approximation of schrodinger eigenvalues and eigenfunctions by canonical perturbation theory the periodically driven quantum rotator a first order approximate formula for eigenvalues and eigenfunctions of the schrodinger operator of the periodically driven quantum rotator is given by means of the classical perturbation theory and an estimate for the remainder is provided in the nonresonant case the remainder is of order 2 while in the resonant case the remainder turns out to be of order 3 2 in the first case to what extent the procedure can be interated to higher orders is also discussed

finitized conformal spectrum of the ising model on the cylinder and torus abstract the spectrum of the critical ising model on a lattice with cylindrical and toroidal boundary conditions is calculated by commuting transfer matrix methods using a simple truncation procedure we obtain the natural finitizations of the conformal spectra recently proposed by melzer these finitizations imply polynomial identities which in the large lattice limit give rise to the rogers ramanujan identities for the c 1 2 virasoro characters

finitized conformal spectra of the ising model on the klein bottle and mobius strip we study the conformal spectra of the critical square lattice ising model on the klein bottle and mobius strip using yang baxter techniques and the solution of functional equations in particular we obtain expressions for the finitized conformal partition functions in terms of finitized virasoro characters this demonstrates that yang baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies the results rely on certain properties of the eigenvalues which are confirmed numerically

canonical perturbation expansions to large order from classical hypervirial and hellmann feynman theorems the classical hypervirial and hellmann feynman theorems are used to formulate a perturbation theory without fourier series that can be used to generate canonical series expansions for the energies of perturbed periodic orbits for separable classical hamiltonians as in the case where these theorems are used to generate quantum mechanical rayleigh schrodinger perturbation series the method is very efficient and may be used to generate expansions to large order either numerically or in algebraic form here the method is applied to one dimensional anharmonic oscillators and radial kepler problems in all cases the classical series for energies and expectation values are seen to correspond to the expansions associated with their quantum mechanical counterparts through an appropriate action preserving classical limit as discussed by turchetti graffi and paul this action fixing is inherent in the classical hellmann feynman theorem applied to periodic orbits

spectral monodromy of non self adjoint operators in the present paper we build a combinatorial invariant called the spectral monodromy from the spectrum of a single non self adjoint h pseudodifferential operator with two degrees of freedom in the semi classical limit our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self adjoint h pseudodifferential operators given by s vu ngoc quantum monodromy in integrable systems commun math phys 203 2 465 479 1999 the first simple case that we treat in this work is a normal operator in this case the discrete spectrum can be identified with the joint spectrum of an integrable quantum system the second more complex case we propose is a small perturbation of a self adjoint operator with a classical integrability property we show that the discrete spectrum in a small band around the real axis also has a combinatorial monodromy the main difficulty in this case is that we do not know the description of the spectrum everywher

ising model on nonorientable surfaces exact solution for the mobius strip and the klein bottle closed form expressions are obtained for the partition function of the ising model on an mxn simple quartic lattice embedded on a mobius strip and a klein bottle the solutions all lead to the same bulk free energy but for finite m and n the expressions are different depending on whether the strip width m is odd or even finite size corrections at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions our results are consistent with the conformal field prediction of a central charge c 1 2 provided that the twisted mobius boundary condition is regarded as a free or fixed boundary

umbilical torus bifurcations in hamiltonian systems we consider perturbations of integrable hamiltonian systems in the neighbourhood of normally umbilic invariant tori these lower dimensional tori do not satisfy the usual non degeneracy conditions that would yield persistence by an adaption of kam theory and there are indeed regions in parameter space with no surviving torus we assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a generalised umbilical catastrophe combining techniques of kam theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large cantor sets applications to gyrostat dynamics are pointed out

on convergence of normal forms for complex frequencies convergence of the normal form for the hamiltonian system is proven xi he yi yi he xi 1 i n where he has the form he x y h0 x eh1 x y x y rn tn and is holomorphic with respect to x y e provided n 2 and the ratio h0 x1 h0 x2 is not real

invariant tori with prescribed frequency for nearly integrable hamiltonian systems in this paper we consider real analytic nearly integrable hamiltonian systems under bruno non degeneracy condition and prove that if the rank of the jacobian of the n dimensional frequency vectors with respect to action variables is n 1 then there exists a one parameter analytic family of invariant tori whose frequencies are the small dilation of a prescribed diophantine frequency vector

quantum mechanics as classical mechanics plus quantum corrections the cubic anharmonic oscillator the rayleigh schrodinger perturbation series for the cubic anharmonic oscillator can be formally rearranged as the classical birkhoff series plus quantum corrections proportional to successive powers of h cross 2 each of the individual quantum corrections is a convergent power series expansion with the same radius of convergence as the classical series of the corresponding term in the jeffreys wentzel kramers brillouin series

