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

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

symmetrically regularized eckart poschl teller and hulthen potentials parity times time reversal symmetry of complex hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for hermiticity perhaps an equally important role is played by the related strengthened analyticity assumptions in a constructive illustration we complexify a few potentials solvable only in s wave then we continue their domain from the semi axis to the whole axis and obtain new exactly solvable models their energies turn out to be real as expected the new one dimensional spectra themselves differ quite significantly from their s wave predecessors

generalized bogoliubov transformations versus d pseudo bosons we demonstrate that not all generalized bogoliubov transformations lead to d pseudo bosons and prove that a correspondence between the two can only be achieved with the imposition of specific constraints on the parameters defining the transformation for certain values of the parameters we find that the norms of the vectors in sets of eigenvectors of two related apparently non selfadjoint number like operators possess different types of asymptotic behavior we use this result to deduce further that they constitute bases for a hilbert space albeit neither of them can form a riesz base when the constraints are relaxed they cease to be hilbert space bases but remain d quasibases

nonuniqueness of the mathcal c operator in mathcal p mathcal t symmetric quantum mechanics the operator in symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations and these equations are difficult to solve exactly so perturbative methods have been used in the past to calculate the usual approach has been to express the hamiltonian as h h0 h1 and to seek a solution for in the form where q q q p is odd in the momentum p even in the coordinate q and has a perturbation expansion of the form q q1 3q3 5q5 in previous work it has always been assumed that the coefficients of even powers of in this expansion would be absent because their presence would violate the condition that q p q is odd in p in an earlier paper it was argued that the operator is not unique because the perturbation coefficient q1 is nonunique here the nonuniqueness of is demonstrated at a more fundamental level it is shown that the perturbation expansion for q actually has the more general form q q0 q1 2q2 in which all powers and not just odd powers of appear for the case in which h0 is the harmonic oscillator hamiltonian q0 is calculated exactly and in closed form and it is shown explicitly to be nonunique the results are verified by using powerful summation procedures based on analytic continuation it is also shown how to calculate the higher coefficients in the perturbation series for q

gazeau klauder coherent states for trigonometric rosen morse potential the gazeau klauder coherent states for the trigonometric rosen morse potential are constructed it is shown that the resolution of unity temporal stability and action identity conditions are satisfied for the coherent states the mandel parameter is also calculated for the weighting distribution function corresponding to the coherent states

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

appearances of pseudo bosons from black scholes equation it is a well known fact that the black scholes equation admits an alternative representation as a schrodinger equation expressed in terms of a non self adjoint hamiltonian we show how pseudo bosons linear or not naturally arise in this context and how they can be used in the computation of the pricing kernel

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

super vertex algebras meromorphic jacobi forms and umbral moonshine abstract the vector valued mock modular forms of umbral moonshine may be repackaged into meromorphic jacobi forms of weight one in this work we constructively solve two cases of the meromorphic module problem for umbral moonshine specifically for the type a niemeier root systems with coxeter numbers seven and thirteen we construct corresponding bigraded super vertex operator algebras equip them with actions of the corresponding umbral groups and verify that the resulting trace functions on canonically twisted modules recover the meromorphic jacobi forms that are specified by umbral moonshine we also obtain partial solutions to the meromorphic module problem for the type a niemeier root systems with coxeter numbers four and five by constructing super vertex operator algebras that recover the meromorphic jacobi forms attached to maximal subgroups of the corresponding umbral groups

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

supersymmetric and shape invariant generalization for nonresonant and intensity dependent jaynes cummings systems a class of shape invariant bound state problems which represent transitions in a two level system introduced earlier are generalized to include arbitrary energy splittings between the two levels as well as intensity dependent interactions we show that the coupled channel hamiltonians obtained correspond to the generalizations of the nonresonant and intensity dependent jaynes cummings hamiltonians widely used in quantized theories of lasers in this general context we determine the eigenstates eigenvalues the time evolution matrix and the population inversion matrix factor pacs numbers 0365f 0370 0530 1130p

a generalized jaynes cummings hamiltonian and supersymmetric shape invariance a class of shape invariant bound state problems which represent two level systems are introduced it is shown that the coupled channel hamiltonians obtained correspond to the generalization of the jaynes cummings hamiltonian

