on characterization of fuzzy vectors and its applications to fuzzy mathematical programming problems in this paper we consider fuzzy mathematical programming problems which involve fuzzy vectors only in the objective function first we define fuzzy vectors on n dimensional euclidean space r n and investigate their properties especially we extend the concepts of the fuzzy max order and the possibility and the necessity measures given in fuzzy number to those of fuzzy vectors next based on the results we consider fuzzy mathematical programming problems including fuzzy multicriteria optimization problems which involve fuzzy vectors only in the objective function and we give the concepts of optimal solutions and characterize their properties

numerical algorithms for eigenvalue assignment by constant and dynamic output feedback algorithms are proposed for eigenvalue assignment eva by constant as well as dynamic output feedback the main algorithm is developed for single input multioutput systems and the results are then extended to multiinput multioutput systems in computing the feedback use is made of the fact that the closed loop eigenvalues can almost always be assigned arbitrarily close to the desired locations in the complex plane provided the system satisfies the condition m p n where m p and n are respectively the number of inputs outputs and states of the system the eva problem has been treated as a converse of the algebraic eigenvalue problem the proposed algorithms are based on the implicitly shifted qr algorithm for solving the algebraic eigenvalue problem the performance of the algorithms is illustrated by several numerical examples

satisficing solutions and duality in interval and fuzzy linear programming in this paper we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions the necessary tools for dealing with such problems in this way we show that the class of crisp classical lp problems can be embedded into the class of flp ones moreover for flp problems we define the concept of duality and prove the weak and strong duality theorems further we define a class of interval linear programming problems as a special subclass of flp problems and apply the previous results to this special case

exact pole assignment by output feedback part 3 abstract this part concludes a series of three papers concerning controllable and observable linear time invariant multivariable control systems in which the number of outputs plus the number of inputs exceeds the number of states following from the second paper fletcher 1987 it remains to consider the assignment by real output feedback of a set of distinct eigenvalues consisting of a mixture of real numbers and complex conjugate pairs of non real numbers we examine the possibility that in the algorithm described in the first paper fletcher and magni 1987 all the real eigenvalues can be assigned before any of the complex conjugate pairs if this is possible then the argument of fletcher 1987 can be used to complete the assignment it is only from the failure of this possibility that the pathological cases detailed in the first paper can arise

survey paper static output feedback a survey this paper reviews the static output feedback problem in the control of linear time invariant lti systems it includes analytical and computational methods and presents in a unified fashion the knowledge gained in the decades of research into this important open problem the paper shows that although many approaches and techniques exist to approach different versions of the problem no efficient algorithmic solutions are available

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

pseudo hermiticity and generalized pt and cpt symmetries we study certain linear and antilinear symmetry generators and involution operators associated with pseudo hermitian hamiltonians and show that the theory of pseudo hermitian operators provides a simple explanation for the recent results of bender brody and jones quant ph 0208076 on the cpt symmetry of a class of pt symmetric non hermitian hamiltonians we present a natural extension of these results to the class of diagonalizable pseudo hermitian hamiltonians h with a discrete spectrum in particular we introduce generalized parity p time reversal t and charge conjugation c operators and establish the pt and cpt invariance of h

symmetric versus hermitian formulations of quantum mechanics a non hermitian hamiltonian that has an unbroken symmetry can be converted by means of a similarity transformation to a physically equivalent hermitian hamiltonian this raises the following question in which form of the quantum theory the non hermitian or the hermitian one is it easier to perform calculations this paper compares both forms of a non hermitian ix3 quantum mechanical hamiltonian and demonstrates that it is much harder to perform calculations in the hermitian theory because the perturbation series for the hermitian hamiltonian is constructed from divergent feynman graphs for the hermitian version of the theory dimensional continuation is used to regulate the divergent graphs that contribute to the ground state energy and the one point green s function the results that are obtained are identical to those found much more simply and without divergences in the non hermitian symmetric hamiltonian the contribution to the ground state energy of the hermitian version of the theory involves graphs with overlapping divergences and these graphs are extremely difficult to regulate in contrast the graphs for the non hermitian version of the theory are finite to all orders and they are very easy to evaluate

