Magnetic Pseudodifferential Operators

Abstract

In previous papers, a generalization of the Weyl calculus was introduced in connection with the quantization of a particle moving in 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 ≥ 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, Q denotes multiplication with the variables Π A = D − A, D is the operator of derivation and A is the vector potential corresponding to a short-range magnetic field.

On the Lipschitz Continuity of Spectral Bands of Harper-Like and Magnetic Schrödinger Operators

“…These phases appear very naturally in the continuous case, see [7,8,19,22,23,24,25,26], where it is shown that if a(x) is the transverse gauge generated by a globally bounded magnetic field |b(x)| ≤ 1, then φ(x, x ′ ) can be chosen to be the path integral of a(x) on the segment linking x ′ with x. This is the same as the magnetic flux of b through the triangle generated by x, x ′ and the origin.…”

On Three Magnetic Relativistic Schrödinger Operators and Imaginary-Time Path Integrals

“…The gauge-covariance of the modified H (2) A in contrast to H (1) A in Ichinose-Tamura [13] was emphasized in Iftimie-Mȃntoiu-Purice [14,15,16] . Let us observe some of these facts in the following.…”

“…A + V is defined with term H (2) A being the pseudo-differential operator modified by Iftimie-Mȃntoiu-Purice [13,14,15]:…”

Symbol calculus of square-integrable operator-valued maps

“…The magnetic Weyl calculus [IMP07,KO02,MP04,MP10,MPR07] has as a background the problem of quantization of a physical system consisting in a spinless particle moving in the euclidean space X ∼ = R n under the influence of a magnetic field, i.e. a closed 2-form B on X (dB = 0), given in a base by matrix-component functions B jk = −B kj : X → R, j, k = 1, .…”

“…The present paper is devoted to developing a set of techniques applicable to operator valued maps on measure spaces π : (Σ, µ) → B(H) that satisfy a square integrability property analogous to that of locally compact group representations (see (2.4) below) although π may not be a group representation and µ may not be a Haar measure. This investigation was motivated by several situations when actually Σ is a group that fails to be locally compact so it does not admit any Haar measure (as for instance in the study of the canonical commutation relations where one has been looking for suitable substitutes of the group algebra for inductive limit groups [Gru97,GN09,GN13]), or when Σ is locally compact but π is not even a projective group representation (see for instance the orthogonality relations for irreducible representations of nilpotent Lie groups: [HM79,Pe94,Wo13], and the references therein; or the magnetic Weyl calculus [MP04,IMP07,MP12,BB09,BB11a]). …”