Weak commutation relations of unbounded operators: Nonlinear extensions

@article{Bagarello2012WeakCR,
  title={Weak commutation relations of unbounded operators: Nonlinear extensions},
  author={Fabio Bagarello and Atsushi Inoue and Camillo Trapani},
  journal={Journal of Mathematical Physics},
  year={2012},
  volume={53},
  pages={123510}
}
We continue our analysis of the consequences of the commutation relation [S,T]=1, where S and T are two closable unbounded operators. The weak sense of this commutator is given in terms of the inner product of the Hilbert space H, where the operators act. We also consider what we call, adopting a physical terminology, a nonlinear extension of the above commutation relations. 
View PDF on arXiv
10 Citations
Background Citations
4

10 Citations

Remarks on Infinite-Dimensional Representations of the Heisenberg Algebra

Infinite-dimensional representations of Lie algebras necessarily invoke the theory of unbounded operator algebras. Starting with the familiar example of the Heisenberg Lie algebra, we sketch the…

Non-self-adjoint hamiltonians defined by Riesz bases

We discuss some features of non-self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that the eigenvectors form a Riesz basis of Hilbert space. Among other things, we…

Extended pseudo-fermions from non commutative bosons

We consider some modifications of the two dimensional canonical commutation relations, leading to non commutative bosons and we show how biorthogonal bases of the Hilbert space of the system can be…

D -Deformed Harmonic Oscillators

We analyze systematically several deformations arising from two-dimensional harmonic oscillators which can be described in terms of D -pseudo bosons. They all give rise to exactly solvable models,…

𝒟$\mathcal {D}$-Deformed Harmonic Oscillators

We analyze systematically several deformations arising from two-dimensional harmonic oscillators which can be described in terms of 𝒟$\mathcal {D}$-pseudo bosons. They all give rise to exactly…

Non‐self‐adjoint operators in quantum physics: ideas, people, and trends

1.1.1.1 Hundred Years of the Bohr’s Model In 2013, on occasion of the centenary of the Bohr’s model of atom (1) (marking, in a way, the birth of quantum theory), one should appreciate the multitude…

Riesz-like bases in rigged Hilbert spaces

The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space $\D[t] \subset \H \subset \D^\times[t^\times]$. A Riesz-like basis, in particular, is…

Everything is possible for the domain intersection dom T ∩ dom T⁎

A Complete Bibliography of Publications in the Journal of Mathematical Physics: 2005{2009

(2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p…

Deformed Canonical (anti‐)commutation relations and non‐self‐adjoint hamiltonians

References

SHOWING 1-10 OF 11 REFERENCES

Weak commutation relations of unbounded operators and applications

Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner…

Pseudo-Bosons, So Far

Nonlinear pseudo-bosons

In a series of recent papers, the author has introduced the notion of (regular) pseudo-bosons showing, in particular, that two number-like operators, whose spectra are N0:=N∪{0}, can be naturally…

A generalized Weyl relation approach to the time operator and its connection to the survival probability

The time operator, an operator which satisfies the canonical commutation relation with the Hamiltonian, is investigated, on the basis of a certain algebraic relation for a pair of operators T and H,…

Partial *- Algebras and Their Operator Realizations

Foreword. Introduction. I: Theory of Partial O*-Algebras. 1. Unbounded Linear Operators in Hilbert Spaces. 2. Partial O*-Algebras. 3.Commutative Partial O*-Algebras. 4. Topologies on Partial…

An introduction to nonharmonic Fourier series

Bases in Banach Spaces - Schauder Bases Schauder's Basis for C[a,b] Orthonormal Bases in Hilbert Space The Reproducing Kernel Complete Sequences The Coefficient Functionals Duality Riesz Bases The…

Generalized Weak Weyl Relation and Decay of Quantum Dynamics

Let $H$ be a self-adjoint operator on a Hilbert space ${\cal H}$, $T$ be a symmetric operator on ${\cal H}$ and $K(t)$ ($t\in \R$) be a bounded self-adjoint operator on ${\cal H}$. We say that…

An introduction to frames and Riesz bases

Frames in Finite-dimensional Inner Product Spaces.- Infinite-dimensional Vector Spaces and Sequences.- Bases.- Bases and their Limitations.- Frames in Hilbert Spaces.- Tight Frames and Dual Frame…

Methods of Modern Mathematical Physics II: Fourier Analysis

  • Self-Adjointness, Academic Press, New York
  • 1975

Commutative Banach Aalgebras, Notas de Matematica n

  • 1959