stabilization for wave equation of variable coefficients with balakrishnan taylor damping and source term abstract in this paper we consider the wave equation of variable coefficients with balakrishnan taylor damping time varying delay and source term by establishing some lyapunov functionals which are equivalent to the energy of the system and using the potential well method to deal with the source term we prove a general stability result

solution of the eigenvalue problem of a n level system coupled to a bosonic degree of freedom without using rwa the eigenvalue problem of a n level system coupled to a bosonic degree of freedom is solved without using rwa for that purpose the bosonic degree of freedom is transformed to bargmann s hilbert space of analytical functions in this representation the schrodinger equation is a system of n coupled linear differential equations of first order using a discrete symmetry these equations are simplified by a suitable transformation of the independent variable starting from the simplified equations the authors develop a method to solve the eigenvalue problem of the n level system in addition they present a simple approximate treatment and compare it with the exact results the approximation turns out to be quite good up to outer level resonance and can be used to explain the differently structured regions in the energy spectra

level crossings and new exact solutions of the two photon rabi model an infinite family of exact solutions of the two photon rabi model was found by investigating the differential algebraic properties of the hamiltonian this family the new states corresponds to energy level crossings not covered by the juddian class which is given by elemetary functions in contrast the new states are expressible in terms of parabolic cylinder or bessel functions we discuss three approaches for discovering this hidden structure factorization of differential equations kimura transformation and a doubly infinite transcendental basis of the bargmann space

analytical treatment of a two level system with transitive and displacive vibronic coupling the eigenvalue problem of a hamiltonian which has been used as a model in exciton dynamics and in the theory of nonradiative transitions is formulated in bargmann s hilbert space of analytical functions 1962 for particle combinations of the parameters of the hamiltonian one finds exact solutions which are elementary transcendental functions in the general case the eigenvalues and the eigenfunctions are determined by matrix continued fractions the eigenfunctions can be represented accurately by neumann expansions with very few terms

a relation between levenshtein type distances and insertion and deletion correcting capabilities of codes a code is a collection of words or strings not necessarily all of the same length over come fixed alphabet a relation is established between the insertion and deletion correcting capability of a code and its minimum distance for suitable levenshtein type distance measures

higher order energy decay for damped wave equations with variable coefficients abstract under appropriate assumptions the higher order energy decay rates for the damped wave equations with variable coefficients c x u t t div a x u a x u t 0 in r n are established the results concern weighted in time and pointwise in time energy decay estimates we also obtain weighted l 2 estimates for spatial derivatives

analytical energy spectrum for hybrid mechanical systems we investigate the energy spectrum for hybrid mechanical systems described by non parity symmetric quantum rabi models a set of analytical solutions in terms of the confluent heun functions and their analytical energy spectrum is obtained the analytical energy spectrum includes regular and exceptional parts which are both confirmed by direct numerical simulation the regular part is determined by the zeros of the wronskian for a pair of analytical solutions the exceptional part is relevant to the isolated exact solutions and its energy eigenvalues are obtained by analyzing the truncation conditions for the confluent heun functions by analyzing the energy eigenvalues for exceptional points we obtain the analytical conditions for the energy level crossings which correspond to two fold energy degeneracy

the holstein polaron problem revisited the holstein hamiltonian was proposed half a century ago since then decades of research have come up empty handed in the pursuit of a closed form solution an exact solution to the two site holstein model is presented in this paper the obtained results provide a clear image of the hamiltonian structure and allow for the investigation of the symmetry energy level crossings and polaronic characteristics of the system the main mathematical tool is a three term recurrence relation between the wave function amplitudes which was obtained using the properties of a family of orthogonal functions namely the poisson charlier polynomials it is shown that with the appropriate choice of basis the eigenfunctions of the problem naturally fall into two families parities associated with the discrete symmetry of the hamiltonian the asymptotic solution to the recurrence relation is found by using the birkhoff expansion the asymptotic sets the truncation criterion for the wave function which ensures the accurate calculation of the energy levels for any strength of electron phonon interaction the level crossing of states with different parities is discussed and the exact points of broken symmetry are found analytically the results are used as the building blocks for studying a four site system the inherited symmetries lead to the formation of a sparse matrix that is convenient for numerical calculations