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

pseudoanti hermitian operators in quaternionic quantum mechanics we introduce the concept of pseudoanti hermitian operators in quaternionic quantum mechanics and give a complete characterization of their spectra we highlight some physical properties related to time reversal symmetry of the pseudoanti hermitian quaternionic hamiltonians

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

steady state errors in nonlinear control systems control systems containing a memoryless nonlinear element are addressed the steady state response to certain inputs when the circle condition for stability is met is characterized

chiral de rham complex we define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi infinite de rham complexes of certain d modules over the loop spaces for calabi yau manifolds they admit n 2 supersymmetry connection with wakimoto modules is discussed

non self adjoint operators with real spectra and extensions of quantum mechanics in this article we review the quantum mechanical setting associated with a non self adjoint hamiltonian with a real spectrum the spectral properties of the hamiltonian of a swanson like model are investigated the eigenfunctions associated with the real simple eigenvalues are shown to form complete systems but not a riesz basis which gives rise to difficulties in the rigorous mathematical formulation of quantum mechanics a new inner product appropriate for the physical interpretation of the model has been consistently introduced the dynamics of the system is described some specificities of the theory of non self adjoint operators with implications in quantum mechanics are discussed in this article we review the quantum mechanical setting associated with a non self adjoint hamiltonian with a real spectrum the spectral properties of the hamiltonian of a swanson like model are investigated the eigenfunctions associated with the real simple eigenvalues are shown to form complete systems but not a riesz basis which gives rise to difficulties in the rigorous mathematical formulation of quantum mechanics a new inner product appropriate for the physical interpretation of the model has been consistently introduced the dynamics of the system is described some specificities of the theory of non self adjoint operators with implications in quantum mechanics are discussed

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

maximal couplings in pt symmetric chain models with the real spectrum of energies the domain of all the coupling strengths compatible with the reality of the energies is studied for a family of non hermitian n n matrix hamiltonians h n with tridiagonal and symmetric structure at all dimensions n the coordinates are found of the extremal points at which the boundary hypersurface touches the circumscribed sphere for odd n 2m 1 or ellipsoid for even n 2k

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

metric operators for quasi hermitian hamiltonians and symmetries of equivalent hermitian hamiltonians we give a simple proof of the fact that every diagonalizable operator that has a real spectrum is quasi hermitian and show how the metric operators associated with a quasi hermitian hamiltonian are related to the symmetry generators of an equivalent hermitian hamiltonian

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

much ado about mathieu abstract eguchi ooguri and tachikawa observed that the coefficients of the elliptic genus of type ii string theory on k3 surfaces appear to be dimensions of representations of the largest mathieu group subsequent work by several people established a candidate for the elliptic genus twisted by each element of m 24 in this paper we prove that the resulting sequence of class functions are true characters of m 24 proving the eguchi ooguri tachikawa mathieu moonshine conjecture the integrality of multiplicities is proved using a small generalisation of sturm s theorem while positivity involves a modification of a method of hooley for finding an effective bound on a family of selberg kloosterman zeta functions at s 3 4 we also prove the evenness property of the multiplicities as conjectured by several authors and use that to investigate the proposal of gaberdiel hohenegger volpato that mathieu moonshine lifts to the conway groups co 0 and co 1 we identify the role group cohomology plays in both mathieu moonshine and monstrous moonshine in particular this gives a cohomological interpretation for the non fricke elements in norton s generalised monstrous moonshine conjecture and gives a lower bound for h 3 b c o 1 c

