arXiv subjects
Publications and source records attributed to Radu Purice.
For families of magnetic pseudodifferential operators defined by symbols and magnetic fields depending continuously on a real parameter $ε$, we show that the corresponding family of spectra also varies continuously with $ε$.
Consider a small sample coupled to a finite number of leads, and assume that the total (continuous) system is at thermal equilibrium in the remote past. We construct a non-equilibrium steady state (NESS) by adiabatically turning on an electrical bias between the leads. The main mathematical challenge is to show that certain adiabatic wave operators exist, and to identify their strong limit when the adiabatic parameter tends to zero. Our NESS is different from, though closely related with the NESS provided by the Jak{\v s}i{ć}-Pillet-Ruelle approach. Thus we partly settle a question asked by Caroli {\it et al} in 1971 regarding the (non)equivalence between the partitioned and partition-free approaches.
We prove rapid decay (even exponential decay under some stronger assumptions) of the eigenfunctions associated to discrete eigenvalues, for a class of self-adjoint operators in $L^2(\mathbb{R}^d)$ defined by ``magnetic'' pseudodifferential operators (studied in \cite{IMP1}). This class contains the relativistic Schrödinger operator with magnetic field.
The aim of these notes is to show how the magnetic calculus developed in \cite{MP, IMP1, IMP2, MPR, LMR} permits to give a new information on the nature of the coefficients of the expansion of the trace of a function of the magnetic Schrödinger operator whose existence was established in \cite{HR2}.
We introduce magnetic coherent states for a particle in a variable magnetic field. They provide a pure state quantization of the phase space R^{2N} endowed with a magnetic symplectic form.
We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schroedinger operators with magnetic fields and scalar potentials, the first one relying on the eigenvalue counting function of operators induced on open bounded sets with Dirichlet boundary conditions, the other one involving the spectral projections of the operator defined on the entire space. In this way one generalizes previous results for non-relativistic operators. The proofs needs the magnetic pseudodifferential calculus, as well as a Feynman-Kac-Ito formula for Levy processes. In addition, in case when both the magnetic field and the scalar potential are periodic, one also proves the existence of the IDS.
We extend the Bargmann transform to the magnetic pseudodifferential calculus, using gauge-covariant families of coherent states. We also introduce modulation mappings, a first step towards adapting modulation spaces to the magnetic case.
The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudo-differential operators of suitable symbol classes. The approach is completely intrinsic; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudo-differential operator.
We review recent results on the magnetic pseudo-differential calculus both in symbolic and in $C^*$-algebraic form. We also indicate some applications to spectral analysis of pseudo-differential operators with variable magnetic fields.
We prove the analog of the Cwickel-Lieb-Rosenblum estimation for the number of negative eigenvalues of a relativistic Hamiltonian with magnetic field $B\in C^\infty_{\rm{pol}}(\mathbb R^d)$ and an electric potential $V\in L^1_{\rm{loc}}(\mathbb R^d)$, $V_-\in L^d(\mathbb R^d)\cap L^{d/2}(\mathbb R^d)$. Compared to the nonrelativistic case, this estimation involves both norms of $V_-$ in $L^{d/2}(\mathbb R^d)$ and in $L^{d}(\mathbb R^d)$. A direct consequence is a Lieb-Thirring inequality for the sum of powers of the absolute values of the negative eigenvalues.
Consider a three dimensional system which looks like a cross-connected pipe system, i.e. a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We finally prove that in virtually all physically interesting situations, the conductivity tensor is given by a Landauer-B{ü}ttiker type formula.
In previous papers, a generalization of the Weyl calculus was introduced in connection with the quantization of a particle moving in $\mathbb R^n$ under the influence of a variable magnetic field $B$. It incorporates phase factors defined by $B$ and reproduces the usual Weyl calculus for B=0. In the present article we develop the classical pseudodifferential theory of this formalism for the standard symbol classes $S^m_{ρ,δ}$. Among others, we obtain properties and asymptotic developments for the magnetic symbol multiplication, existence of parametrices, boundedness and positivity results, properties of the magnetic Sobolev spaces. In the case when the vector potential $A$ has all the derivatives of order $\ge 1$ bounded, we show that the resolvent and the fractional powers of an elliptic magnetic pseudodifferential operator are also pseudodifferential. As an application, we get a limiting absorption principle and detailed spectral results for self-adjoint operators of the form $H=h(Q,Π^A)$, where $h$ is an elliptic symbol, $Π^A=D-A$ and $A$ is the vector potential corresponding to a short-range magnetic field.
We study generalised magnetic Schroedinger operators of the form H(A,V)=h(P^A)+V, where h is an elliptic symbol, P^A is the generator of the magnetic translations, with A a vector potential defining a variable magnetic field B, and V is a scalar potential. We are mainly interested in anisotropic functions B and V. The first step is to show that these operators are affiliated to suitable C*-algebras of (magnetic) pseudodifferential operators. A study of the quotient of these C*-algebras by the ideal of compact operators leads to formulae for the essential spectrum of H(A,V), expressed as a union of spectra of some asymptotic operators, supported by the quasi-orbits of a suitable dynamical system. The quotient of the same C*-algebras by other ideals give localization results on the functional calculus of the operators H(A,V), which can be interpreted as non-propagation properties of their unitary groups.
Recently, we introduced a mathematical framework for the quantization of a particle in a variable magnetic field. It consists in a modified form of the Weyl pseudodifferential calculus and a C*-algebraic setting, these two points of view being isomorphic in a suitable sense. In the present paper we leave Planck's constant vary, showing that one gets a strict deformation quantization in the sense of Rieffel. In the limit h --> 0 one recovers a Poisson algebra induced by a symplectic form defined in terms of the magnetic field.
There is a connection between the Weyl pseudodifferential calculus and crossed product C*-algebras associated with certain dynamical systems. And in fact both topics are involved in the quantization of a non-relativistic particle moving in R^n. Our paper studies the situation in which a variable magnetic field is also present. The Weyl calculus has to be modified, giving a functional calculus for a family of operators (positions and magnetic momenta) with highly non-trivial commutation relations. On the algebraic side, the dynamical system is twisted by a cocyle defined by the flux of the magnetic field, leading thus to twisted crossed products. We outline the interplay between the modified pseudodifferential setting and the C*-algebraic formalism at an abstract level as well as in connection with magnetic field.
In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on ``the minimal coupling principle'' at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.