invariant subspaces and hyper reflexivity for free semigroup algebras in this paper we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras this enables us to prove that they are reflexive and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces such algebras are called hyper reflexive this property is very strong but it has been established in only a very few cases moreover the prototypes of this class of algebras are the natural candidate for a non commutative analytic toeplitz algebra on n variables the case we make for this analogy is very compelling in particular in this paper the key to the invariant subspace analysis is a good analogue of the beurling theorem for invariant subspaces of the unilateral shift this leads to a notion of inner outer factorization in these algebras in a sequel to this paper 13 we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in cn a free semigroup algebra is the weak operator topology closed algebra generated by a set s1 sn of isometries with pairwise orthogonal ranges these conditions are described algebraically by s i sj iji for 1 i j n f

approximation of quantum graph vertex couplings by scaled schrodinger operators on thin branched manifolds we discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds we show that if a neumann type laplacian on such manifolds is amended by suitable potentials the resulting schrodinger operators can approximate non trivial vertex couplings the latter include not only the couplings but also those with wavefunctions discontinuous at the vertex we work out the example of the symmetric couplings and make a conjecture that the same method can be applied to all couplings invariant with respect to the time reversal we conclude with a result that certain vertex couplings cannot be approximated by a pure laplacian

existence of the global solution for fractional logarithmic schrodinger equation abstract in this paper we consider the initial boundary value problem for a class of fractional logarithmic schrodinger equation by using the fractional logarithmic sobolev inequality and introducing a family of potential wells we give some properties of the family of potential wells and obtain existence of global solution

the global analysis on the spectral collocation method for time fractional schrodinger equation abstract in this paper a spectral collocation method is proposed and analyzed for solving the time fractional schrodinger equation the space derivative is discretized using the collocation method and the time fractional derivative using grunwald letnikov formulation the stability and convergence of the full discretization scheme are analyzed based on the z transform the global behavior of the finite difference spectral collocation method is derived numerical examples show a good agreement with the theoretical analysis

asymptotic spectrum for the dirichlet laplacian in thin deformed tubes with scaled geometry it is known that geometric properties do not play a role in the asymptotics of the eigenvalues of the laplacian restricted to thin dirichlet deformed tubular regions in this work it is shown that by properly scaling the geometric properties of the tubes a limit where the geometry is on equal footing with the deformation function is obtained such a result gives an indication of how subtle the confining limits can be

coupling in the singular limit of thin quantum waveguides we analyze the problem of approximating a smooth quantum waveguide with a quantum graph we consider a planar curve with compactly supported curvature and a strip of constant width around the curve we rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve consisting of one vertex and two infinite straight edges i e a broken line we discuss the convergence of the laplacian with dirichlet boundary conditions on the strip in a suitable sense and we obtain two possible limits the laplacian on the line with dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the laplacian on the line the first case generically occurs and corresponds to the decoupling of the two components of the limit curve while in the second case a coupling takes place we present also two families of curves which give rise to coupling

the evolution operator technique in solving the schrodinger equation and its application to disentangling exponential operators and solving the problem of a mass varying harmonic oscillator the authors present a method for finding the evolution operator for the schrodinger equation for the hamiltonian expressible as h t a1 t j4 a2 t j0 a3 t j where j j0 and j are the su 2 group generators such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary as a demonstration of our general approach they solved the problem of a harmonic oscillator with a varying mass

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

on the l convergence of a difference scheme for coupled nonlinear schrodinger equations in this article a finite difference scheme for coupled nonlinear schrodinger equations is studied the existence of the difference solution is proved by brouwer fixed point theorem with the aid of the fact that the difference solution satisfies two conservation laws the finite difference solution is proved to be bounded in the discrete l norm then the difference solution is shown to be unique and second order convergent in the discrete l norm finally a convergent iterative algorithm is presented