relationships between modality constrained programming problems and various fuzzy mathematical programming problems abstract in this paper we focus on the relationships between modality constrained programming problems and various fuzzy mathematical programming problems i e flexible programming problem fuzzy goal programming problem and fuzzy mathematical programming problems proposed by orlovski tanaka et al inuiguchi et al dubois and sakawa et al fuzzy mathematical programming problems discussed in this paper can be interpreted in the framework of modality constrained programming problems thus modality constrained programming problems can be regarded as a base model for unifying various fuzzy mathematical programming problems

stabilization of discrete time linear systems by static output feedback this note proposes a systematic approach for the static output feedback control design for discrete time linear systems it is shown that if the open loop system satisfies some particular structural conditions a static output feedback gain can be calculated easily using a formula only involving the original system matrices among the conditions the system has to satisfy the stronger relies on a minimum phase argument square and nonsquare systems are considered

four easy pieces this paper is a mini summary of four new advances in the theory of pt symmetric quantum mechanics it describes some new calculations that were completed in 2005 the first advance concerns the classical coordinate space trajectories in some pt symmetric theories depending on the initial conditions one can find arbitrarily long periodic pt symmetric classical trajectories the longer trajectories originate from smaller regions of initial conditions there is an interesting resemblance to the so called period doubling route to chaos the second advance concerns the perturbative construction of the c operator for a pt symmetric square well the result has notable similarities to and differences from that for the pt symmetric cubic oscillator the third advance is a detailed comparison of a pt symmetric hamiltonian and the corresponding hermitian hamiltonian it is argued that a perturbative construction of the hermitian hamiltonian leads to an almost intractable theory the fourth advance concerns reflectionless potentials and pt symmetry and a possible connection with problems in cosmology

a geometric framework for pole assignment algorithms the problem of pole assignment by gain output feedback or by low order dynamical compensator is considered from a geometric point of view this makes it possible to unify in a general framework most of the existing pole assignment methods formulated in a state space context such as the minimal order observers the brasch pearson compensator and the methods proposed by h kimura and to simplify their presentation moreover new pole assignment algorithms may be derived from this general formulation

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

large order perturbation theory for a non hermitian pt symmetric hamiltonian a precise calculation of the ground state energy of the complex pt symmetric hamiltonian h p2 14x2 i x3 is performed using high order rayleigh schrodinger perturbation theory the energy spectrum of this hamiltonian has recently been shown to be real using numerical methods here we present convincing numerical evidence that the rayleigh schrodinger perturbation series is borel summable and show that pade summation provides excellent agreement with the real energy spectrum pade analysis provides strong numerical evidence that the once subtracted ground state energy considered as a function of 2 is a stieltjes function the analyticity properties of this stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable hamiltonian h p2 14 x2 ex3

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

fuzzy optimization an appraisal abstract this paper takes a general look at core ideas that make up the burgeoning body of fuzzy mathematical programming emphasizing the methodological view although fuzzy mathematical programming has enjoyed a rapidly increasing acceptance within the scientific community some technical hurdles exist to hinder a unanimity reasons for this as well as possible ways for improvement are also discussed

on the pseudo norm and admissible solutions of the symmetric scarf i potential the physically admissible solutions of the symmetric scarf i potential are identified in the domain of real and complex energies it is found that generally there are no admissible complex energy solutions and there is one with real energy in a limited range of the parameters there are two series of seemingly admissible solutions both with the real and complex energies belonging to quasi parity q however the two sets are not orthogonal in the domain of real energies the sign of the pseudo norm of states with real energy is found to oscillate as 1 n in accordance with the example of other symmetric potentials possessing an infinite number of discrete levels it is argued that the spontaneous breakdown of symmetry cannot be defined for the scarf i potential a comparison with some symmetric extensions of the infinite square well is presented

