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Odd bosonic eigenvalues for the PT -symmetric Hamiltonian (9) in which the parameter g is pure imaginary. The eigenvalues are plotted as functions of Im g . The real (imaginary) parts of the eigenvalues are shown in the left (right) panel. Observe that the eigenvalues are all real when − 3 . 4645 < Im g < 3 . 4645; this is the region of unbroken PT symmetry. There is an infinite sequence of critical points; the next critical points are at Im g = ± 15 . 0485 and at ± 34 . 7994.
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The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=−iu, [u,v]=0. We can construct the Hamiltonian H=J
2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the \(\mathcal{P}\mathcal{T}\)-symme...
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Context 1
... Hamiltonian is no longer Hermitian, some of the eigenvalues are complex. The real (left panel) and imaginary (right panel) parts of the eigenvalues are shown in Figs. 3 and 4, with the odd bosonic eigenvalues in Fig. 3 and the even bosonic eigenvalues in Fig. 4. The key feature of the spectrum is that all of the eigenvalues are real if Im g lies between the critical values − 3 . 4645 and 3 . 4645 for the odd bosonic eigenvalues and between − 0 . 7344 and 0 . 7344 for the even bosonic eigenvalues. This is the region of unbroken PT symmetry. As | Im g | increases past these critical points, the lowest two eigenvalues become degenerate and move into the complex plane as a complex-conjugate pair. Thus, we have entered the regions of broken PT symmetry. In fact, there is an infinite sequence of critical points: The next two lowest pairs of eigenvalues become degenerate and move into the complex plane at the critical points ± 15 . 0485 and ± 34 . 7994 for the odd bosonic eigenvalues and at ± 8 . 2356 and ± 23 . 9030 for the even bosonic eigenvalues. Figures 5 and 6 give detailed plot of the transition from unbroken to broken PT symmetry. Observe that the eigenvalues become degenerate in pairs and that the real and imaginary parts of the eigenvalues make 90 ◦ turns at the critical points. This is a clear indication that the critical points are square-root branch points. We have shown in this paper that the well studied properties of -symmetric quantum mechanical Hamiltonians that are constructed from the elements of the Heisenberg algebra extend to Hamiltonians that are constructed from the elements of the E2 algebra. Both algebras are individually invariant under parity reflection P and under time reversal T . If a Hamiltonian that is constructed from the elements of either of these algebras is Hermitian, then its eigenvalues are all real. However, if the Hamiltonian is non-Hermitian and PT symmetric, then there may be regions of unbroken and unbroken PT symmetry. It is interesting that for the fermionic eigenvalues of the Hamiltonian (9) there is no region of unbroken PT symmetry; that is, the eigenvalues are all complex when g is ...
Context 2
... 5: Blow-up of the region near the critical points at Im g = ± 3 . 4645 on Fig. 3. The imaginary parts of the two lowest energy levels vanish until Im g passes a critical point. At this point the two energy levels become degenerate and Im E ( g ) for each energy level suddenly makes a 90 ◦ turn.
...
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Citations
... This kind of problems have been intensely studied in recent years (see [12] for an earlier review on the issue and also [13,14] for closely related models). ...
... Some time ago Bender and Kalveks [13] and Fernández and Garcia [14] discussed other space-time-symmetric hindered rotors with somewhat different symmetries and calculated several exceptional points. In particular, the latter authors estimated the trend of the location of the exceptional points in terms of the quantum numbers of the coalescing states. ...
We analyze the simple model of a rigid rotor with symmetry and show that the use of parity simplifies considerably the calculation of its eigenvalues. We also consider a non-Hermitian space-time-symmetric counterpart that exhibits real eigenvalues and determine the exceptional point at which the antiunitary symmetry is broken.
... The fact that one can control the band diagram by tuning the non-Hermitian character of the lattice is also not a new result and it was suggested earlier by several authors [7,9,11,20,21]. There are also some very interesting connections between the Mathieu equation and the E2 algebra in non-Hermitian models [22,23,24]. However, to my knowledge, a detailed analysis aimed to obtain analytic expressions for the stability boundaries in the (a, ε) plane for a P T symmetric lattice has not been considered. ...
I've applied multiple-scale perturbation theory to a generalized complex PT-symmetric Mathieu equation in order to find the stability boundaries between bounded and unbounded solutions. The analysis suggests that the non-Hermitian parameter present in the equation can be used to control the shape and curvature of these boundaries. Although this was suggested earlier by several authors, analytic formulas for the boundary curves were not given. This paper is a first attempt to fill this gap in the theory
... This kind of problems have been intensely studied in recent years (see [12] for an earlier review on the issue and also [13,14] for closely related models). ...
... Some time ago Bender and Kalveks [13] and Fernández and Garcia [14] discussed other space-time-symmetric hindered rotors with somewhat different symmetries and calculated several exceptional points. In particular, the latter authors estimated the trend of the location of the exceptional points in terms of the quantum numbers of the coalescing states. ...
We analyze the simple model of a rigid rotor with symmetry and show that the use of parity simplifies considerably the calculation of its eigenvalues. We also consider a non-Hermitian space-time-symmetric counterpart that exhibits real eigenvalues and determine the exceptional point at which the antiunitary symmetry is broken.