the physics of non hermitian operators this special issue contains contributions presented at the workshop on the physics of non hermitian operators held at stellenbosch south africa in november 2005 a few additional articles are included from researchers who could not attend the meeting the workshop was the fourth meeting in a series on pseudo hermitian hamiltonians in quantum physics which commenced in prague in 2003 as a result of growing interest in non hermitian hamiltonians in general and pt symmetric quantum mechanics and quantum field theory in particular naturally hamiltonians of open systems are an obvious part of this topic but the focus here is rather on systems where unitary time evolution as such is not undermined this rather new and perhaps ambitious field thus explores a possible generalization of quantum mechanics which incorporates non hermitian hamiltonians into a consistent quantum mechanical framework new results of a physical and mathematical nature have brought about appreciable progress earlier work has been uncovered or even rediscovered and links between sub fields have been established this issue should provide a stimulus for initiating further advances in the field re focus on earlier perspectives and explore open and controversial questions while non hermitian operators have been encountered in theoretical physics since the advent of quantum mechanics their systematic mathematical investigation and their possible physical implications is a relatively new subject further explored in this issue we would like to thank the university of stellenbosch and the south african national research foundation for their financial support without which it would not have been possible to plan and organize the workshop organizational aspects and technical support were very ably taken care of by christine ruperti and tinus botha whom we would like to thank again finally we also appreciate and acknowledge the invitation from the editorial board of journal of physics a mathematical and general for the opportunity to publish this special issue and trust that this will indeed widen the interest and stimulate progress in the field

hermitian versus non hermitian representations for minimal length uncertainty relations we investigate four different types of representations of deformed canonical variables leading to generalized versions of heisenberg s uncertainty relations resulting from noncommutative spacetime structures we demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces the harmonic oscillator the manifestly non hermitian swanson model and an intrinsically noncommutative model with poschl teller type potential we provide an analytical expression for the metric in terms of quantities specific to the generic solution procedure and show that when it is appropriately implemented expectation values are independent of the particular representation a recently proposed inequivalent representation resulting from jordan twists is shown to lead to unphysical models we suggest an anti pt symmetric modification to overcome this shortcoming

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

finite dimensional pt symmetric hamiltonians this paper investigates finite dimensional pt symmetric hamiltonians it is shown here that there are two ways to extend real symmetric hamiltonians into the complex domain i the usual approach is to generalize such hamiltonians to include complex hermitian hamiltonians ii alternatively one can generalize real symmetric hamiltonians to include complex pt symmetric hamiltonians in the first approach the spectrum remains real while in the second approach the spectrum remains real if the pt symmetry is not broken both generalizations give a consistent theory of quantum mechanics but if d 2 a d dimensional hermitian matrix hamiltonian has more arbitrary parameters than a d dimensional pt symmetric matrix hamiltonian

non hermitian random matrix theory summation of planar diagrams the single ring theorem and the disc annulus phase transition i review aspects of work done in collaboration with a zee and r scalettar 1997 nucl phys b 504 579 1997 nucl phys b 501 643 2001 j math phys 42 5718 on complex non hermitian random matrices i open by explaining why the bag of tools used regularly in analysing hermitian random matrices cannot be applied directly to analyse non hermitian matrices and then introduce the method of hermitization which solves this problem then for rotationally invariant ensembles i derive a master equation for the average density of eigenvalues in the complex plane in the limit of infinitely large matrices this is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the hermitized green function remarkably this resummation can be carried out explicitly for any rotationally invariant ensemble i prove that in the limit of infinitely large matrices the shape of the eigenvalue distribution is either a disc or an annulus this is the celebrated single ring theorem which of these shapes is realized is determined by the parameters coupling constants which determine the ensemble by varying these parameters a phase transition may occur between the two possible shapes i briefly discuss the universal features of this transition as the analysis of this problem relies heavily on summation of planar feynman diagrams i make a special effort at presenting a pedagogical exposition of the diagrammatic method which some readers may find useful