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

parametric output feedback control with response insensitivity a recent parameterization of the class of linear output feedback controllers that assign a set of desired self conjugate eigenvalues to the closed loop system is used to formulate and solve a fundamental response insensitivity problem it is established to what extent output feedback control can be used to render the closed loop system response insensitive to possibly many not necessarily small parameter variations in the open loop state space model a non conservative sequential design procedure is developed for making as many of the closed loop system eigenmodes as possible totally insensitive while retaining arbitrary assignment of the maximum number of closed loop eigenvalues the main result is a class of desensitizing fixed gain output feedback controllers explicitly specified by a set of free parameters which may be chosen to satisfy additional design requirements conditions are given for total modal decoupling insensitivity to possibly many not necessarily small parameter variations in the open l

exact pole assignment by output feedback part 2 abstract we continue the study in fletcher and magni 1987 concerning controllable and observable linear time invariant multivariable control systems in which the number of inputs plus the number of outputs exceeds the number of states in this paper we shall prove that exact assignment of distinct poles by means of a real output feedback is always possible if the set of poles to be assigned consists entirely of complex conjugate pairs of non real numbers we examine the problem of assigning these eigenvalues in complex conjugate pairs by means of the algorithm described in the first paper we reduce the problem to the case in which just one of the observability and controllability indices of the system is larger than 2 and further detailed analysis then disposes of this case

pseudounitary operators and pseudounitary quantum dynamics we consider pseudounitary quantum systems and discuss various properties of pseudounitary operators in particular we prove a characterization theorem for block diagonalizable pseudounitary operators with finite dimensional diagonal blocks furthermore we show that every pseudounitary matrix is the exponential of i 1 times a pseudo hermitian matrix and determine the structure of the lie groups consisting of pseudounitary matrices in particular we present a thorough treatment of 2 2 pseudounitary matrices and discuss an example of a quantum system with a 2 2 pseudounitary dynamical group as other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics demonstrate the coincidence of the symplectic group sp 2n with the real subgroup of a matrix group that is isomorphic to the pseudounitary group u n n and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups

exact pt symmetry is equivalent to hermiticity we show that a quantum system possessing an exact antilinear symmetry in particular pt symmetry is equivalent to a quantum system having a hermitian hamiltonian we construct the unitary operator relating an arbitrary non hermitian hamiltonian with exact pt symmetry to a hermitian hamiltonian we apply our general results to pt symmetry in finite dimensions and give the explicit form of the above mentioned unitary operator and hermitian hamiltonian in two dimensions our findings lead to the conjecture that non hermitian cpt symmetric field theories are equivalent to certain nonlocal hermitian field theories

pt symmetric cubic anharmonic oscillator as a physical model there is a factor of 2 error in equation 61 of this paper correcting this error leads to minor changes in equations 62 and 63 please see pdf for details

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

brief robust stabilization and guaranteed cost control for discrete time linear systems by static output feedback this paper proposes a systematic approach for the static output feedback control design for discrete time uncertain linear systems it is shown that if the open loop system satisfies some particular structural conditions and the uncertainty has a specific structure a static output feedback gain can be calculated easily using a formula only involving the original system matrices among the conditions the system has to satisfy the strongest one relies on a minimum phase argument square and nonsquare systems are considered the performance problem through a quadratic criterion is also discussed guaranteed cost control

interactions of hermitian and non hermitian hamiltonians the coupling of non hermitian symmetric hamiltonians to standard hermitian hamiltonians each of which individually has a real energy spectrum is explored by means of a number of soluble models it is found that in all cases the energy remains real for small values of the coupling constant but becomes complex if the coupling becomes stronger than some critical value for a quadratic non hermitian symmetric hamiltonian coupled to an arbitrary real hermitian symmetric hamiltonian the reality of the ground state energy for small enough coupling constant is established up to second order in perturbation theory

delta function potential with a complex coupling we explore the hamiltonian operator where is the dirac delta function and z is an arbitrary complex coupling constant for a purely imaginary z h has a spectral singularity at for re z 0 h has a real positive continuous spectrum that is free from spectral singularities for this latter case we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive definite inner product that renders h self adjoint this allows us to address the intriguing question of the nonlocal aspects of the equivalent hermitian hamiltonian for the system in particular we compute the energy expectation values for various gaussian wave packets to show that the non hermiticity effect diminishes rapidly outside an effective interaction region

quantization of quadratic lienard type equations by preserving noether symmetries abstract the classical quantization of a family of a quadratic lienard type equation lienard ii equation is achieved by a quantization scheme nucci 2011 28 that preserves the noether point symmetries of the underlying lagrangian in order to construct the schrodinger equation this method straightforwardly yields the schrodinger equation as given in choudhury and guha 2013 6

