Related papers: The source operator method: an overview
A Rodrigues type formula is obtained for the symbols of the covariant bi-differential operators on a simple Jordan algebra.
A family $({\mathbf D}_\lambda)_{\lambda\in \mathbb C}$ of differential operators on the sphere $S^n$ is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of $S^n$ which…
This article explores an algebraic-recursive approach to construct differential operators that commute with a central operator $\hat{H}$ in quantum mechanics. Starting from the Schr\"odinger equation for a free particle, the work derives…
In this article we study differential symmetry breaking operators between principal series representations induced from minimal parabolic subgroups for the pair $(\operatorname{GL}_{n+1}(\mathbb{R}),\operatorname{GL}_n(\mathbb{R}))$. Using…
We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially…
In this paper, we construct and classify all differential symmetry breaking operators between certain principal series representations of the pair $SO_0(4,1) \supset SO_0(3,1)$. In this case, we also prove a localness theorem, namely, all…
The Jordanian deformation of $sl(2)$ bi-algebra structure is studied in view of physical applications to breaking of conformal symmetry in the high energy asymptotics of scattering. Representations are formulated in terms of polynomials,…
We construct and give a complete classification of all the differential symmetry breaking operators $\mathbb{D}_{\lambda, \nu}^{N,m}: C^\infty(S^3, \mathcal{V}_\lambda^{2N+1}) \rightarrow C^\infty(S^2, \mathcal{L}_{m, \nu})$, between the…
The construction of symmetry breaking differential operators, using invariant pluri-harmonic polynomials, due to T. Ibukiyama in the context of the Siegel upper half space, is extended for scalar representations to general Hermitian…
We study conformal symmetry breaking differential operators which map differential forms on $\mathbb{R}^n$ to differential forms on a codimension one subspace $\mathbb{R}^{n-1}$. These operators are equivariant with respect to the conformal…
In a recent paper [J.Math.Phys. vol42, 2236-2265 (2001)], we discussed differential operators within a quaternionic formulation of quantum mechanics. In particular, we proposed a practical method to solve quaternionic and complex linear…
We give a complete classification of conformally covariant differential operators between the spaces of differential $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the restriction of…
In this paper we show variant of the spectral theorem using an algebraic Jordan-Schwinger map. The advantage of this approach is that we don't have restriction of normality on the class of operators we consider. On the other side, we have…
We give a generalization of the Jordan canonical form theorem for a class of bounded linear operators on complex separable Hilbert spaces in terms of direct integrals. Precisely, we study the uniqueness of strongly irreducible…
Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $\mathbb{C}\ell(\mathbb R^n)$, $\mathbb S'$ its dual, $\rho$ and $\rho'$ be the corresponding representations of the spin group ${\rm Spin}(n)$. The group $G= {\rm…
Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We…
Quantum sinh-Gordon model in 1+1 dimensions is one of the simplest and best-studied massive integrable relativistic quantum field theories. We consider this theory on a multi-sheeted Riemann surfaces with a flat metric, which can be seen as…
By using some elementary techniques from operator theory, we prove constructively prove the existence of solutions to Dirichl\'et problems for planar Jordan domains with at least two boundary curves. An iterative method is thus obtained,…
In this paper we classify and construct differential symmetry breaking operators $\mathbb{D}$ from a line bundle over the real projective space $\mathbb{R}\mathbb{P}^n$ to a vector bundle over $\mathbb{R}\mathbb{P}^{n-1}$. We further…
We use the method of similar operators to study a mixed problem for a differential equation with an involution and an operator-valued potential function. The differential operator defined by the equation is transformed into a similar…