# Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty ?

@article{Jana2009NonHermitianQM, title={Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty ?}, author={T. K. Jana and Pinaki Roy}, journal={Symmetry Integrability and Geometry-methods and Applications}, year={2009}, volume={5}, pages={083} }

We study non-Hermitian quantum mechanics in the presence of a minimal length. In particular we obtain exact solutions of a non-Hermitian displaced harmonic oscillator and the Swanson model with minimal length uncertainty. The spectrum in both the cases are found to be real. It is also shown that the models are pseudo-Hermitian and the metric operator is found explicitly in both the cases.

## 9 Citations

### Comment on "Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty"

- Physics
- 2009

We demonstrate that the recent paper by Jana and Roy entitled 'Non-Hermitian quantum mechanics with minimal length uncertainty'[SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various…

### Problem of the coexistence of several non-Hermitian observables in PT -symmetric quantum mechanics

- Physics
- 2017

During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators (with real spectra) which are…

### Minimal and maximal lengths of quantum gravity from non-Hermitian position-dependent noncommutativity

- Physics
- 2021

A minimum length scale of the order of Planck length is a feature of many models of quantum gravity that seek to unify quantum mechanics and gravitation. Recently, Perivolaropoulos in his seminal…

### Hermitian versus non-Hermitian representations for minimal length uncertainty relations

- Mathematics
- 2013

We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg’s uncertainty relations resulting from noncommutative spacetime…

### Strings from position-dependent noncommutativity

- Mathematics
- 2010

We introduce a new set of noncommutative spacetime commutation relations in two space dimensions. The space–space commutation relations are deformations of the standard flat noncommutative spacetime…

### Solvable Models on Noncommutative Spaces with Minimal Length Uncertainty Relations

- Physics
- 2014

Intuitive arguments involving standard quantum mechanical uncertainty relations suggest that at length scales close to the Planck length, strong gravity effects limit the spatial as well as temporal…

### Two-dimensional boson oscillator under a magnetic field in the presence of a minimal length in the non-commutative space

- Physics
- 2021

The studies of the relativistic generalization of the harmonic oscillator has drawn much attention in recent years. The wellknown relativistic model of the harmonic oscillator was revived by…

### Ju l 2 01 0 Position-dependent noncommutativity Strings from position-dependent noncommutativity

- Mathematics
- 2013

We introduce a new set of noncommutative space-time commutation relations in two space dimensions. The space-space commutation relations are deformations of the standard flat noncommutative…

### Reply to Comment on "Non Hermitian Quantum Mechanics with Minimal Length Uncertainty", arXiv:0908.2341

- Physics
- 2009

It is shown that the results of ref [1] are consistent. In ref [1] Swanson model with the following Hamiltonian was considered H = ωaa + λa + δa 2 + ω 2 (1) where λ 6= δ are real numbers and a, a are…

## References

SHOWING 1-10 OF 34 REFERENCES

### Uncertainty relation in quantum mechanics with quantum group symmetry

- Physics
- 1993

The commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann)…

### Algebraic Solution of the Harmonic Oscillator With Minimal Length Uncertainty Relations

- Physics, Mathematics
- 2007

In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates…

### Hilbert space representation of the minimal length uncertainty relation.

- PhysicsPhysical review. D, Particles and fields
- 1995

The quantum mechanical structure which underlies the generalized uncertainty relation which quantum theoretically describes the minimal length as a minimal uncertainty in position measurements is studied.

### Non-pointlike particles in harmonic oscillators

- Physics
- 1997

Quantum mechanics usually describes particles as being pointlike in the sense that, in principle, the uncertainty, , can be made arbitrarily small. Studies on string theory and quantum gravity…

### Pauli-Hamiltonian in the presence of minimal lengths

- Physics
- 2006

We construct the Pauli-Hamiltonian on a space where the position and momentum operators obey generalized commutation relations leading to the appearance of a minimal length. Using the momentum space…

### Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry

- Mathematics
- 1998

The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of $\mathrm{PT}$ symmetry, one obtains new…

### An exact solution of the one-dimensional Dirac oscillator in the presence of minimal lengths

- Physics
- 2006

Using the momentum space representation, we determine the energy eigenvalues, eigenfunctions and the high-temperature thermodynamic properties of the Dirac oscillator in one dimension in the presence…

### Quantum-gravity and minimum length

- Physics
- 1995

The existence of a fundamental scale, a lower bound to any output of a position measurement, seems to be a model-independent feature of quantum gravity. In fact, different approaches to this theory…

### Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum

- Mathematics
- 2001

We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors.