Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.
Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.
In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space L, by a Hamiltonian h which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces h by its isospectral but non-Hermitian avatar H=Ω−1hΩ with Ω†Ω=Θ=I. Although acting in another, manifestly unphysical Hilbert space H, the amended Hamiltonian H=H† can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
After a few generalities we compare fundamental quantum mechanics applied to the harmonic oscillator with unusual oscillatorlike developments dealing with non-hermitian operators, this latter aspect exploiting in particular the property of subnormality. In this last context we can also restore the hermiticity of the Hamiltonian operator and discover a nice property of the new scalar product. General constructions of oscillatorlike Hamiltonians are also considered and new ideas connected with Heisenberg relations are presented.
Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA L?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA H?> H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
In the present paper we study the deformed harmonic oscillator for the non-Hermitian operator [Formula: see text] where λ,θ are real positive parameters, since the parameters α,β,m are for the general case complex. For the case α=1,β=1 and mass m real, we find the eigenfunctions and eigenvalues of energy, the coherent states, the time evolution of the operators [Formula: see text] in the Heisenberg picture and the uncertainty relations. In this case the operator ℋ is Hermitian and PT-symmetric. Also for the case m complex α=1,β=1, the operator ℋ is non-Hermitian and no more PT symmetric, but CPT symmetric with real discrete positive spectrum and the CPT symmetry is preserved. In the general case α,β,m complex, for the non-Hermitian operator ℋ, we obtain complex spectrum and for the special values of the complex parameters α,β the spectrum is real discrete and positive and the CPT symmetry is preserved. The general problem of deformed oscillator for non hermitian operators can be applied to the Solid State Physics.
Recently a f-deformed Fock space which is spanned by ∣n>λ has been introduced. These bases are indeed the eigen-states of a deformed non-Hermitian Hamiltonian. In this contribution, we will use a rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by ∣n>λ bases, to a new one, spanned by an extended two-parameters bases ∣n>λ1,λ2. These bases are now the eigen-states of a non-Hermitian Hamiltonian Hλ1,λ2=aλ1,λ2†a+1/2, where aλ1,λ2†=a†+λ1a+λ2 and a are respectively, the deformed creation and ordinary bosonic annihilation operators. The bases ∣n>λ1,λ2 are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states, in our two-parameters Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states, numerically.
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)