quasiclassical and exact quantization of simple two level model systems the laplace method is considered in application to one dimensional two state eigenvalue problems in the example of simple models including that of coupled parabolic terms and radial oscillators with quadratic linear and constant coupling basic models of renner pseudo jahn teller effects etc in all cases the solution can be presented either as a two state analogue of the bohr sommerfeld quasiclassical quantization rule or in more general form compared with the semiclassical picture of inelastic collisions in the momentum space representation the contour of integration of trajectory equations for the case of bound states is finite and any desired degree of accuracy can easily be achieved because no convergence problems arise the need to compute the equation possessing physical sense e g for adiabatic amplitudes is circumvented suggesting that the method would be applicable also to complicated potentials and to many state problems numerical tests for the energy level positions and calculations of eigenfunctions are presented practical applications and generalizations are outlined

on the solvability of the quantum rabi model and its 2 photon and two mode generalizations we study the solvability of the time independent matrix schrodinger differential equations of the quantum rabi model and its 2 photon and two mode generalizations in bargmann hilbert spaces of entire functions we show that the rabi model and its 2 photon and two mode analogs are quasi exactly solvable we derive the exact closed form expressions for the energies and the allowed model parameters for all the three cases in the solvable subspaces up to a normalization factor the eigenfunctions for these models are given by polynomials whose roots are determined by systems of algebraic equations

short note on the rabi model the spectral density of the rabi model is calculated exactly within a continued fraction approach it is shown that the method provides a simple algorithm for the spectral density with convergent solutions we compare these recursive solutions with the solutions of the jaynes cummings model and discuss the effect of approximations on the spectral properties

boundary feedback stabilization of shallow shells we consider the stabilization of the shallow shell by boundary feedbacks where the model has a middle surface of any shape first we put the shallow shell model in a suitable semigroup scheme the existence the uniqueness and the properties of solutions to the shallow shell are then treated by the semigroup approach and the regularity of elliptic boundary value problems finally we establish the uniform energy decay rate for the shallow shell under some checkable geometric conditions on the middle surface

slice segal bargmann transform the segal bargmann transform is a unitary map between the schrodinger and fock space which is used for example to show the integrability of quantum rabi models slice monogenic functions provide the framework in which functional calculus for quaternionic quantum mechanics can be developed in this paper a generalisation of the segal bargmann transform to the context of slice monogenic functions is constructed and studied in detail it is shown to interact appropriately with the recently constructed slice fourier transform this leads furthermore to a construction of a slice fock space which is shown to be a reproducing kernel space

an algebra of discrete channels that involve combinations of three basic error types recently the author introduced a nonprobabilistic mathematical model of discrete channels the bee channels that involve the error types substitution insertion and deletion this paper defines an important class of bee channels the sid channels which include channels that permit a bounded number of scattered errors and possibly at the same time a bounded burst of errors in any segment of predefined length of a message a formal syntax is defined for generating channel expressions and appropriate semantics is provided for interpreting a given channel expression as a communication channel sid channel that permits combinations of substitutions insertions and deletions of symbols our framework permits one to generalize notions such as error correction and unique decodability and express statements of the form the code k can correct all errors of type and it is decidable whether the code k is uniquely decodable for the channel described by where is any sid channel expression

continued fractions and the rabi model techniques based on continued fractions to numerically compute the spectrum of the quantum rabi model are reviewed they are of two essentially different types in the first case the spectral condition is implemented using a representation in the infinite dimensional bargmann space of analytic functions this approach is shown to approximate the correct spectrum of the full model if the continued fraction is truncated at sufficiently high order in the second case one considers the limit of a sequence of models defined in finite dimensional state spaces in contrast to the first the second approach is ambiguous and can be justified only through recourse to the analyticity argument from the first method

