator and let u(t)=e−itHu₀ be the solution to the Schrödinger equation with the initial condition u₀∈L²(Rd). We show that the wave front set of u(t) in the nontrapping region can be characterized by the wave front set of e−itH₀u₀, where H₀ is the free Schrödinger operator. The characterization of the wave front set is given by the wave operator for the corresponding classical mechanical scattering (or equivalently, by the asymptotics of the geodesic flow).     

On symmetries in the theory of finite rank singular perturbations

For a nonnegative self-adjoint operator A₀ acting on a Hilbert space H singular perturbations of the form A₀+V, are studied under some additional requirements of symmetry imposed on the initial operator A₀ and the singular elements ψj. A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A₀+V. The results are applied for the investigation of singular perturbations of the Schrödinger operator in L₂(R³) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions.     

On holomorphic families of Schrödinger-type operators with singular potentials on manifolds of bounded geometry

We consider a family of Schrödinger-type differential expressions L(κ)=D²+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=^*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L²(E) and L²(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula.     

Floquet operators without singular continuous spectrum

Let U be a unitary operator defined on a infinite-dimensional separable complex Hilbert space H. Assume there exists a self-adjoint operator A on H such that

U^∗AU−A?cI+K     

Rank one perturbations and singular integral operators

We consider rank one perturbations Aα=A+α(⋅,φ)φ of a self-adjoint operator A with cyclic vector φ∈H−1(A) on a Hilbert space H. The spectral representation of the perturbed operator Aα is given by a singular integral operator of special form. Such operators exhibit what we call ‘rigidity’ and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms Tε are uniformly (in ε) bounded operators from L²(μ) to L²(μα), where μ and μα are the spectral measures of A and Aα, respectively. As an application, a sufficient condition for Aα to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of A with respect to φ. Some examples, like Jacobi matrices and Schrödinger operators with L² potentials are considered.     

Rank one perturbations at infinite coupling in Pontryagin spaces

In this paper we relate the operators in the operator representations of a generalized Nevanlinna function N(z) and of the function −N(z)⁻¹ under the assumption that z=∞ is the only (generalized) pole of nonpositive type. The results are applied to the Q-function for S and H and the Q-function for S and H^∞, where H is a self-adjoint operator in a Pontryagin space with a cyclic element w, H^∞ is the self-adjoint relation obtained from H and w via a rank one perturbation at infinite coupling, and S is the symmetric operator given by S=H∩H^∞.     

Time-decay of semigroups generated by dissipative Schrödinger operators

We establish a representation formula for semigroups of contractions in terms of a global limiting absorption principle. As applications, we prove long-time asymptotics and dispersive estimate of the semigroup generated by −iH where H is a dissipative Schrödinger operator.     

The solutions of a variational inequality as critical points of a real valued smooth function

Let H be a real Hilbert space, A : H → H a self-adjoint continuous linear operator. Given a variational inequality related to A, we construct a Lipschitz continuously differentiable function F:H→$\text{R}$ so that the critical points of F are the solutions of the variational inequality.     

Compactness results for Schrödinger equations with asymptotically linear terms

We study the nonlinear problem −Δu+V(x)=f(x,u), x∈RN, lim|x|→∞u(x)=0, where the Schrödinger operator −Δ+V is semibounded and the nonlinearity f is linearly bounded. We establish compactness of Palais-Smale sequences and Cerami sequences for the associated energy functional under general spectral-theoretic assumptions. Applying these results, we obtain existence of three nontrivial solutions if the energy functional has a mountain-pass geometry.     

Self-adjointness of relatively bounded quadratic forms and operators

We consider a Hilbert space $\text{H}$ on which is given a positive self-adjoint operator H. For densely defined bilinear forms or operators A we obtain conditions which ensure that A is an operator, that A is self-adjoint and that e^iAt leaves D(H^r) invariant with H^re^iAt strongly differentiable.     

Two-body threshold spectral analysis, the critical case

We study in dimension d?2 low-energy spectral and scattering asymptotics for two-body d-dimensional Schrödinger operators with a radially symmetric potential falling off like −γr−2, γ>0. We consider angular momentum sectors, labelled by l=0,1,…, for which γ>²(l+d/2−1). In each such sector the reduced Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. We show that the resolvent has a non-trivial oscillatory behaviour as the spectral parameter approaches zero in cones bounded away from the negative half-axis, and we derive an asymptotic formula for the phase shift.     

Jordan zero-product preserving additive maps on operator algebras

Let Φ:A→B be an additive surjective map between some operator algebras such that AB+BA=0 implies Φ(A)Φ(B)+Φ(B)Φ(A)=0. We show that, under some mild conditions, Φ is a Jordan homomorphism multiplied by a central element. Such operator algebras include von Neumann algebras, C^∗-algebras and standard operator algebras, etc. Particularly, if H and K are infinite-dimensional (real or complex) Hilbert spaces and A=B(H) and B=B(K), then there exists a nonzero scalar c and an invertible linear or conjugate-linear operator U:H→K such that either Φ(A)=cUAU⁻¹ for all A∈B(H), or Φ(A)=cUA^∗U⁻¹ for all A∈B(H).     

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).     

Paley-Wiener subspace of vectors in a Hilbert space with applications to integral transforms

The goal of this article is to introduce an analogue of the Paley-Wiener space of bandlimited functions, PWω, in Hilbert spaces and then apply the general result to more specific examples. Guided by the role that the differentiation operator plays in some of the characterizations of the Paley-Wiener space, we construct a space of vectors using a self-adjoint operator D in a Hilbert space H, and denote this space by PWω(D). The article can be virtually divided into two parts. In the first part we show that the space PWω(D) has similar properties to those of the space PWω, including an analogue of the Bernstein inequality and the Riesz interpolation formula. We also develop a new characterization of the abstract Paley-Wiener space in terms of solutions of Cauchy problems associated with abstract Schrödinger equations. Finally, we prove two sampling theorems for vectors in PWω(D), one of which uses the notion of Hilbert frames and the other is based on the notion of variational splines in H. In the second part of the paper we apply our abstract results to integral transforms associated with singular Sturm-Liouville problems. In particular we obtain two new sampling formulas related to one-dimensional Schrödinger operators with bounded potential.     

Dynamical Systems Method of gradient type for solving nonlinear equations with monotone operators

A version of the Dynamical Systems Method (DSM) of gradient type for solving equation F(u)=f where F:H→H is a monotone Fréchet differentiable operator in a Hilbert space H is studied in this paper. A discrepancy principle is proposed and the convergence to the minimal-norm solution is justified. Based on the DSM an iterative scheme is formulated and the convergence of this scheme to the minimal-norm solution is proved.     

A local trace formula for resonances of perturbed periodic Schrödinger operators

We give a local trace formula for the pair (P₁(h)=P₀+W(hy),P₀), where P₀ is a periodic Schrödinger operator, W is a decreasing perturbation and h is a small positive parameter. We apply this result to establish the existence of ∼h⁻ⁿ resonances near some energy λ of σ(P₀).     

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?     

Estimates for fundamental solutions and spectral bounds for a class of Schrödinger operators

We study a class of Schrödinger operators of the form , where is a nonnegative function singular at 0, that is V(0)=0. Under suitable assumptions on the potential V, we derive sharp lower and upper bounds for the fundamental solution hε. Moreover, we obtain information on the spectrum of the self-adjoint operator defined by Lε in L²(R). In particular, we give a lower bound for the eigenvalues.     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.     

An abstract Szeg? theorem

Given a semi-group U(t) of bounded linear operators with bounded self-adjoint generator A we estimate the logarithm of the section determinants of U(t) in terms of A. When A is subject to an additional condition, which is related to so-called Følner sequences of orthogonal projections, this estimate implies a Szeg? type theorem for bounded, self-adjoint, and strictly positive operators. We show that the condition mentioned is satisfied when A is a Toeplitz operator or a compact operator.