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?     

On Schrödinger operators with multipolar inverse-square potentials

Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established.     

Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices

We consider C=A+B where A is selfadjoint with a gap (a,b) in its spectrum and B is (relatively) compact. We prove a general result allowing B of indefinite sign and apply it to obtain a (δV)^d/2 bound for perturbations of suitable periodic Schrödinger operators and a (not quite) Lieb–Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices.     

On a factorization of the Schrödinger and Klein–Gordon operators

A general scheme for factorizing second‐order time‐dependent operators of mathematical physics is given, which allows a reduction of corresponding second‐order equations to biquaternionic equations of first order. Examples of application of the proposed scheme are presented for both constant and variable coefficients. Copyright © 2008 John Wiley & Sons, Ltd.     

Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential on , namely We assume that and is a nonnegative function. By definition, a function belongs to the Hardy space if Under certain assumptions on V we characterize the space in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to for with no additional assumptions on the potential V.     

Scattering for Schrödinger Operators with Magnetic Fields

We study the existence and completeness of the wave operators W_ω(A(b),-Δ) for general Schrodinger operators of the form is a magnetic potential.     

Infinitely many solutions for nonlinear Schrödinger systems with magnetic potentials in

We study the following nonlinear Schrödinger system with magnetic potentials in : where μ₁>0, μ₂>0, and is a coupling constant. Under some weak symmetry conditions on A(y), P(y), and Q(y), which are given in the introduction, we prove that the nonlinear Schrödinger system has infinitely many non‐radial complex‐valued segregated and synchronized solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

A compactness criterion and application to the commutators associated with Schrödinger operators

Let T be an integral operator. In this paper, we introduce a ‐compactness criterion of , where . As an application, we apply this criterion to deal with ‐compactness of commutators associated to Schrödinger operators with potentials in the reverse Hölder's class.     

A Calderón–Zygmund operator of higher order Schrödinger type

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We show that an operator of higher order Schrödinger type is a Calderón–Zygmund operator. We also show that there exist potentials which satisfy our assumptions but are not nonnegative polynomials.     

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

For the following singularly perturbed problem we construct a solution which concentrates at several given isolated positive local minimum components of V as . Here, the nonlinearity f is of critical growth. Moreover, the monotonicity of and the so‐called Ambrosetti–Rabinowitz condition are not required.     

Integral representation for the solution of the stationary Schrödinger equation in a cone

In this paper, we consider the Dirichlet problem for the stationary Schrödinger equation in a cone with continuous boundary data. For a solution u of the stationary Schrödinger equation in a cone, we prove that if its positive part u⁺ satisfying a slowly growing condition, then its negative part u^? can also be dominated by a similar slowly growing condition. Meanwhile, u can be represented by its integral in the boundary of the cone.     

The attractor of the dissipative coupled fractional Schrödinger equations

In the present paper, we consider the dissipative coupled fractional Schrödinger equations. The global well‐posedness by the contraction mapping principle is obtained. We study the long time behavior of solutions for the Cauchy problem. We prove the existence of global attractor. Copyright © 2013 John Wiley & Sons, Ltd.     

Spectral Theory for Slowly Oscillating Potentials II. Schrödinger Operators

This paper continues the investigation about the singularity theory in dual rich quasi–Banach spaces given in T. Runst [Ru 2]. The abstract results are applied to the study of the solution structure of semilinear elliptic boundary value problems in spaces of Besov – Triebel – Lizorkin type.     

Quasilinear Schrödinger equations with unbounded or decaying potentials

We study the existence of nonnegative and nonzero solutions for the following class of quasilinear Schrödinger equations: where V and Q are potentials that can be singular at the origin, unbounded or vanishing at infinity. In order to prove our existence result we used minimax techniques in a suitable weighted Orlicz space together with regularity arguments and we need to obtain a symmetric criticality type result.     

Commutation methods for Schrödinger operators with strongly singular potentials

We explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrödinger operators (also known as Bessel operators). We also investigate the connections with the generalized Bäcklund–Darboux transformation.     

On the fractional Schrödinger problem with non‐symmetric potential

We study the semilinear equation where 0 < s < 1, , V(x) is a sufficiently smooth non‐symmetric potential with , and ? > 0 is a small number. Letting U be the radial ground state of (?Δ)^sU + U ? U^p=0 in , we build solutions of the form for points ?_j,j = 1,?,m, using a Lyapunov–Schmidt variational reduction. Copyright © 2014 John Wiley & Sons, Ltd.     

The higher order Riesz transform and BMO type space associated to Schrödinger operators

Let L = ?Δ + V be a Schrödinger operator on $\mathbb {R}^nLet L = ?Δ + V be a Schrödinger operator on $\mathbb {R}^n$ (n ≥ 3), where $V \not\equiv 0$ is a nonnegative potential belonging to certain reverse Hölder class B_s for $s \ge \frac{n}{2}$. In this article, we prove the boundedness of some integral operators related to L, such as L^?1?², L^?1V and L^?1( ? Δ) on the space $BMO_L(\mathbb {R}^n)$.