stabilization of the variable coefficient structural acoustic model with curved middle surface and delay effects in the structural component abstract this paper is concerned with the uniform energy decay rates of a structural acoustic model which describes the interactions between the acoustic medium and the elastic structural in an acoustic chamber where one wall is flexible and curved the coupled system considered consists of a variable coefficient wave equation and a variable coefficient plate equation defined on the riemannian manifold with the coupling on the interface between the acoustic medium and the flexible wall both the components of the dynamics are subject to nonlinear boundary damping furthermore a nonlinear delay term acting in the boundary feedbacks of the structure component is considered the riemannian geometry method is applied to derive the energy estimates of the nonlinear coupled system with variable coefficients the uniform energy decay rates of the nonlinear variable coefficient structural acoustic system with delay effects on the riemannian manifold are quantified by a solution to a constructed nonlinear ode

the quantum rabi model for two qubits we study a system composed of two nonidentical qubits coupled to a single mode quantum field we calculate the spectra of the system in the deep strong coupling regime via perturbation theory up to second order corrections and show that it converges to two forced oscillator chains for cases well into that regime our predictions are confirmed by the numerical calculation of the spectra using a parity decomposition of the corresponding hilbert space the numerical results point to two interesting types of behavior in the ultra strong coupling regime the rotating wave approximation is valid for some particular cases and there exist crossings in the spectra within each parity subspace we also present the normal modes of the system and give an example of the time evolution of the mean photon number population inversion von neumann entropy and wootters concurrence in the ultra strong and deep strong coupling regimes

lee metric bch codes and their application to constrained and partial response channels shows that each code in a certain class of bch codes over gf p specified by a code length n spl les p sup m 1 and a runlength r spl les p 1 2 of consecutive roots in gf p sup m has minimum lee distance spl ges 2r for the very high rate range these codes approach the sphere packing bound on the minimum lee distance furthermore for a given r the length range of these codes is twice as large as that attainable by berlekamp s 1984 extended negacyclic codes the authors present an efficient decoding procedure based on euclid s algorithm for correcting up to r 1 errors and detecting r errors that is up to the number of lee errors guaranteed by the designed minimum lee distance 2r bounds on the minimum lee distance for r spl ges p 1 2 are provided for the reed solomon case i e when the bch code roots are in gf p the authors present two applications first lee metric bch codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and or deletion of zeros in d k constrained channels second the code construction with its decoding algorithm can be formulated over the integer ring providing an algebraic approach to correcting errors in partial response channels where matched spectral null codes are used

spectral analysis of non commutative harmonic oscillators the lowest eigenvalue and no crossing abstract the lowest eigenvalue of non commutative harmonic oscillators q 0 0 1 is studied it is shown that q can be decomposed into four self adjoint operators q p 1 2 q p and all the eigenvalues of each operator q p are simple we show that the lowest eigenvalue of q is simple whenever furthermore a jacobi matrix representation of q p is given and spectrum of q p is considered numerically

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

on the observability inequalities for exact controllability of wave equations with variable coefficients this paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part subject to dirichlet or neumann boundary controls the observability inequalities are established by the riemannian geometry method under some geometric condition for the dirichlet problem and for the neumann problem respectively next a number of nontrivial examples are presented to verify the observability inequality in particular a counterexample is given without boundary exact controllability where the control is exerted on the whole boundary

stabilization of the transmission wave plate equation with variable coefficients abstract we consider the stabilization of the transmission wave plate equation with variable coefficients and a general transmission boundary by virtue of the riemannian geometric approach we obtain some stability results for nonlinear feedbacks acting on the regions under suitable geometric conditions

controllability of the linear one dimensional wave equation with inner moving forces we address the null inner controllability of the one dimensional wave equation with supports of the controls that may vary with respect to the time variable we consider theoretical and numerical aspects assuming a geometric optic condition on the support of the controls we first prove a generalized observability inequality for the homogeneous wave equation implying the controllability we then introduce and prove the well posedness of a mixed formulation that characterizes the controls of minimal square integrable norm a numerical approximation of this formulation based on a conformal space time finite element method is then presented several numerical experiments are finally reported

