The Weyl Calculus: Finite Dimensional Aspects

Journal of Mathematical Analysis and Applications, 2004

We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl-Titchmarsh matrixvalued functions as well as a new version of the functional model in such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension generating periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.

2010

We outline an abstract approach to the pseudo-differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of representations associated with infinite-dimensional coadjoint orbits. We illustrate the approach by the case of infinite-dimensional Heisenberg groups. The classical Weyl-Hörmander calculus is recovered for the Schrödinger representations of the finite-dimensional Heisenberg groups.

arXiv (Cornell University), 2011

2015

We show that the Schrödinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.

We give necessary and sufficient conditions for a regular generalized Nevanlinna function $Q$ ($Q\in N_{\kappa }\left( \mathcal{H} \right)$) to be a Weyl (a.k.a. Weyl-Titchmarch) function. We also study an important subclass of $N_{\kappa }(\mathcal{H})$, the functions that have a boundedly invertible derivative at infinity $Q'\left( \infty \right):=\lim \limits_{z \to \infty}{zQ(z)}$. Those functions are regular and have the operator representation $Q\left( z \right)=\tilde{\Gamma}^{+}\left( A-z \right)^{-1}\tilde{\Gamma} ,z\in \rho \left( A \right)$, where $A$ is a bounded self-adjoint operator in a Pontryagin space $\mathcal{K}$. We prove that every strict such function $Q$ is a Weyl function associated with the symmetric operator $S:=A_{\vert (I-P)\mathcal{K}}$, where $P$ is the orthogonal projection, $P:=\tilde{\Gamma} \left( \tilde{\Gamma}^{+} \tilde{\Gamma} \right)^{-1} \tilde{\Gamma}^{+} $. We also give relation matrices of the adjoint relation $S^{+}$ and of $\hat{A}$, ...

1994

2006

A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We characterize all finite order linear differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the multivariate generalization of the notion of multiplier sequence. We give a complete description of all multivariate multiplier sequences as well as those of finite order. Next, we formulate and prove a natural analog of the Lax conjecture for real stable polynomials in two variables and use it to classify all finite order linear differential operators that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. As a further consequence of our methods we establish symbol curve theorems and a duality theorem showing that a differential operator preserves stability if and only if its Fischer-Fock adjoint has the same property. These are vast generalizations of the classical Hermite-Poulain-Jensen theorem, Pólya's curve theorem and Schur-Maló-Szegö type composition theorems in the univariate case as well as natural multivariate extensions of the aforementioned theorems. We also give several other applications of our results and discuss further directions and open problems.

Proceedings of the American Mathematical Society

The Kato spectrum of an operator is deployed to give necessary and sufficient conditions for Browder’s theorem to hold.

Journal of Mathematical Physics, 2014

In this work we recall the definition of matrix immanants, a generalization of the determinant and permanent of a matrix. We use them to generalize families of symmetric and antisymmetric orbit functions related to Weyl groups of the simple Lie algebras An. The new functions and their properties are described, in particular we give their continuous orthogonality relations. Several examples are shown.

Experimental Mathematics, 2007