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

branched quantum wave guides with dirichlet boundary conditions the decoupling case we consider a family of open sets m which shrinks with respect to an appropriate parameter to a graph under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted dirichlet spectrum of m converges to the spectrum of the differential laplacian on the graph with dirichlet boundary conditions at the vertices i e a graph operator without coupling between different edges the smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood the lower bound is given by the first transversal mode of the edge neighbourhood we also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a laplacian acting on the edge with an additional potential coming from the curvature

recursive boson system in the cuntz algebra o bosons and fermions are often written by elements of other algebras abe private communication gave a realization of bosons by formal infinite sums of the canonical generators of the cuntz algebra o we show that such formal infinite sum always makes sense on a certain dense subspace of any permutative representation of o in this meaning we can regard as if the algebra b of bosons was a unital subalgebra of o on a given permutative representation according to this relation we compute branching laws arising from restrictions of representations of o on b for example it is shown that the fock representation of b is given as the restriction of the standard representation of o on b

convergence of spectra of uniformly fattened open book structures abstract we consider a stratified compact surface m in r 3 which we call an open book structure whose singular points form simple closed smooth curves and its neighborhood m which we call a fattened open book structure assuming a transversality condition at the singularities and absence of zero dimensional strata i e corners we show that the discrete spectrum of the neumann laplacian in m converges when 0 to the spectrum of a differential operator on m similar results have been obtained before for the case of fattened graphs i e m being one dimensional in the case of a smooth surfaces m the problem has been studied well however having singularities along strata of lower dimensions significantly complicates considerations as in the quantum graph case such considerations are triggered by various applications

dynamical evolution of the projector induced from the generalized squeezed state the quantum and classical time evolution of the approximate simultaneous observable on phase space it is merely called a projector induced from the generalized squeezed state is considered herein the purpose of this article is to give a trace norm estimate of the difference between these two types of time evolutions of the projectors as a consequence of this result it is shown that this trace norm estimate converges to 0 as 0

generalized mehler formula for time dependent non selfadjoint quadratic operators and propagation of singularities we study evolution equations associated to time dependent dissipative non selfadjoint quadratic operators we prove that the solution operators to these non autonomous evolution equations are given by fourier integral operators whose kernels are gaussian tempered distributions associated to non negative complex symplectic linear transformations and we derive a generalized mehler formula for their weyl symbols some applications to the study of the propagation of gabor singularities characterizing the lack of schwartz regularity for the solutions to non autonomous quadratic evolution equations are given

on δ like potential scattering on star graphs we discuss the potential scattering on the noncompact star graph the schrodinger operator with the short range potential localized in a neighborhood of the graph vertex is considered we study the asymptotic behavior of the corresponding scattering matrix in the zero range limit it has been known for a long time that in dimension 1 there is no non trivial hamiltonian with the distributional potential i e the potential acts as a totally reflecting wall several authors have in recent years studied the scattering properties of the regularizing potentials e 2q x e approximating the first derivative of the dirac delta function a non zero transmission through the regularized potential has been shown to exist as e 0 we extend these results to star graphs with the point interaction which is an analog of the potential on the line we prove that generically such a potential on the graph is opaque we also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs this set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary sturm liouville problem associated with q on the graph

branching laws for endomorphisms of fermions and the cuntz algebra o2 fermions are expressible in terms of polynomials in the canonical generators and their conjugates of the cuntz algebra o2 a canonical construction of such fermions is called a recursive fermion system in this paper various unital endomorphisms of the algebra of fermions are explicitly constructed by using the recursive fermion system we investigate how such endomorphisms transform the fock representation the infinite wedge representation and their duals

interval maps from cuntz krieger algebras abstract markov interval maps f naturally produce transition aperiodic 0 1 matrices with extra features we characterize the 0 1 matrices that can be realized as markov transition matrices of interval maps and parametrize the orbit representations yielded in correia ramos et al 2008 2 of the cuntz krieger algebra o a obtained from interval maps with the same matrix a