global smooth solutions for the quasilinear wave equation with boundary dissipation abstract we consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system our main tool is the geometrical analysis the main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system we show that the existence of global solutions depends not only on this dissipation structure but also on a riemannian metric given by the coefficients and the equilibrium of the system some geometrical conditions on this riemannian metric are presented to guarantee the existence of global solutions in particular we prove that the norm of the state of the system decays exponentially if the input stops after a finite time which implies the exponential stabilization of the system by boundary feedback

inverse observability estimates for second order hyperbolic equations with variable coefficients abstract we consider a general second order hyperbolic equation defined on an open bounded domain r n with variable coefficients in both the elliptic principal part and in the first order terms as well at first no boundary conditions b c are imposed our main result theorem 3 5 is a reconstruction or inverse estimate for solutions w under checkable conditions on the coefficients of the principal part the h1 l2 energy at time t t or at time t 0 is dominated by the l2 norms of the boundary traces w a and wt modulo an interior lower order term once homogeneous b c are imposed our results yield under a uniqueness theorem needed to absorb the lower order term continuous observability estimates for both the dirichlet and neumann case with an explicit sharp observability time hence by duality exact controllability results moreover no artificial geometrical conditions are imposed on the controlled part of the boundary in the neumann case in contrast with existing literature the first step of our method employs a riemann geometry approach to reduce the original variable coefficient principal part problem in r n to a problem on an appropriate riemann manifold determined by the coefficients of the principal part where the principal part is the laplacian in our second step we employ explicit carleman estimates at the differential level to take care of the variable first order energy level terms in our third step we employ micro local analysis yielding a sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the neumann case

representation and properties of para bose oscillator operators ii coherent states and the minimum uncertainty states the energy position and momentum eigenstates of a para bose oscillator system were considered in paper i here we consider the bargmann or the analytic function description of the para bose system this brings in in a natural way the coherent states z defined as the eigenstates of the annihilation operator the transformation functions relating this description to the energy position and momentum eigenstates are explicitly obtained possible resolution of the identity operator using coherent states is examined a particular resolution contains two integrals one containing the diagonal basis z z and the other containing the pseudodiagonal basis z z we briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations the problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible dependence of this minimum value is considered

general stability of a viscoelastic variable coefficients plate equation with time varying delay in localized nonlinear internal feedback abstract this paper is concerned with the decay properties of a memory type euler bernoulli plate equation with variable coefficients and time varying delay in localized nonlinear internal feedback by using piecewise multiplier method and riemannian geometric analysis we establish the explicit and general decay rates of system whose nonlinear internal feedbacks act on a suitable subregion

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

exact solution of the e e jahn teller and rabi hamiltonian by generalised spheroidal wavefunctions the schrodinger equation for the e e jahn teller and rabi systems in bargmann s hilbert space 1962 is a system of two ordinary differential equations of first order for the spin up and down components of the wavefunctions this system has two regular and one irregular singular points the energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions the differential equations of the generalised spheroidal wavefunctions have the same singular points and the same exponents at each singular point it is therefore conjectured that the component wavefunctions in the excited state i can be expanded in i 4 generalised spheroidal wavefunctions the energy eigenvalues i i 0 1 5 calculated with the conjectured component wavefunctions agree with numerical values within the computational error the same is true for the coefficients of the neumann expansion of the component wavefunctions a proof is still missing

a generalized potential in the theory of the rabi and e x epsilon jahn teller system the eigenvalue problem for the rabi and the e x epsilon jahn teller hamiltonian in bargmanns hilbert space of analytical functions is a system of two first order differential equations for the two component wavefunctions whose entire solutions the eigenstates are sought we show that each eigenstate is a terminating series in the derivatives of a scalar entire function d z called the generalized potential which satisfies a higher order differential equation the coefficients of the terminating series depend on the physical parameters and are polynomials in the independent variable z the coefficients are identical in all eigenstates

