Mean field dynamics of a quantum tracer particle interacting with a boson gas

We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in ${\mathbb{R}}^3$. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to $\frac{1}{N}$ where N is the expected particle number. Assuming that the mass of the tracer particle is proportional to N, we derive generalized Hartree equations in the limit $N\to \infty$. Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials.     

Discrete spectrum of electromagnetic Dirac operators

We consider the Dirac operators with electromagnetic fields on 2-dimensional Euclidean space. We offer the sufficient conditions for electromagnetic fields that the associated Dirac operator has only discrete spectrum.

     

On the spectrum of magnetic Dirac operators with Coulomb-type perturbations

We carry out the spectral analysis of singular matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the perturbations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Constant, periodic as well as diverging magnetic fields are covered, and Coulomb potentials up to the physical nuclear charge Z<137 are allowed. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.     

Hausdorff operators on the Heisenberg group

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H n. The sharp bounds for the strong type(p, p)(1 ≤ p ≤∞) estimates of n-dimensional Hausdorff operators on H n are obtained. The sharp bounds for strong(p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on H n. The weak type(p, p)(1 ≤ p ≤∞) estimates are also obtained.     

Multiplicities of the eigenvalues of periodic Dirac operators

Let us consider the Dirac operator

     

Spectral asymptotics and regularized traces for Dirac operators on a star-shaped graph

In this article, we consider the spectral problems for Dirac operators on a star-shaped graph. First, we derive the asymptotic expressions of the eigenvalues and then explicitly establish the regularized trace formulae for the operators in terms of the coefficients of the operators.     

An approach to the spectrum structure of Dirac operators by the local-compactness method

The purpose of this paper is to investigate the spectra of the Dirac operator . The local compactness of is shown under some assumption on . This method enables us to prove that if as , then and to give a significant sufficient condition that or has a purely discrete spectrum.

     

Minimal kernels of Dirac operators along maps

Let M be a closed spin manifold and let N be a closed manifold. For maps and Riemannian metrics g on M and h on N, we consider the Dirac operator of the twisted Dirac bundle . To this Dirac operator one can associate an index in . If M is 2‐dimensional, one gets a lower bound for the dimension of the kernel of out of this index. We investigate the question whether this lower bound is obtained for generic tupels .     

Heisenberg evolution of quantum observables represented by unbounded operators

This paper deals with open quantum systems. In particular, we focus on the adjoint quantum master equations with initial conditions given by unbounded operators. Examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. The article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the Born-Markov approximations. To this end, we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of Schrödinger type. Furthermore, we examine quantum dynamical semigroup properties of the Heisenberg evolutions of general classes of observables.     

Inverse problems for impulsive Dirac operators with spectral parameters contained in the boundary and multitransfer conditions

An inverse spectral problem is considered for Dirac operators with parameter‐dependent transfer conditions inside the interval, and parameter appears also in one boundary condition. The approach that was used in the investigation of uniqueness theorems of inverse problems for Weyl function or two eigenvalue sets is employed. Copyright © 2014 John Wiley & Sons, Ltd.     

Far‐field patterns of solutions of the perturbed Dirac equation

The purpose of the paper is to study the asymptotic behavior at infinity of solutions of a perturbed Dirac equation in called k‐monogenic. Every such solution is a solution of the Helmholtz equation with values in a complex Clifford algebra. The main goal is to use the far‐field pattern to characterize the radiating (outgoing) k‐monogenic functions among the radiating solutions of the Helmholtz equation. It will be shown that an algebraic condition characterizes these far‐field patterns. Copyright © 2012 John Wiley & Sons, Ltd.     

On a refinement of Heisenberg uncertainty relation by means of quantum Fisher information

We show that an uncertainty relation proved by Luo and Yanagi and related to Wigner-Yanase-Dyson information cannot hold for an arbitrary quantum Fisher information. We show that, by changing a costant, a similar trace inequality holds in full generality.     

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

We extend classical results of Kostant et al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.     

Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group

We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group , of the form

where is a complex matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that and their commutator are linearly independent, we show that is not locally solvable, even in the presence of lower-order terms, provided that . In the case we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group a phenomenon first observed by Karadzhov and Müller in the case of It is interesting to notice that the analysis of the exceptional operators for the case turns out to be more elementary than in the case When the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.

     

Deforming the point spectra of one-dimensional Dirac operators

We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original and deformed operators are unitarily equivalent when restricted to the complement of the subspace spanned by the newly inserted eigenvalue. Moreover, the unitary transformation operator which links the original operator to its deformed version is explicitly determined.

     

Inverse problems with partial data for a Dirac system: A Carleman estimate approach

We prove that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary. The proof is based on a combination of Carleman estimates for first and second order systems, and involves a reduction of the boundary measurements to the second order case. For this reduction a certain amount of decoupling is required. To effectively make use of the decoupling, the Carleman estimates are established for coefficients which may become singular in the asymptotic limit.     

Lower Bounds for the Eigenvalues of the Dirac Operator: Part I. The Hypersurface Dirac Operator

We give optimal lower bounds for the hypersurface Diracoperator in terms of the Yamabe number, the energy-momentum tensor andthe mean curvature. In the limiting case, we prove that the hypersurfaceis an Einstein manifold with constant mean curvature.