extensions of representations of the car algebra to the cuntz algebra o2 the fock and the infinite wedge fermions are expressed by polynomials of canonical generators of the cuntz algebra o2 and they generate the u 1 fixed point subalgebra a o2u 1 of o2 by the canonical gauge action we extend the fock and the infinite wedge representations of a to permutative representations of o2 by these extensions the boson fermion correspondence is rewritten by canonical generators of o2

quantum graphs as holonomic constraints we consider the dynamics on a quantum graph as the limit of the dynamics generated by a one particle hamiltonian in r2 with a potential having a deep strict minimum on the graph when the width of the well shrinks to zero for a generic graph we prove convergence outside the vertices to the free dynamics on the edges for a simple model of a graph with two edges and one vertex we prove convergence of the dynamics to the one generated by the laplacian with dirichlet boundary conditions in the vertex

approximate inverse circulant plus diagonal preconditioners for toeplitz plus diagonal matrices we consider the solutions of hermitian positive definite toeplitz plus diagonal systems t d x b where t is a toeplitz matrix and d is diagonal and positive however unlike the case of toeplitz systems no fast direct solvers have been developed for solving them in this paper we employ the preconditioned conjugate gradient method with approximate inverse circulant plus diagonal preconditioners to solving such systems the proposed preconditioner can be constructed and implemented efficiently using fast fourier transforms we show that if the entries of t decay away exponentially from the main diagonals the preconditioned conjugate gradient method applied to the preconditioned system converges very quickly numerical examples including spatial regularization for image deconvolution application are given to illustrate the effectiveness of the proposed preconditioner

unitary isomorphism of fock spaces of bosons and fermions arising from a representation of the cuntz algebra o2 bosons and fermions are described by using canonical generators of cuntz algebras on any permutative representation according to branching laws associated with these descriptions a certain representation of the cuntz algebra o2 induces fock representations hb and hf of bosons and fermions simultaneously from this a unitary operator u from hb to hf is obtained we show the explicit formula of the action of u on the standard basis of hb it is shown that u preserves the particle number of hb and hf

crank nicolson difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative in this paper the crank nicolson cn difference scheme for the coupled nonlinear schrodinger equations with the riesz space fractional derivative is studied the existence of this difference solution is proved by the brouwer fixed point theorem the stability and convergence of the cn scheme are discussed in the l 2 norm when the fractional order is two all those results are in accord with the difference scheme developed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also presented

analysis of time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise abstract in this paper we establish the existence and uniqueness of the mild solution and the holder continuity for the time fractional and space nonlocal stochastic nonlinear schrodinger equation driven by multiplicative white noise a full discrete scheme with spectral galerkin method in space and exponential method in time is proposed it is rigorously proved that the numerical scheme admits an unique solution finally we present the optimal error estimates of the proposed scheme based on the regularity analysis

on partially inexact hss iteration methods for the complex symmetric linear systems in space fractional cnls equations the space fractional coupled nonlinear schrdinger cnls equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix and a symmetric positive definite diagonal plus toeplitz matrix the hermitian and skew hermitian splitting hss method and the partially inexact hss pihss method are employed to solve the discretized linear system in the inner iteration processes of the hss method we only need to solve the linear sub systems associated with the hermitian part inexactly by the conjugate gradient cg method resulting in pihss iteration method theoretical analyses show that both hss and pihss methods are unconditionally convergent numerical examples are given to demonstrate the effectiveness of the hss iteration and the pihss iteration

preconditioned modified hermitian and skew hermitian splitting iteration methods for fractional nonlinear schrodinger equations abstract a variant of preconditioned modified hermitian and skew hermitian splitting pmhss iteration method is proposed for a class of toeplitz like complex linear equations arising from the space fractional coupled nonlinear schrodinger equations theoretical analysis show that the new method is convergent unconditionally moreover the choice of the theoretical optimal parameter is further studied in detail which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the toeplitz matrix and the extreme absolute values of the diagonal matrix in addition the splitting can be used as a preconditioner for krylov subspaces method numerical experiments show that the proposed method is efficient and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem

a linearly implicit conservative difference scheme for the space fractional coupled nonlinear schrodinger equations abstract in this paper a linearly implicit conservative difference scheme for the coupled nonlinear schrodinger equations with space fractional derivative is proposed this scheme conserves the mass and energy in the discrete level and only needs to solve a linear system at each step the existence and uniqueness of the difference solution are proved the stability and convergence of the scheme are discussed and it is shown to be convergent of order o 2 h 2 in the discrete l 2 norm with the time step and mesh size h when the fractional order is two all those results are in accord with the difference scheme proposed for the classical non fractional coupled nonlinear schrodinger equations some numerical examples are also reported

semiclassical sum rules and generalized coherent states various semiclassical sum rules are obtained for matrix elements of smooth observables using semiclassical estimates of time evolved coherent states together with stationary phase theorems

universal fermionization of bosons on permutative representations of the cuntz algebra o2 bosons and fermions are described by using canonical generators of cuntz algebras on any permutative representation we show a fermionization of bosons which universally holds on any permutative representation of the cuntz algebra o2 as examples we show fermionizations on the fock space and the infinite wedge

hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the jordan schwinger map the aim of this paper is to show that there is a hopf structure of the parabosonic and parafermionic algebras and this hopf structure can generate the well known hopf algebraic structure of the lie algebras through a realization of lie algebras using the parabosonic and parafermionic extension of the jordan schwinger map the differences between the hopf algebraic and the graded hopf superalgebraic structure on the parabosonic algebra are discussed

some extensions of semiclassical limit ℏ 0 for wigner functions on phase space the semiclassical asymptotics of the unitary quantum propagator applied to the wigner functions which are associated with the initial quantum state belonging to schwartz class are considered herein the limit 0 of the semiclassical asymptotics of these wigner functions is investigated as a consequence of this result there exists a simple semiclassical asymptotic which obeys the unitary conservation law over any finite time interval 0 t

asymptotic behavior of stochastic fractional power dissipative equations on rn abstract we study a reaction fractional diffusion equation with additive noise on the entire space r n with particular interest in the asymptotic behavior of solutions we first transform the equation into a random equation whose solutions generate a random dynamical system a priori estimates for solutions are derived when the nonlinearity satisfies certain growth conditions using estimates for far field values of solutions and a cut off technique asymptotic compactness is proved thus the existence of a random attractor in l 2 r n is established

features of moyal trajectories we study the moyal evolution of the canonical position and momentum variables we compare it with the classical evolution and show that contrary to what is commonly found in the literature the two dynamics do not coincide we prove that this divergence is quite general by studying hamiltonians of the form p2 2m v q several alternative formulations of moyal dynamics are then suggested we introduce the concept of star function and use it to reformulate the moyal equations in terms of a system of ordinary differential equations on the noncommutative moyal plane we then use this formulation to study the semiclassical expansion of moyal trajectories which is cast in terms of a order by order in recursive hierarchy of i first order partial differential equations as well as ii systems of first order ordinary differential equations the latter formulation is derived independently for analytic hamiltonians as well as for the more general case of locally integrable ones we present various examples i

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

highly accurate numerical schemes for multi dimensional space variable order fractional schrdinger equations as a natural generalization of the fractional schrdinger equation the variable order fractional schrdinger equation has been exploited to study fractional quantum phenomena in this paper we develop an exponentially accurate jacobigausslobatto collocation jgl c method to solve the variable order fractional schrdinger equations in one dimension 1d and two dimensions 2d in this method the aforementioned problem is reduced to a system of ordinary differential equations odes in the time variable as a result we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations one based on the implicit rungekutta irk method of fourth order and the other based on jacobigaussradau collocation jgr c method the validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1d and 2d the convergence of the methods is graphically analyzed the results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable order nonlinear partial differential equations

cuntz krieger algebras representations from orbits of interval maps abstract let f be an expansive markov interval map with finite transition matrix a f then for every point we yield an irreducible representation of the cuntz krieger algebra o a f and show that two such representations are unitarily equivalent if and only if the points belong to the same generalized orbit the restriction of each representation to the gauge part of o a f is decomposed into irreducible representations according to the decomposition of the orbit