hill s determinant approach to single mode spin boson model an efficient and stable numerical scheme based on hill s determinant is proposed for computing the whole spectrum of the single mode spin boson model its shown that for the low lying levels a simple version of hill s determinant method works well while for high levels a normalization of the determinant is necessary and this normalization results in an entire function whose zeros correspond to the energy levels it is very convenient to use this entire function to compute all the energy levels numerically some high levels of the model are computed and compared with irish s general rotating wave approximation a scaling law in the large photon number limit is numerically tested and several statistical properties of the spectrum are investigated

observability inequalities for the transmission of shallow shells in this paper we establish some observability inequalities from both boundary and distributed controls for the transmission of shallow shell with a middle surface of any shape in particular if only the boundary controls are allowed we obtain the condition for the observability inequality on young s modulus and poisson s coefficient of the material which is similar as those in the case of wave and plate equations

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

on exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback abstract the uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered the existence of both weak and strong solutions to the system is proven by the galerkin method the exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the riemannian manifold this equivalent energy function shows particularly that the system is essentially a dissipative system this result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in a guesmia a new approach of stabilization of nondissipative distributed systems siam j control optim 42 2003 24 52 where the system is claimed to be nondissipative

a generalized potential in the theory of the rabi and e x epsilon jahn teller systems ii for pt i see reik ibid vol 26 p 6549 1993 the eigenvalue problem for the rabi and e x epsilon jahn teller hamiltonians in bargmann s hilbert space is a system of two first order differential equations for the two component wavefunctions for which entire solutions are sought the concept of the generalized potential has been introduced in a previous paper together with a particular example here we treat a simpler potential n z which satisfies a second order ordinary differential equation closely related to the differential equation of the confluent heun functions the component wavefunctions are linear in the potential and its first derivative the coefficients of n z and dn z dz are functions of z and the physical parameters which are identical in all eigenstates the relation to the previous example is fully discussed

functional integral treatment of some quantum nondemolition systems in the scheme of a quantum nondemolition qnd measurement an observable is measured without perturbing its evolution in the context of studies of decoherence in quantum computing we examine the open quantum system of a two level atom or equivalently a spin 1 2 system in interaction with quantum reservoirs of either oscillators or spins under the qnd condition of the hamiltonian of the system commuting with the system reservoir interaction for completeness we also examine the well known non qnd spin bose problem for all these many body systems we use the methods of functional integration to work out the propagators the propagators for the qnd hamiltonians are shown to be analogous to the squeezing and rotation operators respectively for the two kinds of baths considered squeezing and rotation being both phase space area preserving canonical transformations this brings out an interesting connection between the energy preserving qnd hamiltonians and the homogeneous linear canonical transformations

differential equations for singular values of products of ginibre random matrices it was proved by akemann et al 2013 phys rev e 88 052118 that squared singular values of products of m complex ginibre random matrices form a determinantal point process whose correlation kernel is expressible in terms of meijer s g functions kuijlaars and zhang arxiv 1308 1003 recently showed that at the edge of the spectrum this correlation kernel has a remarkable scaling limit which can be understood as a generalization of the classical bessel kernel of random matrix theory in this paper we investigate the fredholm determinant of the operator with the kernel where j is a disjoint union of intervals and is the characteristic function of the set j this fredholm determinant is equal to the probability that j contains no particles of the limiting determinantal point process defined by the gap probability we derive the hamiltonian differential associated with the corresponding fredholm determinant and relate them with the monodromy preserving deformation equations of the jimbo miwa mori ueno and sato theory in the special case we give a formula for the gap probability in terms of a solution of a system of nonlinear ordinary differential equations

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

a solution of the bosonic and algebraic hamiltonians by using an aim we apply the notion of an asymptotic iteration method aim to determine the eigenvalues of the bosonic hamiltonians that include a wide class of quantum optical models we consider the solutions of the hamiltonians which are even the polynomials of the fourth order with respect to the boson operators we also demonstrate the applicability of the method for obtaining the eigenvalues of the simple lie algebraic structures the eigenvalues of the multi boson hamiltonians have been obtained by transforming them into the form of the single boson hamiltonian in the framework of the aim

full spectrum of the two photon and the two mode quantum rabi models this paper is concerned with the rigorous analytical determination of the spectrum of the two photon and the two mode quantum rabi models to reach this goal we exploit the hidden symmetries in these models by means of the unitary and similarity transformations in addition to the bargmann fock space description in each case the purely quantum mechanical problem of the rabi model studied is reduced to solutions for differential equations this eventually gives a third order differential equation for each of these models which is reduced to a second order differential equation by additional transformations the analytical expressions of the wave functions describing the energy levels are obtained in terms of the confluent hypergeometric functions

algebraic equations for the exceptional eigenspectrum of the generalized rabi model we obtain the exceptional part of the eigenspectrum of the generalized rabi model also known as the driven rabi model in terms of the roots of a set of algebraic equations this approach provides a product form for the wavefunction components and allows an explicit connection with recent results obtained for the wavefunction in terms of truncated confluent heun functions other approaches are also compared for particular parameter values the exceptional part of the eigenspectrum consists of doubly degenerate crossing points we give a proof for the number of roots of the constraint polynomials and discuss the number of crossing points

non commutative harmonic oscillators and fuchsian ordinary differential operators the spectral problem for non commutative harmonic oscillators is shown to be equivalent to solve fuchsian ordinary differential equations with four regular singular points in a complex domain

quasi exact solution of the rabi hamiltonian a method is suggested to obtain the quasi exact solution of the rabi hamiltonian it is conceptually simple and can be easily extended to other systems the analytical expressions are obtained for eigenstates and eigenvalues in terms of orthogonal polynomials it is also demonstrated that the rabi system in a particular case coincides with the quasi exactly solvable poschl teller potential

addendum to algebraic equations for the exceptional eigenspectrum of the generalized rabi model in our recent paper li and batchelor 2015 j phys a math theor 48 454005 we obtained exceptional points in the eigenspectrum of the generalized rabi model in terms of a set of algebraic equations we also gave a proof for the number of roots of the constraint polynomials defining these exceptional solutions as a function of the system parameters and discussed the number of crossing points in the eigenspectrum this approach however only covered a subset of all exceptional points in the eigenspectrum in this addendum we clarify the distinction between the exceptional parts of the eigenspectrum for this model and discuss the subset of exceptional points not determined in our paper

the canonical form of the jahn teller hamiltonian the jahn teller hamiltonian in its bargmann fock representation is transformed by the birkhoff method into a canonical form in which all regular singularities between zero and infinity have been removed the resulting equation is of the kummer type and identical to the previously obtained canonical form of the rabi hamiltonian szopa m and ceulemans a 1996 j math phys 37 5402 the isolated exact solutions of the and jahn teller hamiltonians are identified as special symmetries of the canonical form

two level system in a one mode quantum field numerical solution on the basis of the operator method accurate eigenvalues and eigenfunctions of a two level system interacting with a one mode quantum field are calculated numerically a special iteration procedure based on the operator method permits one to consider the solution within a wide range of the hamiltonian parameters and to find the uniformly approximating analytical formula for the eigenvalues characteristic features of the model are considered such as the level intersections the population of the field states and the chaotization in the system through the doubling of the frequencies

an inverse problem for the schrodinger equation with variable coefficients and lower order terms abstract we study the uniqueness and stability for an inverse problem of schrodinger equation with variable coefficients and lower order terms we obtain a carleman estimate by which we recover the potential from partial boundary term

observability inequalities for shallow shells we consider some observability inequalities from boundary for a general shallow shell with a middle surface of any shape at first an estimate is established by the geometric multiplier method in the case that no boundary conditions are imposed under some checkable geometric conditions then our results yield continuous observability estimates for two kinds of boundary conditions which have a physical meaning with an explicit observability time and hence by duality exact controllability results

construction of fixed length insertion deletion correcting runlength limited codes an algorithm is presented for the construction of fixed length insertion deletion correcting runlength limited rll codes in this construction binary d k constrained codewords are generated by codewords of a q ary lee metric based code it is shown that this new construction always yields better codes than known constructions the use of a q ary lee 1987 metric code q odd is based on the assumption that an error insertion deletion or peak shift has maximal size q 1 2 it is shown that a decoding algorithm for the lee metric code can be extended so that it can also be applied to insertion deletion correcting rll codes furthermore such an extended algorithm can also correct some error patterns containing errors of size more than q 1 2 as a consequence if s denotes the maximal size of an error then in some cases the alphabet size of the generating code can be s 1 instead of 2 spl middot s 1

the canonical form of the rabi hamiltonian the rabi hamiltonian describing the coupling of a two level system to a single quantized boson mode is studied in the bargmann fock representation the corresponding system of differential equations is transformed into a canonical form in which all regular singularities between zero and infinity have been removed the canonical or birkhoff transformed equations give rise to a two dimensional eigenvalue problem involving the energy and a transformational parameter which affects the coupling strength the known isolated exact solutions of the rabi hamiltonian are found to correspond to the uncoupled form of the canonical system

spin 1 2 system coupled to a single mode a resolvent expansion of the time dependent solution in order to describe the dynamics of a spin oscillator hamiltonian analytically a second order differential equation for the transition probability amplitude is established by means of a transcription into the bargmann space of entire functions the solution of the differential equation is given by a resolvent expansion whose zeroth order member is in general exact up to fourth order in time for various degenerate parameter values however it is exact for all times the laplace transform of any higher order member is given explicitly

uniform decay of the solution to a wave equation with memory conditions on the boundary abstract a wave equation with variable coefficients in principal part and memory conditions on the boundary is considered the riemannian geometry method is applied to prove the exponential decay of the energy provided the relaxation function also decays exponentially

berry phase of the tavis cummings model with three modes of oscillation in this paper we develop a general method to obtain the berry phase of time dependent hamiltonians with a linear structure given in terms of the su 1 1 and su 2 groups this method is based on the similarity transformations of the displacement operator performed to the generators of each group and let us diagonalize these hamiltonians then we introduce a trilinear form of the tavis cummings model to compute the su 1 1 and su 2 berry phases of this model

control and stabilization for the wave equation with variable coefficients in domains with moving boundary abstract the study of stabilization and control for pdes with variable coefficients involves higher level of complexity than the corresponding case of constant coefficients further compounding the complexity are the concern and effects of dynamic boundary motion the problem in its general form is extremely challenging to treat but under certain specific physical and geometric conditions such as the time likeness of the boundary and a limited speed of domain expansion energy decay estimates can be established and the exact controllability can also be obtained by control theoretic and riemannian geometric methods our approach here is based on the bochner technique of differential geometry in terms of riemannian metric and geometric multipliers by generalizing an energy identity method used earlier in bardos and chen 1981 concrete examples are also given to illustrate the geometric conditions and the theorems

the quantum rabi model and lie algebra representations of mathfrak s mathfrak l _ 2 the aim of the present paper is to understand the spectral problem of the quantum rabi model in terms of lie algebra representations of we define a second order element of the universal enveloping algebra of which through the image of a principal series representation of provides a picture equivalent to the quantum rabi model drawn by confluent heun differential equations by this description in particular we give a representation theoretic interpretation of the degenerate part of the spectrum i e judd s eigenstates of the rabi hamiltonian due to ku in 1985 which is a part of the exceptional spectrum parameterized by integers we also discuss the non degenerate part of the exceptional spectrum of the model in addition to the judd eigenstates from a viewpoint of infinite dimensional irreducible submodules or subquotients of the non unitary principal series such as holomorphic discrete series representations of