... P; j being the eigenvalues of K (k) P given by (52). Exactly the same equation has been studied as the Schödinger equation with a PT-symmetric non-Hermitian Hamiltonian containing a complexvalued potential [27]. ...
We consider the prevalent phenomenon that a pair of eigenvalues of the Liouville-von Neumann operator (Liouvillian) changes from pure imaginary to complex values with a common imaginary part for resonance states in an extended function space outside the Hilbert space. Such a transition point is an exceptional point, where non-Hermitian degeneracy occurs and both the pairs of eigenvalues and of eigenvectors coalesce. The transition can be attributed to a spontaneous breakdown of a parity and time-reversal (PT)-symmetry. This PT-symmetry in the Liouvillian dynamics results from the microscopic dynamics based on the fundamental physical laws. The kinetic equation of the Boltzmann type for a particle weakly coupled with a bath consists of the collision term, which is similar to a Hermitian operator and has even parity, and the flow term, which is anti-Hermitian and has odd parity. As a result of the competition between the two terms, a pair of PT-symmetric eigenstates of the effective Liouvillian converts to a PT-symmetry related pair as the flow term becomes more dominant than the collision term beyond an exceptional point.
... We identified various types of PT -symmetries for the E 2 -algebra, which for concrete non-Hermitian models served to explain the reality of their spectra in part of the parameter space. Similar features were also previously observed for special cases of the E 2 [5] and E 3 [6] Euclidean Lie algebra. Further interest in these kind of models stems from the fact that for specific representations the models become identical to some complex potential systems currently investigated in optics [7,8,9,10,11,12] and solid state physics [13]. ...
We propose a noncommutative version of the Euclidean Lie algebra E2 . Sev-
eral types of non-Hermitian Hamiltonian systems expressed in terms of generic combi-
nations of the generators of this algebra are investigated. Using the breakdown of the
explicitly constructed Dyson maps as a criterium, we identify the domains in the pa-
rameter space in which the Hamiltonians have real energy spectra and determine the
exceptional points signifying the crossover into the different types of spontaneously bro-
ken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find excep-
tional points which remain invariant under the deformation as well as exceptional points
becoming dependent on the deformation parameter of the algebra.
... The smallest critical value at N → ∞ is in agreement with the one reported in [36], whereas our values resulting from the smallest zero for N = 5, 7 differ slightly from those reported in [24]. Our values for N → ∞ also agree precisely with those reported in [37] when divided by a factor 2. We observe, that the rate of convergence becomes very poor for larger values of the coupling constant and even the lowest value requires fairly large values of N to reach a good precision. A much better convergence can be obtained by taking the limit directly on the level of the three-term recurrence relation. ...
We propose the notion of -quasi-exact solvability and apply this idea
to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system
depending on two parameters. The model considered reduces to the complex
Mathieu Hamiltonian in a double scaling limit, which enables us to compute
the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for
some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions
are univariate polynomials in the energy obeying a three-term recurrence relation. The latter
property guarantees the existence of a linear functional such that the
polynomials become orthogonal. The polynomials are shown to factorize for all levels above
the quantization condition leading to vanishing norms rendering them to be weakly orthogonal.
In two concrete examples we compute the explicit expressions for the Stieltjes measure.
... On the other hand, we recall that Heisenberg (Lie) algebras play an important role in mathematical physics and geometry, in particular in Quantum Mechanics (see for instance [1,8,9,19,20,24,26,27,29,30,31,32,36,42,44]). Indeed, the Heisenberg Principle of Uncertainty implies the non-compatibility of position and momentum observables acting on fermions. ...
... . Let the induction hypothesis true for i = j and we will show it for i = j + 1. Taking into account (8) we have ...
We introduce and provide a classification theorem for the class of
Heisenberg-Fock Leibniz algebras. This category of algebras is formed by those
Leibniz algebras L whose corresponding Lie algebras are Heisenberg algebras
and whose -modules I, where I denotes the ideal generated by
the squares of elements of L, are isomorphic to Fock modules. We also
consider the three-dimensional Heisenberg algebra and study three classes
of Leibniz algebras with as corresponding Lie algebra, by taking certain
generalizations of the Fock module. Moreover, we describe the class of Leibniz
algebras with as corresponding Lie algebra and such that the action gives rise to a minimal faithful representation of . The
classification of this family of Leibniz algebras for the case of n=3 is
given.
... It was recently shown that for Euclidean-algebra in two, E 2 [211] and three dimensions, E 3 [212], some simple non-Hermitian versions also possess real spectra. Here [65] we will follow the line of thoughts of [163] and investigate systematically the analogous of quasi-exactly solvable models of Lie algebraic type, that in those models, which can be written as bilinear combinations in terms of the Euclidean algebra generators. ...
... Each of these symmetries may be utilized to describe different types of physical scenarios. For instance, PT 1 was considered in [211] with P 1 : θ → θ + π corresponding to a reflection of the particle to the opposite side of the circle for the representation (3.40). For the same representation we can identify the remaining symmetries as P 2 : θ → θ + 2π, P 3 : θ → π/2 − θ, P 4 : θ → π − θ and P 5 : θ → −θ. ...
