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

bifurcation analysis of two predator prey models bifurcation analysis is used to systematically detect unexpected dynamic behaviors of two predator prey models that have recently received support and attention comparisons with other existing models are also made in order to show that the singularities of the two examined models are due to some biological weakness of their functional forms

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

general aspects of pt symmetric and p self adjoint quantum theory in a krein space in our previous work we proposed a mathematical framework for symmetric quantum theory and in particular constructed a krein space in which symmetric operators would naturally act in this work we explore and discuss various general consequences and aspects of the theory defined in the krein space not only spectral properties and symmetry breaking but also several issues crucial for the theory to be physically acceptable such as time evolution of state vectors probability interpretation uncertainty relation classical quantum correspondence completeness existence of a basis and so on in particular we show that for a given real classical system we can always construct the corresponding symmetric quantum system which indicates that symmetric theory in the krein space is another quantization scheme rather than a generalization of the traditional hermitian one in the hilbert space we propose a postulate for an operator to be a physical observable in this framework

pseudo hermitian versus hermitian position dependent mass hamiltonians in a perturbative framework we formulate a systematic algorithm for constructing a whole class of hermitian position dependent mass hamiltonians which to lowest order of perturbation theory allow a description in terms of symmetric hamiltonians the method is applied to the hermitian analogue of the symmetric cubic anharmonic oscillator a new example is provided by a hamiltonian approximately equivalent to a symmetric extension of the one parameter trigonometric poschl teller potential

a physical interpretation for the non hermitian hamiltonian we explore a way of finding the link between a non hermitian hamiltonian and a hermitian one based on the analysis of bethe ansatz solutions for a class of non hermitian hamiltonians and the scattering problems for the corresponding hermitian hamiltonians it is shown that a scattering state of an arbitrary hermitian lattice embedded in a chain as the scattering center shares the same wavefunction with the corresponding non hermitian tight binding lattice which consists of the hermitian lattice with two additional on site complex potentials no matter whether the non hermitian is broken symmetry or even non symmetry exactly solvable models are presented to demonstrate the main points of this paper

metric operator in pseudo hermitian quantum mechanics and the imaginary cubic potential we present a systematic perturbative construction of the most general metric operator and positive definite inner product for quasi hermitian hamiltonians of the standard form h 1 2 p 2 v x in one dimension we show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in 1 1 dimensions for the case that v x is purely imaginary the latter have the form of a nonhomogeneous wave equation which admits an exact solution we apply our general method to obtain the most general metric operator for the imaginary cubic potential v x i x 3 this reveals an infinite class of previously unknown cpt as well as non cpt inner products we compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system our results for the imaginary cubic potential show that unlike the quantum system the corresponding classical system is not sensitive to the choice of the metric operator as another application of our method we give a complete characterization of the pseudo hermitian canonical quantization of a free particle moving in r that is consistent with the usual choice for the quantum hamiltonian finally we discuss subtleties involved with higher dimensions and systems having a fixed length scale

quantization of the lienard ii equation and jacobi s last multiplier in this paper the role of jacobi s last multiplier in mechanical systems with a position dependent mass is unveiled in particular we map the lienard ii equation to a position dependent mass system the quantization of the lienard ii equation is then carried out using the point canonical transformation method together with the von roos ordering technique finally we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated laguerre and exceptional laguerre functions

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

spectral problems for the weyl ordered form of operators 1pnqn in this paper we consider quantization of powers of the ratio between the hamiltonian coordinates for position and momentum in one dimensional systems the domain of the operators consists of square integrable functions over a finite real interval to ensure boundedness and self adjointness the spectral problems for the operators that result from using weyl ordering are discussed by introducing fredholm integral operator forms in position representation and the symmetry of the actions of the parity and time reversal operators on the kernels is discussed finally the general structures and properties of the eigenfunctions and eigenvalues are also derived and analyzed

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

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

studies on defect groups abstract the main results of this paper are as follows theorem 1 if g is a finite group with a strongly p embedded subgroup then g has a p block of defect zero the theorem solves a problem of alperin theorem 5 let g be a finite group and d a p subgroup of g such that ng d id has a strongly p embedded subgroup then d is a defect group for some p block of g if and only if there exists a p element x in g such that d is a sylow p subgroup of c g x corollary 6 if g is a finite group with am abelian sylow p subgroup then every strong p subgroup of g is a defect group for some p block of g in particular every maximal sylow p intersection of g is a defect group

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

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

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

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

dade s inductive conjecture for the ree groups of type g2 in the defining characteristic abstract we verify the inductive form of dade s conjecture for the finite simple groups 2 g 2 3 2 m 1 where m is a positive integer for the prime p 3 together with work by j an 1994 indian j math 36 7 27 this completes the verification of the conjecture for this series of groups

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

coupled channel version of the pt symmetric square well a coupled pair of pt symmetric square wells is studied as a prototype of a quantum system characterized by two manifestly non hermitian commuting observables via the diagonalization of our hamiltonian h h and spin like observable we demonstrate that there exists a domain of couplings where both the respective sets of eigenvalues en n 0 1 energies and 1 spin projections remain real in such a measurable regime the model acquires a consistent probabilistic interpretation mediated by our selection of one of many available interaction dependent scalar products

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

spectrum generating algebras for position dependent mass oscillator schrodinger equations the interest of quadratic algebras for position dependent mass schrodinger equations is highlighted by constructing spectrum generating algebras for a class of d dimensional radial harmonic oscillators with d 2 and a specific mass choice depending on some positive parameter via some minor changes the one dimensional oscillator on the line with the same kind of mass is included in this class the existence of a single unitary irreducible representation belonging to the positive discrete series type for d 2 and of two of them for d 1 is proved the transition to the constant mass limit 0 is studied and deformed su 1 1 generators are constructed these operators are finally used to generate all the bound state wavefunctions by an algebraic procedure

on finite groups of p local rank one and conjectures of dade and robinson abstract we use the classification of finite simple groups to demonstrate that conjecture 4 1 of g r robinson 1996 proc london math soc 3 72 312 330 and dade s projective conjecture hold for finite groups of p local rank one

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

symmetric effective mass schrodinger equations we outline a general method of obtaining exact solutions of symmetric schrodinger equations with a position dependent effective mass using this method exact solutions of some symmetric potentials have been obtained we have also discussed the construction of isospectral symmetric potentials

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

closed formula for the metric in the hilbert space of a symmetric model we introduce a very simple exactly solvable symmetric non hermitian model with a real spectrum and derive a closed formula for the metric operator which relates the problem to a hermitian one

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

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