quasi exactly solvable potentials on the line and orthogonal polynomials in this paper we show that a quasi exactly solvable normalizable or periodic one dimensional hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three term recursion relation in particular we prove that normalizable exactly solvable one dimensional systems are characterized by the fact that their associated polynomials satisfy a two term recursion relation we study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one dimensional quasi exactly solvable hamiltonian showing in particular that its associated stieltjes measure is supported on a finite set from this we deduce that the corresponding moment problem is determined and that the kth moment grows like the kth power of a constant as k tends to infinity we also show that the moments satisfy a constant coefficient linear difference equation and that this property actually characterizes weakly orthogonal polynomial systems

dependences related to strict binary relations in this paper we study dependences on a free monoid x and their relations to strict binary relations on x as a consequence we characterize strict binary relations by the notion of dependence moreover some results concerning the relationship between dependences and codes are also obtained a visualization of this work is provided by the theory of graph

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

control and stabilization for the wave equation part iii domain with moving boundary we use energy invariants to study the growth and decay estimates for solutions of the wave equation in a domain with moving boundary sufficient conditions are formulated which insure the exact distributed parameter controllability of the wave equation

global existence and uniform decay of a damped klein gordon equation in a noncylindrical domain abstract in this paper we consider a damped klein gordon equation in a noncylindrical domain this work is devoted to proving the existence of global solutions and decay for the energy of solutions for a damped klein gordon equation in a noncylindrical domain

explicit observability inequalities for the wave equation with lower order terms by means of carleman inequalities in this paper by means of carleman estimates and the usual energy estimate we obtain directly two observability inequalities for the linear wave equation with time variant nonsmooth lower order terms we do not need any unique continuation property of the linear equation a priori since this is actually one of the by products of our analysis furthermore the constant in the observability inequality is estimated by an explicit function of the norm of the involved coefficients in the equation also we apply our observability estimates to exact controllability for wave equations

exact solution of non adiabatic model hamiltonians in solid state physics and optics simple non adiabatic model hamiltonians are treated using the realisation of the vibrational modes on bargmann s hilbert space of analytic functions in this formulation the schrodinger equation is a system of linear first order differential equations the energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions the solutions are given in terms of neumann series the recurrence relations for the expansion coefficients have a simple structure under particular conditions for the interaction constant they allow for terminating neumann series isolated exact solutions in the general case the conditions for the eigenvalues are transcendental equations involving a continued fraction the continued fraction can be approximated to any desired degree of accuracy in a rapidly convergent process based on worpitzky s theorem and its relation to conformal mapping the eigenvalues are calculated a physical interpretation of the solutions which makes use of intuitive arguments is also given

error detecting properties of languages the language property of error detection ensures that the communications medium cannot transform a word of the language to another word of the language in this paper we provide some insights on the notion of error detection from a language theoretic point of view we define certain error detecting properties of languages and codes including the notion of error detection with finite delay which is a natural extension of unique decodability with finite delay we obtain results about the error detecting capabilities of regular and other languages and of known classes of codes moreover we consider the problem of estimating the optimal redundancy of infinite languages with the property of detecting errors of the deletion type

boundary feedback stabilization of the transmission problem of naghdi s model we consider the stabilization of the transmission problem of naghdi s model by boundary feedbacks where the model has a middle surface of any shape the exponential decay rate for the problem is established under some checkable geometric conditions on the middle surface

structural analysis of error correcting codes for discrete channels that involve combinations of three basic error types a nonprobabilistic mathematical model is introduced of discrete channels that involve the error types substitution insertion and deletion the model is based on the novelty that errors can be expressed as strings over an alphabet of basic error symbols some general conditions on errors are defined which bound the error effects on messages obtaining thus the class of bounded error effects channels bee channels these channels can be used to model for instance scattered errors and bursts of errors of any combination of the three error types a general notion of error correcting code is defined and a characterization of the error correcting codes for a given bee channel is obtained then an algorithm is presented that tests for a given finite code not necessarily of fixed length and a given description of a bee channel whether the code is error correcting for the channel defined by the given description this result can be considered as an extension of the well known theorem of sardinas and patterson 1953 for testing the unique decodability of a given finite code in this sense it can be said that unique decodability is decidable also in the presence of errors