time dependent darboux supersymmetric transformations for non hermitian quantum systems we propose time dependent darboux supersymmetric transformations that provide a scheme for the calculation of explicitly time dependent solvable non hermitian partner hamiltonians together with two hermitian hamilitonians the latter form a quadruple of hamiltonians that are related by two time dependent dyson equations and two intertwining relations in form of a commutative diagram our construction is extended to the entire hierarchy of hamiltonians obtained from time dependent darboux crum transformations as an alternative approach we also discuss the intertwining relations for lewis riesenfeld invariants for hermitian as well as non hermitian hamiltonians that reduce the time dependent equations to auxiliary eigenvalue equations the working of our proposals is discussed for a hierarchy of explicitly time dependent rational hyperbolic airy function and nonlocal potentials

moyal products a new perspective on quasi hermitian quantum mechanics the rationale for introducing non hermitian hamiltonians and other observables is reviewed and open issues identified we present a new approach based on moyal products to compute the metric for quasi hermitian systems this approach is not only an efficient method of computation but also suggests a new perspective on quasi hermitian quantum mechanics which invites further exploration in particular we present some first results which link the berry connection and curvature to non perturbative properties and the metric

a note on the pq theorem and the extrapolation of signals the problem of determining a bandlimited function from its values on a finite interval is ill conditioned in the sense that although the pertinent inverse map exists it is discontinuous at every point whenever certain closely related general problems are well conditioned in the sense that the inverse operator is continuous they can be solved using a special case of a known algorithm in particular attention is directed to the relation between the pq theorem its hilbert space projection operator setting and later work

non self adjoint hamiltonians defined by sesquilinear forms and their physical applications in this paper we introduce a theory of sesquilinear forms with respect to a non self adjoint hamiltonian and its physical operators in particular we discuss some features of non self adjoint operators and sesquilinear forms which are defined starting from biorthogonal sequences that are not riesz bases generalized riesz systems d quasi bases and d e quasi bases

unified algebraic method to non hermitian systems with lie algebraic linear structure we suggest a generic algebraic method to solve non hermitian hamiltonian systems with lie algebraic linear structure such method can not only unify the non hermitian hamiltonian and the hermitian hamiltonian with the same structure but also keep self consistent between similarity transformation and unitary transformation to clearly reveal the correctness and physical meaning of such algebraic method we illustrate our method with two different types of non hermitian hamiltonians the non hermitian hamiltonian with heisenberg algebraic linear structure and the non hermitian hamiltonian with su 1 1 algebraic linear structure we obtain energy eigenvalues and the corresponding eigenstates of non hermitian forced harmonic oscillator with heisenberg algebra structure via a proper similarity transformation we also calculate the eigen problems of general non hermitian hamiltonian with su 1 1 structure in terms of the similarity transformation as an application we focus on studying the non hermitian sing

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

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

equivalent hamiltonians for pt symmetric versions of dual 2d field theories in the context of non hermitian but pt symmetric hamiltonians we explore the systematics of the coefficients of the multiple commutators that appear in the perturbative expansions for q the exponent of the metric operator and h the equivalent hermitian hamiltonian we find exact expressions for both q and h for pt symmetric versions of the dual two dimensional quantum field theories the sine gordon model and the massive thirring model and elucidate how these solutions arise from the structure of the multiple commutator coefficients

remarks on the pseudo norm in symmetric quantum mechanics this paper presents an underlying analytical relationship between the pseudo norm associated with the symmetric hamiltonian h p2 v q and the stokes multiplier of the differential equation corresponding to this hamiltonian we show that the sign alternation of the pseudo norm which has been observed as a generic feature of the inner product is essentially controlled by the derivative of a stokes multiplier with respect to the eigenparameter

solvable non hermitian discrete square well with closed form physical inner product a new hermitizable quantum model is proposed in which the bound state energies are real and given as roots of an elementary trigonometric expression while the wave function components are expressed as superpositions of two chebyshev polynomials as an site lattice version of square well with complex robin type two parametric boundary conditions the model is unitary with respect to the hilbert space metric which becomes equal to the most common dirac s metric in the conventional textbook hermitian hamiltonian limit this metric is constructed in closed form at all

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

calculation of the hidden symmetry operator for a symmetric square well it has been shown that a hamiltonian with an unbroken symmetry also possesses a hidden symmetry that is represented by the linear operator this symmetry operator guarantees that the hamiltonian acts on a hilbert space with an inner product that is both positive definite and conserved in time thereby ensuring that the hamiltonian can be used to define a unitary theory of quantum mechanics in this paper it is shown how to construct the operator for the symmetric square well using perturbative techniques

continuum states in generalized swanson models a one to one correspondence is known to exist between the spectra of the discrete states of the non hermitian swanson type hamiltonian and an equivalent hermitian schrodinger hamiltonian h the two hamiltonians being related through a similarity transformation in this work we consider the continuum states of h and examine the nature of the corresponding states of h

position dependent effective mass dirac equations with pt symmetric and non pt symmetric potentials we present a new procedure to construct the one dimensional non hermitian imaginary potential with a real energy spectrum in the context of the position dependent effective mass dirac equation with the vector coupling scheme in 1 1 dimensions in the first example we consider a case for which the mass distribution combines linear and inversely linear forms the dirac problem with a pt symmetric potential is mapped into the exactly solvable schrodinger like equation problem with the isotonic oscillator by using the local scaling of the wavefunction in the second example we take a mass distribution with smooth step shape the dirac problem with a non pt symmetric imaginary potential is mapped into the exactly solvable schrodinger like equation problem with the rosen morse potential the real relativistic energy levels and corresponding wavefunctions for the bound states are obtained in terms of the supersymmetric quantum mechanics approach and the function analysis method

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

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

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

milne quantization for non hermitian systems we generalize the milne quantization condition to non hermitian systems in the general case the underlying nonlinear ermakov milne pinney equation needs to be replaced by a nonlinear integral differential equation however when the system is pt symmetric or and quasi pseudo hermitian the equations simplify and one may employ the original energy integral to determine its quantization we illustrate the working of the general framework with the swanson model and two explicit examples for pairs of supersymmetric hamiltonians in one case both partner hamiltonians are hermitian and in the other a hermitian hamiltonian is paired by a darboux transformation to a non hermitian one

some spectral equivalences between schrodinger operators spectral equivalences of the quasi exactly solvable sectors of two classes of schrodinger operators are established using gaudin type bethe ansatz equations in some instances the results can be extended leading to full isospectrality in this manner we obtain equivalences between pt symmetric problems and hermitian problems we also find equivalences between some classes of hermitian operators

generalized swanson models and their solutions we analyse a class of non hermitian quadratic hamiltonians which are of the form where are real constants with and and are generalized creation and annihilation operators respectively thus these hamiltonians may be classified as generalized swanson models it is shown that the eigenenergies are real for a certain range of values of the parameters a similarity transformation mapping the non hermitian hamiltonian h to a hermitian one h is also obtained it is shown that h and h share identical energies as explicit examples the solutions of a couple of models based on the trigonometric rosen morse i and the hyperbolic rosen morse ii type potentials are obtained we also study the case where the non hermitian hamiltonian is symmetric

non hermitian hamiltonians of lie algebraic type we analyse a class of non hermitian hamiltonians which can be expressed in terms of bilinear combinations of generators in the lie algebra or their isomorphic su 1 1 counterparts the hamiltonians are prototypes for solvable models of lie algebraic type demanding a real spectrum and the existence of a well defined metric we systematically investigate the constraints these requirements impose on the coupling constants of the model and the parameters in the metric operator we compute isospectral hermitian counterparts for some of the original non hermitian hamiltonians alternatively we employ a generalized bogoliubov transformation which allows us to compute explicitly real energy eigenvalue spectra for these type of hamiltonians together with their eigenstates we compare the two approaches

exactly solvable non hermitian jaynes cummings type hamiltonian admitting entirely real spectra from supersymmetry it is shown that for a given hermitian hamiltonian possessing supersymmetry there is always a non hermitian jaynes cummings type hamiltonian jcth admitting entirely real spectra the parent supersymmetric hamiltonian and the corresponding non hermitian jcth are simultaneously diagonalizable the exact eigenstates of these non hermitian hamiltonians are constructed algebraically for certain shape invariant potentials including a non hermitian version of the standard jaynes cummings jc model for which the parent supersymmetric hamiltonian is the superoscillator it is also shown that a non hermitian version of the several physically motivated generalizations of the jc model admits entirely real spectra the positive definite metric operator in the hilbert space is constructed explicitly along with the introduction of a new inner product structure so that the eigenstates form a complete set of orthonormal vectors and the time evolution is unitary

space of state vectors in symmetric quantum mechanics the space of states of symmetric quantum mechanics is examined the requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to a space with an indefinite metric self consistent expressions for the probability amplitude and the average value of operator are suggested further specification of the space of state vectors yields a superselection rule redefining the notion of the superposition principle an expression for the probability current density satisfying the equation of continuity and vanishing for the bound state is proposed

calculation of the metric in the hilbert space of a cal p cal t symmetric model via the spectral theorem in a previous paper 1 we introduced a very simple symmetric non hermitian hamiltonian with a real spectrum and derived a closed formula for the metric operator relating the problem to a hermitian one in this paper we propose an alternative formula for the metric operator which we believe is more elegant and whose construction based on a backward use of the spectral theorem for self adjoint operators provides new insights into the nature of the model

complete set of inner products for a discrete pt symmetric square well hamiltonian a discrete n point runge kutta version h n of one of the simplest non hermitian square well hamiltonians with real spectrum is studied its possible hermitizations mediated by nontrivial often called non dirac metrics i are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model a complete set of these alternative multiparametric metrics a b n defining all the eligible hamiltonian dependent representations of the physical hilbert space of states is constructed in closed form for any coupling 1 1 and for any matrix dimension n

quantum and classical statistical mechanics of a class of non hermitian hamiltonians this paper investigates the thermodynamics of a large class of non hermitian pt symmetric oscillators whose energy spectrum is entirely real the spectrum is estimated by second order wkb approximation which turns out to be very accurate even for small quantum numbers and used to generate the quantum partition function graphs showing the thermal behavior of the entropy and the specific heat at all regimes of temperature are given to obtain the corresponding classical partition function it turns out to be necessary in general to integrate over a complex phase space for the wrong sign quartic whose equivalent hermitian hamiltonian is known exactly how this formulation arises starting from the hermitian case is demonstrated explicitly

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

pseudospectra in non hermitian quantum mechanics we propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non hermitian operators we relate pseudospectral properties to quasi hermiticity similarity to self adjoint operators and basis properties of eigenfunctions the abstract results are illustrated by unexpected wild properties of operators familiar from pt symmetric quantum mechanics

quasistationary quaternionic hamiltonians and complex stochastic maps we show that the complex projections of time dependent quasianti hermitian quaternionic hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi hermitian density matrices if and only if a quasistationarity condition is fulfilled i e if and only if is an hermitian positive time independent complex operator an example is also discussed

weak pseudo hermiticity and antilinear commutant we inquire into some properties of diagonalizable pseudo hermitian operators showing that their definition can be relaxed and that the pseudo hermiticity property is strictly connected with the existence of an antilinear symmetry this result is then illustrated by considering the particular case of the complex morse potential

the quantum effective mass hamilton jacobi problem in this paper the quantum hamilton jacobi theory based on the position dependent mass model is studied two effective mass functions having different singularity structures are used to examine the morse and poschl teller potentials the residue method is used to obtain the solutions of the quantum effective mass hamilton jacobi equation further it is shown that the eigenstates of the generalized non hermitian swanson hamiltonian for morse and poschl teller potentials can be obtained by using the riccati equation without solving a differential equation

metric operators for non hermitian quadratic su 2 hamiltonians a class of non hermitian quadratic su 2 hamiltonians having an anti linear symmetry is constructed this is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra in fact different realizations for this type of symmetry are obtained including the natural occurrence of charge conjugation together with parity and time reversal once specified the underlying anti linear symmetry of the hamiltonian the former if unbroken leads to a purely real spectrum and the latter can be mapped to a hermitian counterpart by amongst other possibilities a similarity transformation here lie algebraic methods which were used to investigate the generalized swanson hamiltonian assis and fring 2009 j phys a math theor 42 015203 are employed to identify the class of quadratic hamiltonians that allow for such a mapping to the hermitian counterpart whereas for the linear su 2 system every hamiltonian of this type can be mapped to a hermitian counterpart by a transformation which is itself an exponential of a linear combination of su 2 generators the situation is more complicated for quadratic hamiltonians therefore the possibility of more elaborate similarity transformations including quadratic exponents is also explored in detail the existence of finite dimensional representations for the su 2 hamiltonian as opposed to the su 1 1 studied before allows for comparison with explicit diagonalization results for finite matrices finally the similarity transformations constructed are compared with the analogue of swanson s method for exact diagonalization of the problem establishing a simple relation between both approaches

transition elements for a non hermitian quadratic hamiltonian the non hermitian quadratic hamiltonian h a a a2 a 2 is analyzed where a and a are harmonic oscillator creation and annihilation operators and and are real constants for the case that 2 4 0 it is shown using operator techniques that the hamiltonian possesses real and positive eigenvalues a generalized bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator the eigenstates are shown to possess an imaginary norm for a large range of the parameter space finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of bender et al for non hermitian hamiltonians transition probabilities governed by h are shown to be manifestly unitary when the complexification procedure is followed a specific transition element between harmonic oscillator states is evaluated for both the hermitian and non hermitian cases to identify the differences in time evolution

exact diagonalization of non hermitian so 3 2 models generalized two mode boson systems we propose a unified approach to exactly diagonalize generalized non hermitian so 3 2 models this approach is a series of similarity transformations which is constructed by some similarity transformation operators associated with su 1 1 and su 2 subalgebras of so 3 2 lie algebra during this diagonalization it is worth noting that a key step is to get rid of the terms e and f together via the proper similarity transformations first in this way exact solutions of the non hermitian so 3 2 models are obtained meanwhile we give the corresponding eigenstates which are regarded as lie algebra so 3 2 coherent like number states the results can cover the generic form of the eigenvalues and eigenstates to the generalized non hermitian two mode boson systems with the discrete spectrum including 2d pt symmetric and non pt symmetric oscillators as the special cases also they are true for the hermitian case

perturbation theory of symmetric hamiltonians in the framework of perturbation theory the reality of the perturbed eigenvalues of a class of symmetric hamiltonians is proved using stability techniques we apply this method to symmetric unperturbed hamiltonians perturbed by symmetric additional interactions

reflectionless potentials and pt symmetry large families of hamiltonians that are non hermitian in the conventional sense have been found to have all eigenvalues real a fact attributed to an unbroken pt symmetry the corresponding quantum theories possess an unconventional scalar product the eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane for a special class of such systems it is possible to impose the pt symmetric boundary conditions on the real axis which lies on the edges of the wedges the pt symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless

meromorphic jacobi forms of half integral index and umbral moonshine modules in this work we consider an association of meromorphic jacobi forms of half integral index to the pure d type cases of umbral moonshine and solve the module problem for four of these cases by constructing vertex operator superalgebras that realise the corresponding meromorphic jacobi forms as graded traces we also present a general discussion of meromorphic jacobi forms with half integral index and their relationship to mock modular forms