semiclassical analysis of constrained quantum systems we study the dynamics of a quantum particle in constrained by a strong potential force to stay within a distance of order in suitable units from a smooth n dimensional submanifold m we prove that in the semiclassical limit the evolution of the wavefunction is approximated in norm up to terms of order 1 2 by the evolution of a semiclassical wave packet centred on the trajectory of the corresponding classical constrained system

a fast linearized conservative finite element method for the strongly coupled nonlinear fractional schrodinger equations abstract in this paper a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional schrodinger equations we prove that the scheme preserves both the mass and energy which are defined by virtue of some recursion relationships using the sobolev inequalities and then employing the mathematical induction the discrete scheme is proved to be unconditionally convergent in the sense of l 2 norm and h 2 norm which means that there are no any constraints on the grid ratios then the prior bound of the discrete solution in l 2 norm and l norm are also obtained moreover we propose an iterative algorithm by which the coefficient matrix is independent of the time level and thus it leads to toeplitz like linear systems that can be efficiently solved by krylov subspace solvers with circulant preconditioners this method can reduce the memory requirement of the proposed linearized finite element scheme from o m 2 to o m and the computational complexity from o m 3 to o m log m in each iterative step where m is the number of grid nodes finally numerical results are carried out to verify the correction of the theoretical analysis simulate the collision of two solitary waves and show the utility of the fast numerical solution techniques

remarks on the inhomogeneous fractional nonlinear schrodinger equation using a sharp gagliardo nirenberg type inequality well posedness issues of the initial value problem for a fractional inhomogeneous schrodinger equation are investigated

orbit representations from matrices abstract each markov interval map f naturally produces a transition 0 1 matrix of interval type in every row the entries equal to 1 should be consecutive we show that any 0 1 matrix a can be transformed into an interval type matrix a i by a careful use of the state splitting we then prove that a i can be realized as a transition matrix of an interval map f a i a i arising from the perron frobenius eigenvalue a i and eigenvector of a i finally we construct orbit representations associated with a from those of a i arising from the dynamical system 0 1 f a i a i

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

maximum norm error analysis of a difference scheme for the space fractional cnls the difference method for the space fractional coupled nonlinear schrodinger equations cnls is studied the fractional centered difference is used to approximate the space fractional laplacian this scheme conserves the discrete mass and energy due to the nonlocal nature of fractional laplacian in the classic sobolev space it is hard to obtain the error estimation in l to overcome this difficulty the fractional sobolev space h 2 and a fractional norm equivalence in h 2 are introduced then the convergence of order o h 2 2 in l is proved by fractional sobolev inequality where h is the mesh size and is the time step numerical examples are given to illustrate the theoretical results at last

on representations of cuntz algebras and pure isometries abstract we prove that every representation of the cuntz algebra o n on a separable hilbert space h arises from a pure isometry v whose wandering space h im v has dimension n we identify the permutative representations in this construction

the global solution for a class of systems of fractional nonlinear schrodinger equations with periodic boundary condition in this paper we consider a class of systems of fractional nonlinear schrodinger equations we prove the existence and uniqueness of the global solution to the periodic boundary value problem by using the faedo galerkin method

on hss like iteration method for the space fractional coupled nonlinear schrodinger equations the implicit conservative difference scheme with the fractional centered difference formula which is unconditionally stable is employed to discretize the space fractional coupled nonlinear schrodinger equations the coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric diagonal plus toeplitz matrix in this paper the hss like iteration method is proposed to solve the discretized linear system theoretical analyses show that the hss like iteration method is unconditionally convergent numerical examples are presented to illustrate the effectiveness of the hss like iteration method

representations of thompson groups from cuntz algebras abstract we associate a representation of the thompson group v and thus of f and t by considering the restriction to every representation of the cuntz algebra o 2 the well developed theory of representations of the cuntz algebras leads us to exhibit an uncountable family of unitary representations of v which are pairwise non equivalent and two others for f and t

