topological bf theories in 3 and 4 dimensions in this paper we discuss topological bf theories in 3 and 4 dimensions observables are associated to ordinary knots and links in 3 dimensions and to 2 knots in 4 dimensions the vacuum expectation values of such observables give a wide range of invariants here we consider mainly the 3 dimensional case where these invariants include alexander polynomials homfly polynomials and kontsevich integrals

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

ym2 continuum expectations lattice convergence and lassos the two dimensional yang mills theory ym2 is analyzed in both the continuum and the lattice in the complete axial gauge the continuum theory may be defined in terms of a lie algebra valued white noise and parallel translation may be defined by stochastic differential equations this machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesc the expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group the time parameters of the heat kernels are determined by the areas enclosed by the collectionc and the arguments are determined by the crossing topologies of the curves inc the expectations for the wilson lattice models have a similar structure and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations it is also shown that the lasso variables advocated by l gross 36 exist and are sufficient to generate all the measurable functions on the ym2 measure space

on the deficiency indices and self adjointness of symmetric hamiltonian systems abstract the main purpose of this paper is to investigate the formal deficiency indices n of a symmetric first order system jf bf h f on an interval i where i r or i r here j b h are n n matrix valued functions and the hamiltonian h 0 may be singular even everywhere we obtain two results for such a system to have minimal numbers n 0 if i r resp n n if i r and a criterion for their maximality n 2n for i r as well as the quasi regularity this covers the kac krein and de branges trans amer math soc 99 1961 118 theorems on 2 2 canonical systems and some results from kogan and rofe beketov proc roy soc edinburgh sect a 74 1974 75 5 some conditions for a canonical system to have intermediate formal deficiency indices are presented too we also obtain a generalization of the well known titchmarsh sears theorem for second order sturm liouville type equations this contains results due to lidskii and krein as special cases we present two approaches to the above problems one dealing with formal deficiency indices and one dealing with ordinary deficiency indices our main non formal approach is based on the investigation of a symmetric linear relation smin which is naturally associated to a first order system this approach works in the framework of extension theory and therefore we investigate in detail the domain d s min of s min in particular we prove the so called regularity theorem for d s min as a byproduct of the regularity result we obtain very short proofs of generalizations of the main results of the paper by kogan and rofe beketov 1974 75

the wilson loop observables of chern simons theory on ℝ3 in axial gauge we define the wilson loop observables wlos for pure chern simons models with base manifold m 3 rigorously as infinite dimensional oscillatory integrals by exploiting an axial gauge fixing and applying certain regularization techniques like loop smearing and framing the values of the wlos can be computed explicitly if the structure group g of the model considered is abelian one obtains well known linking number expressions for the wlos if g is non abelian one obtains expressions which are similar but not identical to the state sum representations for the homfly and kauffman polynomials by jones and turaev

weyl solutions and j selfadjointness for dirac operators we consider a non selfadjoint dirac type differential expression 0 1 d q y j n dy over dx q x y with a non selfadjoint potential matrix q epsilon l 1 loc scr j bbb c n times n and a signature matrix j n j j n j n epsilon bbb c n times n here scr j denotes either the line bbb r or the half line bbb r with this differential expression one associates in l 2 scr j bbb c n the closed maximal and minimal operators d max q and d min q respectively one of our main results for the whole line case states that d max q d min q in l 2 bbb r bbb c n moreover we show that if the minimal operator d min q in l 2 bbb r bbb c n is j symmetric with respect to an appropriate involution j then it is j selfadjoint similar results are valid in the case of the semiaxis bbb r in particular we show that if n 2p and the minimal operator d min q in l 2 bbb r bbb c 2p is j symmetric then there exists a 2p times p weyl type matrix solution psi z cdot epsilon l 2 bbb r bbb c 2p times p of the equation d max q psi z cdot z psi z cdot a similar result is valid for the expression 0 1 whenever there exists a proper extension tilde a with dim dom tilde a dom d min q p and nonempty resolvent set in particular it holds if a potential matrix q has a bounded imaginary part this leads to the existence of a unique weyl function for the express 0 1 the main results are proven by means of a reduction to the self adjoint case by using the technique of dual pairs of operators the differential expression 0 1 is of significance as it appears in the lax formulation of the vector valued nonlinear schrodinger equation

pattern formation in a model of competing populations with nonlocal interactions abstract we analyze and compute an extension of a previously developed population model based on the well known diffusive logistic equation with nonlocal interaction to a system involving competing species our model involves a system of nonlinear integro differential equations with the nonlocal interaction characterized by convolution integrals of the population densities against specified kernel functions the extent of the nonlocal coupling is characterized by a parameter so that when 0 the problem becomes local we consider critical points of the model i e spatially homogeneous equilibrium solutions there is generally one critical point in the first quadrant i e both population densities positive denoting coexistence of the two species we show that this solution can be destabilized by the nonlocal coupling and obtain general conditions for stability of this critical point as a function of the specific kernel function and parameters of the model we study the nonlinear behavior of the model and show that the populations can evolve to localized cells or islands we find that the stability transition is supercritical near the stability boundary solutions are small amplitude nearly sinusoidal oscillations however when increases large amplitude nonlinear states are found we find a multiplicity of stable steady state patterns we further show that with a stepfunction kernel function the structure of these islands a highly nonlinear phenomenon can be described analytically finally we analyze the role of the kernel function and show that for some choices of kernel function the resulting population islands can exhibit tip splitting behavior and island amplitude modulation

j self adjoint extensions of a class of hamiltonian differential systems abstract this paper is concerned with j self adjoint extensions for a class of hamiltonian differential systems the domains of the corresponding minimal and maximal operators are described a classification of limit types of the systems at each of endpoints is given a property of the regularity field of the corresponding minimal operator is derived and an expression of the j defect index of the minimal operator by those of the left and right minimal operators is presented based on them a characterization of all j self adjoint extensions for the hamiltonian system is obtained in terms of square integrable solutions of a certain equation which is generated by this hamiltonian system and its formal adjoint as a consequence characterizations of all j self adjoint extensions are given for the systems in some special cases in particular all j self adjoint extensions are characterized in terms of square integrable solutions of the system itself in certain special cases

norm estimates of complex symmetric operators applied to quantum systems this paper communicates recent results in the theory of complex symmetric operators and shows through two non trivial examples their potential usefulness in the study of schrodinger operators in particular we propose a formula for computing the norm of a compact complex symmetric operator this observation is applied to two concrete problems related to quantum mechanical systems first we give sharp estimates on the exponential decay of the resolvent and the single particle density matrix for schrodinger operators with spectral gaps second we provide new ways of evaluating the resolvent norm for schrodinger operators appearing in the complex scaling theory of resonances

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

j self adjoint operators with mathcal c symmetries an extension theory approach a well known tool in conventional von neumann quantum mechanics is the self adjoint extension technique for symmetric operators it is used e g for the construction of dirac hermitian hamiltonians with point interaction potentials here we reshape this technique to allow for the construction of pseudo hermitian j self adjoint hamiltonians with complex point interactions we demonstrate that the resulting hamiltonians are bijectively related to the so called hypermaximal neutral subspaces of the defect krein space of the symmetric operator this symmetric operator is allowed to have arbitrary but equal deficiency indices n n general properties of the operators for these hamiltonians are derived a detailed study of operator parametrizations and krein type resolvent formulae is provided for j self adjoint extensions of symmetric operators with deficiency indices 2 2 the technique is exemplified on 1d pseudo hermitian schr dinger and dirac hamiltonians with complex point interaction potentials

a mathematical construction of the non abelian chern simons functional integral we construct rigorously an infinite dimensional distribution which corresponds to the chern simons cs functional integral associated with a principal fiber bundle over r 3 with structure group a compact connected lie group we determine the moments of the cs distribution and show that these coincide with those used in informal studies of the cs integral a locality property of the cs distribution is proven the complexified theory of frohlich and king is also discussed within our framework

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

segregation process and phase transition in cyclic predator prey models with an even number of species we study a spatial cyclic predator prey model with an even number of species for n 4 6 and 8 that allows the formation of two defensive alliances consisting of the even and odd label species the species are distributed on the sites of a square lattice the evolution of spatial distribution is governed by iteration of two elementary processes on neighboring sites chosen randomly if the sites are occupied by a predator prey pair then the predator invades the prey s site otherwise the species exchange their sites with a probability x for low x values a self organizing pattern is maintained by cyclic invasions if x exceeds a threshold value then two types of domain grow up that are formed by the odd and even label species respectively monte carlo simulations indicate the blocking of this segregation process within a range of x for n 8

on the strong limit point and dirichlet properties of second order differential expressions second order differential expressions of the form w 1 pf qf are considered at a singular end point some of the known relationships between properties such as dirichlet weak dirichlet and strong limit point are extended to incorporate an arbitrary positive weight function and complexvalued coefficients

on the boundary conditions characterizing j selfadjoint extensions of j symmetric operators the study of boundary value problems involving linear differentiai equations with real valued coefficients is by now a well established area of analysis on the other hand much less is known about the solvability of such problems when the coefficients or boundary conditions are known to be complex examples of this latter type arise naturally for example in nuclear physics e g the so called optical mode1 for low energy scattering 1 p ijo electromagnetic field theory dielectric waveguides with heat loss c f 19 i or the propagation of radio waves through inhomogeneous media s and elsewhere in cases like these the relevant differential expressions are no longer formally symmetric and hence the powerful methods associated with the spectral theory of selfadjoint operators are not available to facilitate the study of such problems glazman introduced in 4 the concept of a j symmetric operator in a complex hilbert space r let j be a given conjugation operator on 2 i e j is a conjugate linear involution with jx 3 u x for all x and y in 2 a closed densely defined linear operator t in 3 is said to be j symmetric if

the chern simons theory and knot polynomials the chern simons gauge theory is studied using a functional integral quantization this leads to a differential equation for expectations of wilson lines the solution of this differential equation is shown to be simply related to the two variable jones polynomial of the corresponding link in the case where the gauge group issu n a similar equation has been used before to get the jones polynomial from a braid representation of the link the main novelty of our approach is that we get the jones polynomial from a plat representation of the link

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

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

p mathcal t symmetric non self adjoint operators diagonalizable and non diagonalizable with a real discrete spectrum consider in the operator family h0 a 1a1 a dad d 2 is the quantum harmonic oscillator with rational frequencies w is a symmetric bounded potential and g is a real coupling constant we show that if g being an explicitly determined constant the spectrum of h g is real and discrete moreover we show that the operator has a real discrete spectrum but is not diagonalizable

oscillatory integrals on hilbert spaces and schrodinger equation with magnetic fields the theory of oscillatory integrals on hilbert spaces the mathematical version of feynman path integrals is extended to cover more general integrable functions preserving the property of the integrals to have converging finite dimensional approximations an application to the representation of solutions of the time dependent schrodinger equation with a scalar and a magnetic potential by oscillatory integrals on hilbert spaces is given a relation with ramer s functional in the corresponding probabilistic setting is found

complex symmetric differential operators on fock space abstract the space of entire functions which are integrable with respect to the gaussian weight known also as the fock space is one of the preferred functional hilbert spaces for modeling and experimenting harmonic analysis quantum mechanics or spectral analysis phenomena this space of entire functions carries a three parameter family of canonical isometric involutions we characterize the linear differential operators acting on fock space which are complex symmetric with respect to these conjugations in parallel as a basis of comparison we discuss the structure of self adjoint linear differential operators the computation of the point spectrum of some of these operators is carried out in detail

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

abelian chern simons theory and linking numbers via oscillatory integrals a rigorous mathematical model of abelian chern simons theory based on the theory of infinite dimensional oscillatory integrals developed by albeverio and ho egh krohn is introduced a gauge fixed chern simons path integral is constructed as a fresnel integral in a certain hilbert space wilson loop variables are defined as fresnel integrable functions and it is shown in this context that the expectation value of products of wilson loops with respect to the chern simons path integral is a topological invariant which can be computed in terms of pairwise linking numbers of the loops as conjectured by witten furthermore a lattice chern simons action is proposed which converges to the continuum limit

complex symmetric operators and applications ii a bounded linear operator t on a complex hilbert space h is called complex symmetric if t ct c where c is a conjugation an isometric antilinear involution of h we prove that t cj t where j is an auxiliary conjugation commuting with t t t we consider numerous examples including the poincare neumann singular integral bounded operator and the jordan model operator compressed shift the decomposition t cj t also extends to the class of unbounded c selfadjoint operators originally introduced by glazman in this context it provides a method for estimating the norms of the resolvents of certain unbounded operators