... , 9. Clearly the Hamiltonian H PT 1 is non-Hermitian with regard to the standard inner product when considering it for a Hermitian representation with J † = J, v † = v and u † = u, unless µ 2 = 0, µ 5 = −2µ 4 , µ 6 = 2µ 3 . The specific case H BK = J 2 +igv when µ i = 0 for i = 1, 4 was studied in [211], where partially real spectra were found but no isospectral counterparts were constructed. Using the relations (3.44)-(3.46), ...
Our main focus is to explore different models in noncommutative spaces in
higher dimensions. We provide a procedure to relate a three dimensional
q-deformed oscillator algebra to the corresponding algebra satisfied by
canonical variables describing non-commutative spaces. The representations for
the corresponding operators obey algebras whose uncertainty relations lead to
minimal length, areas and volumes in phase space, which are in principle
natural candidates of many different approaches of quantum gravity. We study
some explicit models on these types of noncommutative spaces, first by
utilising the perturbation theory, later in an exact manner. In many cases the
operators are not Hermitian, therefore we use PT -symmetry and
pseudo-Hermiticity property, wherever applicable, to make them self-consistent.
Apart from building mathematical models, we focus on the physical implications
of noncommutative theories too. We construct Klauder coherent states for the
perturbative and nonperturbative noncommutative harmonic oscillator associated
with uncertainty relations implying minimal lengths. In both cases, the
uncertainty relations for the constructed states are shown to be saturated and
thus imply to the squeezed coherent states. They are also shown to satisfy the
Ehrenfest theorem dictating the classical like nature of the coherent
wavepacket. The quality of those states are further underpinned by the
fractional revival structure. More investigations into the comparison are
carried out by a qualitative comparison between the dynamics of the classical
particle and that of the coherent states based on numerical techniques. The
qualitative behaviour is found to be governed by the Mandel parameter
determining the regime in which the wavefunctions evolve as soliton like
structures.
... It was recently shown that for E 2 [15] and E 3 [16] some simple non-Hermitian versions also possess real spectra. Here we will follow the line of thought of [3] and investigate systematically the analogues of quasi-exactly solvable models of Lie algebraic type, that is those models which can be written as bilinear combinations in terms of the Euclidean algebra generators. ...
... Each of these symmetries may be utilized to describe different types of physical scenarios. For instance, PT 1 was considered in [15] with P 1 : θ → θ + π corresponding to a reflection of the particle to the opposite side of the circle for the representation (2.2). For the same representation we can identify the remaining symmetries as P 2 : θ → θ + 2π, P 3 : θ → π/2 − θ, P 4 : θ → π − θ and P 5 : θ → −θ. ...
... The specific case H BK = J 2 + igv when µ i = 0 for i = 1, 4 was studied in [15], where partially real spectra were found but no isospectral counterparts were constructed. Using the relations (2.6)-(2.8), ...
We study several classes of non-Hermitian Hamiltonian systems, which can be
expressed in terms of bilinear combinations of Euclidean Lie algebraic
generators. The classes are distinguished by different versions of antilinear
(PT)-symmetries exhibiting various types of qualitative behaviour. On the basis
of explicitly computed non-perturbative Dyson maps we construct metric
operators, isospectral Hermitian counterparts for which we solve the
corresponding time-independent Schroedinger equation for specific choices of
the coupling constants. In these cases general analytical expressions for the
solutions are obtained in the form of Mathieu functions, which we analyze
numerically to obtain the corresponding energy eigenspectra. We identify
regions in the parameter space for which the corresponding spectra are entirely
real and also domains where the PT symmetry is spontaneously broken and
sometimes also regained at exceptional points. In some cases it is shown
explicitly how the threshold region from real to complex spectra is
characterized by the breakdown of the Dyson maps or the metric operator. We
establish the explicit relationship to models currently under investigation in
the context of beam dynamics in optical lattices.
... There has recently been interest in PT-symmetric Hamiltonians that exhibit real eigenvalues for a range of values of a potential parameter. Some of them are anharmonic oscillators [1-9] as well as models with Dirichlet [10][11][12] periodic and anti-periodic boundary conditions [13,14]. ...
... In section 5 we apply perturbation theory to one of the models and discuss the convergence of the perturbation series for the eigenvalues by comparison with the accurate results produced by the DM. In section 6 we discuss a PT-symmetric perturbed planar rigid rotor that was studied earlier as an example with E2 algebra [14]. In section 7 we discuss a non-hermitian perturbed three-dimensional rigid rotor that was not treated before as far as we know. ...
... Bender and Kalvecks [14] studied the eigenvalues of − ψ ′′ (θ) + g cos(θ)ψ(θ) = Eψ(θ), ...
We calculate accurate critical parameters for a class of non-hermitian
Hamiltonians by means of the diagonalization method. We study three
one-dimensional models and two perturbed rigid rotors with PT symmetry. One of
the latter models illustrates the necessity of a more general condition for the
appearance of real eigenvalues that we also discuss here.
