METAPLECTIC REPRESENTATION, CONLEY–ZEHNDER INDEX, AND WEYL CALCULUS ON PHASE SPACE

2007 ◽  
Vol 19 (10) ◽  
pp. 1149-1188 ◽  
Author(s):  
MAURICE DE GOSSON
Keyword(s):  
Phase Space ◽  
Differential Operators ◽  
Heisenberg Algebra ◽  
Symplectic Space ◽  
Weyl Symbol ◽  
Metaplectic Representation ◽  
Weyl Calculus ◽  
Schrödinger Representation ◽  
Precise Calculation ◽  
Pseudo Differential Operators

We define and study a metaplectically covariant class of pseudo-differential operators acting on functions on symplectic space and generalizing a modified form of the usual Weyl calculus. This construction requires a precise calculation of the twisted Weyl symbol of a class of generators of the metaplectic group and the use of a Conley–Zehnder type index for symplectic paths, defined without restrictions on the endpoint. Our calculus is related to the usual Weyl calculus using a family of isometries of L2(ℝn) on closed subspaces of L2(ℝ2n) and to an irreducible representation of the Heisenberg algebra distinct from the usual Schrödinger representation.

scholarly journals Structure Constants of the Weyl Calculus

2013 ◽  
Vol 112 (1) ◽  
pp. 112 ◽  
Author(s):  
Wen Deng
Keyword(s):  
Phase Space ◽  
Differential Operators ◽  
Weyl Calculus ◽  
Pseudo Differential Operators ◽  
Structure Constants ◽  
Explicit Bounds

We find some explicit bounds on the $\mathcal{L}(L^2)$-norm of pseudo-differential operators with symbols defined by a metric on the phase space. In particular, we prove that this norm depends only on the "structure constants" of the metric and a fixed semi-norm of the symbol. Analogous statements are made for the Fefferman-Phong inequality.

scholarly journals -BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS

2017 ◽  
Vol 18 (3) ◽  
pp. 531-559 ◽  
Author(s):  
Julio Delgado ◽  
Michael Ruzhansky
Keyword(s):  
Phase Space ◽  
Vector Fields ◽  
Differential Operators ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Pseudo Differential Operators ◽  
Different Types ◽  
Unitary Dual ◽  
Laplacian Operators ◽  
Noncommutative Analogue

Given a compact Lie group$G$, in this paper we establish$L^{p}$-bounds for pseudo-differential operators in$L^{p}(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the noncommutative analogue of the phase space$G\times \widehat{G}$, where$\widehat{G}$is the unitary dual of$G$. We obtain two different types of$L^{p}$bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using$\mathscr{S}_{\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF}}^{m}(G)$classes which are a suitable extension of the well-known$(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF})$ones on the Euclidean space. The results herein extend classical$L^{p}$bounds established by C. Fefferman on$\mathbb{R}^{n}$. While Fefferman’s results have immediate consequences on general manifolds for$\unicode[STIX]{x1D70C}>\max \{\unicode[STIX]{x1D6FF},1-\unicode[STIX]{x1D6FF}\}$, our results do not require the condition$\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$. Moreover, one of our results also does not require$\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$. Examples are given for the case of$\text{SU}(2)\cong \mathbb{S}^{3}$and vector fields/sub-Laplacian operators when operators in the classes$\mathscr{S}_{0,0}^{m}$and$\mathscr{S}_{\frac{1}{2},0}^{m}$naturally appear, and where conditions$\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$and$\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$fail, respectively.

scholarly journals METAPLECTIC FORMULATION OF THE WIGNER TRANSFORM AND APPLICATIONS

2013 ◽  
Vol 25 (10) ◽  
pp. 1343010 ◽  
Author(s):  
NUNO COSTA DIAS ◽  
MAURICE A. DE GOSSON ◽  
JOÃO NUNO PRATA
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Differential Operators ◽  
Space Representation ◽  
Wigner Transform ◽  
Frequency Space ◽  
Time Frequency ◽  
Pseudo Differential Operators ◽  
The Fourier Transform ◽  
Metaplectic Operator

We show that the cross Wigner function can be written in the form [Formula: see text] where [Formula: see text] is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics.

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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