# 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.

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## References

SHOWING 1-10 OF 11 REFERENCES

### Weak commutation relations of unbounded operators and applications

- Mathematics
- 2011

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…

### Nonlinear pseudo-bosons

- Mathematics
- 2011

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

- Mathematics
- 2000

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

- Mathematics
- 2002

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

- Mathematics
- 1980

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

- Mathematics
- 2005

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

- Mathematics
- 2002

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