exact pole assignment by output feedback part 1 abstract we shall prove that exact assignment of distinct poles using output feedback is possible in any controllable and observable linear time invariant multivariable system in which the number of inputs plus the number of outputs exceeds the number of states except in clearly identified circumstances furthermore we give a construction for the entire class of pathological cases the main part of our argument is concerned with the assigning of complex conjugate eigenvalues by means of a real feedback matrix indeed we begin by giving a simple algorithm which is effective over any field although this problem has been discussed by numerous authors in the past they have usually presented only generic results we examine the complete class of systems satisfying the hypotheses mentioned

on the reality of the eigenvalues for a class of symmetric oscillators we study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays where is a real polynomial and we prove that if for some we have for all then the eigenvalues are all positive real we then sharpen this to a larger class of polynomial potentials

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum 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

a squeeze like operator approach to position dependent mass in quantum mechanics we provide a squeeze like transformation that allows one to remove a position dependent mass from the hamiltonian methods to solve the schrodinger equation may then be applied to find the respective eigenvalues and eigenfunctions as an example we consider a position dependent mass that leads to the integrable morse potential and therefore to well known solutions

output feedback eigenstructure assignment using two sylvester equations the eigenstructure assignment problem with output feedback is studied for systems satisfying the condition p m n the main tool used is the concept of c a b invariance and two coupled sylvester equations the solution of which leads to the computation of an output stabilizing feedback a computationally efficient algorithm for the solution of these two coupled equations which leads to the computation of a desired output feedback is presented

some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant comparison between the exact value of the spectral zeta function zh 1 5 6 5 3 2cos 5 2 1 5 3 5 and the results of numeric and wkb calculations supports the conjecture by daniel bessis 1995 private communication that all the eigenvalues of this pt invariant hamiltonian are real for one dimensional schrodinger operators with complex potentials having a monotonic imaginary part the eigenfunctions and the imaginary parts of their logarithmic derivatives have no real zeros

on existence of a biorthonormal basis composed of eigenvectors of non hermitian operators we present a set of necessary conditions for the existence of a biorthonormal basis composed of eigenvectors of non hermitian operators as an illustration we examine these conditions in the case of normal operators we also provide a generalization of the conditions which is applicable to non diagonalizable operators by considering not only eigenvectors but also all root vectors

the bezoutian the hankel matrix output feedback and system eigenstructure abstract given a minimal linear time invariant system for which the transfer matrix has a right rp 1 and a left d 1 n coprime matrix fraction description an inverse relation is established between the block hankel matrix formed from the markov parameters of the eigenstructure of p and d and the generalized bezoutian matrix formed from the polynomial matrices r p n and d furthermore it is shown that all eigenstructures that are assignable by output feedback form the same markov parameters and this suggests a new approach to eigenstructure assignment

further characterizations of possibility theoretical indices in fuzzy optimization abstract in this paper we investigate six types of fuzzy relations based on zadeh s possibility theory such relations were originally proposed by dubois and prade as comparison indices for fuzzy numbers and were extended by inuiguchi ichihashi and kume to general cases the fuzzy relations we consider are defined for fuzzy sets in a vector space preordered by a convex cone we reveal the connection of the fuzzy relations with set relations order like crisp relations known in the area of set optimization specifically we prove some equalities and equivalences involving the fuzzy relations and set relations under three different assumptions we finally consider a fuzzy optimization problem in a general form and define its solution concepts applying the above equivalences to the problem we obtain its relationship to certain set optimization problems

pseudo hermiticity and some consequences of a generalized quantum condition we exploit the hidden symmetry structure of a recently proposed non hermitian hamiltonian and of its hermitian equivalent one this sheds new light on the pseudo hermitian character of the former and allows access to a generalized quantum condition special cases lead to hyperbolic and morse like potentials in the framework of a coordinate dependent mass model

pseudo hermiticity for a class of nondiagonalizable hamiltonians we give two characterization theorems for pseudo hermitian possibly nondiagonalizable hamiltonians with a discrete spectrum that admit a block diagonalization with finite dimensional diagonal blocks in particular we prove that for such an operator h the following statements are equivalent 1 h is pseudo hermitian 2 the spectrum of h consists of real and or complex conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex conjugate eigenvalues are identical 3 h is hermitian with respect to a positive semidefinite inner product we further discuss the relevance of our findings for the merging of a complex conjugate pair of eigenvalues of diagonalizable pseudo hermitian hamiltonians in general and the pt symmetric hamiltonians and the effective hamiltonian for a certain closed frw minisuperspace quantum cosmological model in particular