j self adjointness of a class of dirac type operators in this note we prove that the maximally defined operator associated with the dirac type differential expression m q iddxim q q ddxim where q represents a symmetric m m matrix i e q x q x a e with entries in l1loc r is j self adjoint where j is the antilinear conjugation defined by j 1c 1 0imim0 and c a1 am b1 bm a1 am b1 bm the differential expression m q is of significance as it appears in the lax formulation of the non abelian matrix valued focusing nonlinear schrodinger hierarchy of evolution equations

on the limit point classification of a class of non self adjoint ordinary differential operators the limit point classification of a class of non symmetric ordinary differential expressions on half line is considered which is related to the well posedness of certain boundary value problems

self organization of five species in a cyclic competition game abstract cyclic competition game models particularly the rock paper scissors model play important roles in exploring the problem of multi species coexistence in spatially ecological systems we propose an extended rock paper scissors game to model cyclic interactions among five species and find that two of the five can coexistent when biodiversity disappears which is different from the rock paper scissors game as the number of fingers is five we named the new model the fingers game where the thumb forefinger middle finger ring finger and little finger cyclically dominate their subsequent species and are dominated by their former species we investigate the fingers model in two ways direct simulations and nonlinear partial differential equations an important finding is that the number of species in a cyclic competition game has an influence on the emergence of biodiversity to be specific the rock paper scissors model is in favor of maintaining biodiversity in comparison with the fingers model when the variables population size reproduction rate selection rate and migration rate are the same it is also shown that the mobility and reproduction rate can promote or jeopardize biodiversity

complex symmetric weighted composition operators on h2 d abstract in this paper we study complex symmetric weighted composition operators on the hardy space we provide some characterizations of and when a weighted composition operator w is complex symmetric we investigate which combinations of weights and maps of the open unit disk give rise to complex symmetric weighted composition operators with a special conjugation as some applications we obtain several examples for nonnormal complex symmetric operators in addition we give spectral properties of complex symmetric weighted composition operators we examine eigenvalues and eigenvectors of such operators and find some conditions for which a complex symmetric weighted composition operator is hilbert schmidt finally we consider cyclicity hypercyclicity and the single valued extension property for complex symmetric weighted composition operators

complex symmetry of weighted composition operators on the fock space abstract with the techniques of weighted composition operators we introduce a new concept of general weighted composition conjugations on the fock space f 2 c and completely solve the following problems the description of weighted composition operators which are conjugations the criteria for weighted composition operators to be complex symmetric the spectrum and the point spectrum of complex symmetric operators and the criteria for schatten class membership as well as for normality of such operators

on the asymmetric may leonard model of three competing species in this paper we analyze the global asymptotic behavior of the asymmetric may leonard model of three competing species frac dx i dt x i 1 x i beta ix i 1 alpha ix i 1 x i 0 0 i 1 2 3 with x 0 x 3 x 4 x 1 under the assumption 0 b1b2b3 and p is a saddle point with one dimensional stable manifold gamma if a1a2a3 b1b2b3 then p is global asymptotically stable in int r3 ii if a1a2a3 b1b2b3 then for each initial condition x 0 not in gamma the solution v

vassiliev knot invariants and chern simons perturbation theory to all orders at any order the perturbative expansion of the expectation values of wilson lines in chern simons theory gives certain integral expressions we show that they all lead to knot invariants moreover these are finite type invariants whose order coincides with the order in the perturbative expansion together they combine to give a universal vassiliev invariant

complex predator invasion waves in a holling tanner model with nonlocal prey interaction abstract we consider predator invasions for the nonlocal holling tanner model predators are introduced in a small region adjacent to an extensive predator free region in its simplest form an invasion front propagates into the predator free region with a predator prey coexistence state displacing the predator free state however patterns may form in the wake of the invasion front due to instability of the coexistence state the coexistence state can be subject to either oscillatory or cellular instability depending on parameters furthermore the oscillatory instability can be either at zero wave number or finite wave number in addition the unstable predator free state can be subject to additional cellular instabilities when the extent of the nonlocality is sufficiently large we perform numerical simulations that demonstrate that the invasion wave may have a complex structure in which different spatial regions exhibit qualitatively different behaviors these regions are separated by relatively narrow transition regions that we refer to as fronts we also derive analytic approximations for the speeds of the fronts and find qualitative and quantitative agreement with the results of computations

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

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

complex symmetric c0 semigroups on the fock space abstract in this paper we introduce a new concept in dynamical systems the complex symmetric operator c 0 semigroups groups acting on hilbert space using techniques of weighted composition operators we study these semigroups on the fock space f 2 and show in particular that complex symmetric operator c 0 semigroups contain the familiar unitary semigroups as a proper subclass

classification of sturm liouville differential equations with complex coefficients and operator realizations in this paper a new classification of sturm liouville differential equations with complex coefficients is given compared with the corresponding result of brown et al this classification reveals the great effects of rotation angle and it is independent of the rotation angles moreover the asymptotic behaviours of functions in the maximal domain are presented and j self adjoint extensions associated with the differential equations are characterized

perturbative expansion of chern simons theory with non compact gauge group naive imitation of the usual formulas for compact gauge group in quantizing three dimensional chern simons gauge theory with non compact gauge group leads to formulas that are wrong or unilluminating in this paper an appropriate modification is described which puts the perturbative expansion in a standard manifestly unitary format the one loop contributions which differ from naive extrapolation from the case of compact gauge group are computed and their topological invariance is verified

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

on j selfadjoint extensions of j symmetric ordinary differential operators on donne une caracterisation des conditions aux limites qui determinent le domaine d une extension j autoadjointe d un operateur differentiel j symetrique t 0 qui a une extremite singuliere

quadratic mathcal p mathcal t symmetric operators with real spectrum and similarity to self adjoint operators it is established that a symmetric elliptic quadratic differential operator with real spectrum is similar to a self adjoint operator precisely when the associated fundamental matrix has no jordan blocks this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

cyclic competition of mobile species on continuous space pattern formation and coexistence we propose a model for cyclically competing species on continuous space and investigate the effect of the interplay between the interaction range and mobility on coexistence a transition from coexistence to extinction is uncovered with a strikingly nonmonotonic behavior in the coexistence probability about the minimum in the probability switches between spiral and plane wave patterns arise a strong mobility can either promote or hamper coexistence depending on the radius of the interaction range these phenomena are absent in any lattice based model and we demonstrate that they can be explained using nonlinear partial differential equations our continuous space model is more physical and we expect the findings to generate experimental interest

beyond rock paper scissors systems deterministic models of cyclic ecological systems with more than three species abstract we consider cyclic ecological systems consisting of more than three species extending the well studied three species rock paper scissors rps system for such systems alliance and partial alliance states consisting of non competing species can form a phenomenon not present in classical rps systems we develop analyze and simulate a may leonard type model accounting for cyclic dynamics and spatial diffusion we analyze in detail four species and five species systems determine dynamics for both spatially homogeneous initial conditions i e ode models and the propagation and interaction of patches of different species driven by diffusion and the cyclic dynamics pde models we show markedly different behavior between the four and five species models with alliance states as stable steady states for the former whereas both coexistence of all species and heteroclinic cycles are possible stable outcomes for the latter finally we discuss the extension of this even odd split to systems with more than five species

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

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible

chern simons theory on r3 in axial gauge a rigorous approach we study pure chern simons models on m r3 using a functional integral quantization approach which is based on axial gauge fixing it is well known see e g comm math phys 126 1989 167 comm math phys 186 1997 563 that in axial gauge the chern simons action function is quadratic and that the faddeev popov determinant of this gauge fixing procedure is a constant function this means that the wilson loop observables wlos of the model considered can be obtained heuristically by integrating certain quantities against a functional measure of gaussian type we demonstrate that although these heuristic integral expressions look rather singular it is possible to give a rigorous meaning to them by combining constructions from white noise analysis with certain regularization techniques like loop smearing and framing for the special case g u 1 g being the group of the model we carry out the details of this approach which is also applicable to the case of non abelian g and find well known linking number expressions for the wlos

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