universal entropy of conformal critical theories on a klein bottle we show that rational conformal field theories in 1 1 dimensions on a klein bottle with a length l and width satisfying l have a universal entropy this universal entropy depends on the quantum dimensions of the primary fields and can be accurately extracted by taking a proper ratio between the klein bottle and torus partition functions enabling the characterization of conformal critical theories the result is checked against exact calculations in quantum spin 1 2 xy and ising chains

estimates on invariant tori near an elliptic equilibrium point of a hamiltonian system abstract we give a precise statement for the kam theorem in a neighbourhood of an elliptic equilibrium point of a hamiltonian system if the frequencies of the elliptic point are nonresonant up to a certain order k 4 and a nondegeneracy condition is fulfilled we get an estimate for the measure of the complement of the kam tori in a neighbourhood of given radius moreover if the frequencies satisfy a diophantine condition with exponent we show that in a neighbourhood of radius r the measure of the complement is exponentially small in 1 r 1 1 we also give a related result for quasi diophantine frequencies which is more useful for practical purposes the results are obtained by putting the system in birkhoff normal form up to an appropiate order and the key point relies on giving accurate bounds for its terms

the geography problem for irreducible spin four manifolds we study the geography problem for smooth irreducible simplyconnected spin four manifolds for a large class of hoinotopy types we exhibit both symplectic and non symplectic representatives we also compute the seiberg witten invariants of all the four manifolds we construct

quantum birkhoff gustavson normal form in some completely resonant cases abstract the goal of this paper is to present an explicit computation of the birkhoff gustavson normal form in the 1 1 1 2 and 1 3 resonances by using the concept of weyl quantization especially when it is about the first terms we use in particular the weyl symbolic calculus and the abstract theorem of the quantum birkhoff gustavson normal form in infinite dimension

asymptotic density of eigenvalue clusters for the perturbed landau hamiltonian we consider the landau hamiltonian i e the 2d schrodinger operator with constant magnetic field perturbed by an electric potential v which decays sufficiently fast at infinity the spectrum of the perturbed hamiltonian consists of clusters of eigenvalues which accumulate to the landau levels applying a suitable version of the anti wick quantization we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity we obtain an explicit description of the asymptotic density of the eigenvalues in terms of the radon transform of the perturbation potential v

poincare contraction of su 1 1 fock bargmann structure some aspects of the contraction process so0 1 2 to poincare are studied the starting point is the choice of a suitable parametrization for the de sitterian phase space so0 1 2 so 2 approximately su 1 1 u 1 the authors show that the contraction to poincare must be realized by restricting the fock bargmann space to a specific subspace this constraint is necessary to make the divergent terms disappear in particular the classical result according to which the discrete series representation of su 1 1 contracts onto the wigner representation p m is described at a global level

a posteriori kam theory with optimal estimates for partially integrable systems abstract in this paper we present a posteriori kam results for existence of d dimensional isotropic invariant tori for n dof hamiltonian systems with additional n d independent first integrals in involution we carry out a covariant formulation that does not require the use of action angle variables nor symplectic reduction techniques the main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates which may cause difficulties and underperformance when quantifying the hypotheses of the kam theorem in such reduced coordinates the results include ordinary and generalized iso energetic kam theorems the approach is suitable to perform numerical computations and computer assisted proofs

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

superconformal invariance and the geography of four manifolds the correlation functions of supersymmetric gauge theories on a four manifold x can sometimes be expressed in terms of topological invariants of x we show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four manifold topology in particular in the example of gauge group su 2 with one doublet hypermultiplet we derive a theorem relating classical topological invariants such as the euler character and signature to sum rules for seiberg witten invariants a short account of this paper can be found in 1

on the diagonalization of quantum birkhoff gustavson normal form an application to quantum mechanics of one of classical perturbation theory methods the birkhoff gustavson normal form bgnf is described in the quantum case it results in the van vleck perturbation theory performed upon wick normal ordered operators algebraic aspects of this procedure and formal construction of invariants integrals of motion for a perturbed system are considered it turned out that a larger set of such operators existed in the quantum mechanics rather than in the classical one it is demonstrated that according to general results of the quantum mechanical perturbation theory the quantum bgnf may always be diagonalized and two formal processes for such diagonalization are constructed in the opposite case the classical bgnf is in general nondiagonalizable this reflects the fact that the classical perturbation theory cannot handle a system with two or more resonances possible reasons for such different behavior of two very close in spirit perturbation procedures are discussed results of the described procedure entirely performed upon the wick normal ordered operators are equivalent to those of rayleigh schrodinger perturbation expansion

kam theory the legacy of kolmogorov s 1954 paper kolmogorov arnold moser or kam theory was developed for conservative dynamical systems that are nearly integrable integrable systems in their phase space usually contain lots of invariant tori and kam theory establishes persistence results for such tori which carry quasi periodic motions we sketch this theory which begins with kolmogorov s pioneering work

berry s phase hannay s angle and quantum normal forms for any polynomial perturbation of the generalized harmonic oscillator the berry phase is explicitly computed in all orders of perturbation theory as a polynomial in h cross whose term of order zero is the hannay angle this construction is a direct application of the technique of quantum normal forms

stability of hamiltonian systems at high energy let h t p u p be the hamiltonian for a classical mechanical system of l degrees of freedom where t is an integrable hamiltonian generating quasiperiodic motion in the first l 1 degrees of freedom which has the form

the schrodinger equation and canonical perturbation theory lett0 ev be the schrodinger operator corresponding to the classical hamiltonianh0 ev whereh0 is thed dimensional harmonic oscillator with non resonant frequencies 1 d and the potentialv q1 qd is an entire function of order d 1 1 we prove that the algorithm of classical canonical perturbation theory can be applied to the schrodinger equation in the bargmann representation as a consequence each term of the rayleigh schrodinger series near any eigenvalue oft0 admits a convergent expansion in powers of of initial point the corresponding term of the classical birkhoff expansion moreover ifv is an even polynomial the above result and the kam theorem show that all eigenvalues n e oft0 ev such thatn coincides with a kam torus are given up to order e by a quantization formula which reduces to the bohr sommerfeld one up to first order terms in

partition function zeroes of a self dual ising model abstract we consider the ising model on an m n rectangular lattice with an asymmetric self dual boundary condition and derive a closed form expression for its partition function we show that zeroes of the partition function are given by the roots of a polynomial equation of degree 2m 1 which trace out certain loci in the complex temperature plane particularly it is shown that a real solutions of the polynomial equations always lead to zeroes on the unit circle and a segment of the negative real axis and b all temperature zeroes lie on two circles in the limit of m for any n closed form expressions of the loci as well as the density of zero distributions in the limit of n are derived for m 1 and 2 in addition we explain the reason of and establish the criterion for partition function zeroes of any self dual spin model to reside precisely on the unit circle this elucidates a recent finding in the case of the self dual potts model

quantum degeneracy and classical perturbation theory the degenerate rayleigh schrodinger perturbation theory for the eigenvalues of the class of schrodinger operators s e s0 ev where s0 is the laplace beltrami operator on the one dimensional torus t1 and v is an analytic potential perturbed rotator is explicitly constructed by means of the algorithm of the classical resonant birkhoff normal form as a consequence the degenerate quantum perturbation theory tends to the classical one in the classical limit

semiclassical estimates for weyl quantisation abstract we discuss weyl quantisation for operators with holomorphic symbols in the bargmann representation as an application we extend to this class of operators the semiclassical analysis of perturbation theory as introduced by degli esposti et al ann phys 209 1991 364 392

a proof of the isoenergetic kam theorem from the ordinary one a proof is given of the isoenergetic kam theorem for hamiltonian systems using the ordinary kam theorem and a transversality argument

diophantine tori and spectral asymptotics for nonselfadjoint operators we study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint h pseudodifferential operators in dimension 2 assuming that the classical flow of the unperturbed part possesses several invariant lagrangian tori enjoying a diophantine property we get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases in the first case we assume that the strength e of the perturbation is o hd for some d 0 and is bounded from below by a fixed positive power of h in the second case e is assumed to be sufficiently small but independent of h and we describe the eigenvalues completely in a fixed h independent domain in the complex spectral plane

semiclassical approximation for schrodinger operators on a two sphere at high energy let h s v be a schrodinger operator acting in l2 s with s the two dimensional unit sphere s the spherical laplacian and v a smooth potential approximate eigenfunctions and eigenvalues for h are obtained involving expansions in inverse powers of the classical angular momentum variables provided that these variables are in a region of phase space where the corresponding classical hamiltonian is nearly integrable the analysis is carried out in a bargmann representation where s becomes a quadratic expression in the sum of two quantum harmonic oscillator hamiltonians and v becomes a pseudodifferential operator

classical limit of the harmonic oscillator wigner functions in the bargmann representation writing the wigner functions of any pair of harmonic oscillator eigenstates in the bargmann representation a direct and detailed proof is given of their convergence in the sense of distributions to at the classical limit m n fixed

a quantum cherry theorem for perturbations of the plane rotator we consider on l2 t2 the schrodinger operator family l r with domain and action defined as d l h2 t2 l u 12 2 1 12 2 22 u i 1 1 2 2 u vu here r 1 2 1 2 are vectors of complex non real frequencies and v a pseudodifferential operator of order zero l represents the weyl quantization of the hamiltonian family l x 12 1 12 2 22 1 1 2 2 v x defined on the phase space r2 t2 where v x c2 r2 t2 r we prove the uniform convergence with respect to 0 1 of the quantum normal form which reduces to the classical one for 0 this result simultaneously entails an exact quantization formula for the quantum spectrum as well as a convergence criterion for the classical birkhoff normal form generalizing a well known theorem of cherry

a uniform quantum version of the cherry theorem consider in l 2 mathbb r 2 the operator family h epsilon p 0 hbar omega epsilon f 0 p 0 is the quantum harmonic oscillator with diophantine frequency vector f 0 a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space and epsilon in mathbb c then there exist epsilon ast 0 independent of hbar and an open set gamma subset mathbb c 2 setminus mathbb r 2 such that if epsilon epsilon ast and omega in gamma the quantum normal form near p 0 converges uniformly with respect to hbar this yields an exact quantization formula for the eigenvalues and for hbar 0 the classical cherry theorem on convergence of birkhoff s normal form for complex frequencies is recovered

a version of thirring s approach to the kolmogorov arnold moser theorem for quadratic hamiltonians with degenerate twist we give a proof of the kolmogorov arnold moser kam theorem on the existence of invariant tori for weakly perturbed hamiltonian systems based on thirring s approach for hamiltonians that are quadratic in the action variables the main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic hamiltonians we show that thirring s proof for nondegenerate hamiltonians can be adapted to hamiltonians with degenerate twist this case in fact drastically simplifies thirring s proof

quantum evolution and classical flow in complex phase space for a class of holomorphic perturbations of the harmonic oscillator inn degrees of freedom a local solution of the time dependent schrodinger equation in the bargmann representation is constructed which pointwise propagates to leading order in along the classical trajectories in complex phase space

convergence of a quantum normal form and a generalization of cherry s theorem we consider on the schrodinger operator family with the domain and action defined as follows where 1 2 is a vector of complex frequencies and v is a pseudodifferential operator of order zero he represents the weyl quantization of the hamiltonian family defined on the phase space where we prove the uniform convergence with respect to 0 1 of the quantum normal form which reduces to the classical one for 0 as a consequence we simultaneously obtain an exact quantization formula for the quantum spectrum as well as a convergence criterion for the classical birkhoff normal form generalizing a well known theorem of cherry

resonant harmonic oscillators and eigenvalue multiplicity explicit formulae are worked out for the eigenvalue multiplicity of a system of n independent quantum harmonic oscillators in the general case of 1 or s or n 1 resonance relations among the frequencies omega 1 omega n as a particular case we prove that even though the quantum numbers are always less than the degrees of freedom the eigenvalues are in general intrinsically degenerate only in the completely resonant case s n 1

third harmonic generation as a third order quasi exactly solvable system the two degrees of freedom in the usual third harmonic generation effective hamiltonian can be separated both classically and quantum mechanically into a harmonic oscillator and a nonlinear oscillator in turn the quantum hamiltonian of the nonlinear oscillator can be written as a cubic polynomial in the generators of a quasi exactly solvable lie algebra and the physically relevant eigenstates are precisely those that can be determined exactly although in general not explicitly since the standard jeffreys wentzel kramers brillouin methods are not easily applicable to general third order differential equations we use a bohr sommerfeld quantization of the classical orbits to obtain approximate explicit formulas for the corresponding quantum eigenvalues

effective stability and kam theory abstract the two main stability results for nearly integrable hamiltonian systems are revisited nekhoroshev theorem concerning exponential lower bounds for the stability time effective stability and kam theorem concerning the preservation of a majority of the nonresonant invariant tori perpetual stability to stress the relationship between both theorems a common approach is given to their proof consisting of bringing the system to a normal form constructed through the lie series method the estimates obtained for the size of the remainder rely on bounds of the associated vectorfields allowing one to get the optimal stability exponent in nekhoroshev theorem for quasiconvex systems on the other hand a direct and complete proof of the isoenergetic kam theorem is obtained moreover a modification of the proof leads to the notion of nearly invariant torus which constitutes a bridge between kam and nekhoroshev theorems

semiclassical analysis of a quasi exactly solvable system second harmonic generation the usual second harmonic generation effective hamiltonian is shown to be equivalent to a one dimensional schrodinger operator with a sextic polynomial potential this operator belongs to the class of quasi exactly solvable models for which a finite number of eigenvalues and eigenvectors can be determined exactly a jeffreys wentzel kramers brillouin analysis provides accurate asymptotic expansions for these eigenvalues in the limit of large unperturbed energies