modality constrained programming problems a unified approach to fuzzy mathematical programming problems in the setting of possibility theory abstract in this paper fuzzy mathematical programming problems are formulated based on the idea analogous with the chance constrained programming problem the difference in meaning between the ambiguity of the coefficients and that of the decision maker s preference is emphasized the constraints with fuzzy coefficients are treated as the restriction that should be satisfied properly rather than perfectly the objective functions with fuzzy coefficients are treated variously depending on the interpretations i e the optimization of the modalities the optimization of the fractile the minimization of the ambiguity and so forth the deterministic equivalent constraints and the deterministic equivalent problems are shown when the constraints and the objective functions are linear a numerical example is given to illustrate the proposed formulations

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

exact pole assignment with regularity by output feedback in descriptor systems part 2 in this paper the first of two we examine the problem of modal control of linear time invariant multivariable descriptor systems in particular for systems satisfying a certain inequality we will give necessary and sufficient conditions for the existence of an output feedback matrix possibly complex which exactly assigns an arbitrary set of r the number of independent semi state variables of the system distinct complex numbers to a generalized eigenspectrum of the closed loop system while ensuring that the resulting system has a unique smooth solution for each admissible initial condition furthermore it is shown that if the set of proposed poles contains only real elements the assignment can be carried out using real feedback next we present conditions that are necessary and sufficient for pole assignability by real feedback of a self conjugate set of complex numbers provided that the system satisfies a further inequality

calculation of the hidden symmetry operator in symmetric quantum mechanics in a recent paper it was shown that if a hamiltonian h has an unbroken symmetry then it also possesses a hidden symmetry represented by the linear operator the operator commutes with both h and the inner product with respect to is associated with a positive norm and the quantum theory built on the associated hilbert space is unitary in this paper it is shown how to construct the operator for the non hermitian symmetric hamiltonian h 1 2p2 1 2x2 ix3 using perturbative techniques it is also shown how to construct the operator for h 1 2p2 1 2x2 x4 using nonperturbative methods

a bilinear formulation for the output feedback problem in linear systems this note ties together a geometric theory and previous algebraic results for the closed loop eigenstructure assignment by output feedback in multivariable systems necessary and sufficient conditions for the closed loop stabilization eigenstructure assignment are presented in terms of a bilinear sylvester equation the solutions of which completely characterize the closed loop eigenstructure problem with output feedback some preliminary results for the solution of this bilinear equation are presented

some properties of extended fuzzy preference relations using modalities abstract a fuzzy preference relation is extended to six kinds of relations between fuzzy sets the properties of the six extended relations are investigated especially a sufficient condition for reducing the extended relations to four kinds is discussed when fuzzy sets are bounded and normal and some examples are given to illustrate it further a sufficient condition for the extended negative relations to coincide with the extended inverse relations is described

symmetric quantum theory defined in a krein space we provide a mathematical framework for symmetric quantum theory which is applicable irrespective of whether a system is defined on or a complex contour whether symmetry is unbroken and so on the linear space in which symmetric quantum theory is naturally defined is a krein space constructed by introducing an indefinite metric into a hilbert space composed of square integrable complex functions in a complex contour we show that in this krein space every symmetric operator is hermitian if and only if it has transposition symmetry as well from which the characteristic properties of the symmetric hamiltonians found in the literature follow some possible ways to construct physical theories are discussed within the restriction to the class k h

metrics and isospectral partners for the most generic cubic pt symmetric non hermitian hamiltonian we investigate properties of the most general symmetric non hermitian hamiltonian of cubic order in the annihilation and creation operators as a ten parameter family for various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and moyal products we elaborate on the subtleties of this approach for special choices of the ten parameters the hamiltonian reduces to various models previously studied such as to the complex cubic potential the so called swanson hamiltonian or the transformed version of the from below unbounded quartic x4 potential in addition it also reduces to various models not considered in the present context namely the single site lattice reggeon model and a transformed version of the massive sextic x6 potential which plays an important role as a toy model to identify theories with vanishing cosmological constant