a circulant preconditioner for fractional diffusion equations the implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable is employed to discretize fractional diffusion equations the resulting systems are toeplitz like and then the fast fourier transform can be used to reduce the computational cost of the matrix vector multiplication the preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant hence the convergence rate of the proposed iterative algorithm is superlinear numerical experiments are carried out to demonstrate that our circulant preconditioner works very well even though for cases of variable diffusion coefficients

existence of the global smooth solution to the period boundary value problem of fractional nonlinear schrodinger equation in this paper we prove the existence and uniqueness of the global smooth solution to the period boundary value problem of fractional nonlinear schrodinger equation by using energy method

scattering solutions in networks of thin fibers small diameter asymptotics small diameter asymptotics is obtained for scattering solutions in a network of thin fibers the asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph we calculate the lagrangian gluing conditions at vertices v in gamma for the problems on the limiting graph if the frequency of the incident wave is above the bottom of the absolutely continuous spectrum the gluing conditions are formulated in terms of the scattering data for each individual junction of the network

generalized fractional schrodinger equation with space time fractional derivatives in this paper the generalized fractional schrodinger equation with space and time fractional derivatives is constructed the equation is solved for free particle and for a square potential well by the method of integral transforms fourier transform and laplace transform and the solution can be expressed in terms of mittag leffler function the green function for free particle is also presented in this paper finally we discuss the relationship between the cases of the generalized fractional schrodinger equation and the ones in standard quantum

some physical applications of fractional schrodinger equation the fractional schrodinger equation is solved for a free particle and for an infinite square potential well the fundamental solution of the cauchy problem for a free particle the energy levels and the normalized wave functions of a particle in a potential well are obtained in the barrier penetration problem the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined in the quantum scattering problem according to the fractional schrodinger equation the green s function of the lippmann schwinger integral equation is given

existence and upper semicontinuity of random attractors for non autonomous fractional stochastic ginzburg landau equations this paper deals with the dynamical behavior of solutions for nonautonomous stochastic fractional ginzburg landau equations driven by multiplicative noise with 0 1 we first transform the stochastic equation into a random equation the solutions of which generate a random dynamical system consequently we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain finally we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero

bound state for the fractional schrodinger equation with unbounded potential this paper considers the fractional schrodinger equation with unbounded potential we prove the existence of ground state by lagrange multiplier method the standing wave with prescribed frequency is also obtained by nehari s manifold approach in the last part we obtain that the standing wave is bound state

two body scattering on a graph and application to simple nanoelectronic devices the two body system on a graph with one junction is considered the effective three body scattering problem turns out to be exactly solvable for pointwise interactions additional degrees of freedom corresponding to a dynamics of some structure e g an atomic cluster located in the junction point of common contact of three thin electrodes are considered these degrees of freedom bring effective energy dependent interaction into the effective schrodinger equation in the scattering channel the wave function of the system is constructed in the explicit form using the extension theory methods the obtained results are applied to the qualitative description of a simple three electrode nanoelectronic device the perturbation theory approach based on the analysis of the liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

recursive fermion system in cuntz algebra i embeddings of the car canonical anticommutation relations algebra of fermions into the cuntz algebra 2 or 2 d more generally are presented by using recursive constructions as a typical example an embedding of car onto the u 1 invariant subalgebra of 2 is constructed explicitly generalizing this construction to the case of 2 p an embedding of car onto the u 1 invariant subalgebra of 2 p is obtained restricting a permutation representation of the cuntz algebra we obtain the fock representation of car we apply the results to embed the algebra of parafermions of order p into 2 p according to green s ansatz

dynamical behavior of non autonomous fractional stochastic reaction diffusion equations abstract in this paper we investigate the asymptotic behavior of solutions for the non autonomous fractional stochastic reaction diffusion equations driven by multiplicative noise with s 0 1 we prove the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain u by the compactness of sobolev embedding h s u l 2 u which is different from the previous work 46

the squeezed state approach of the semiclassical limit of the time dependent schrodinger equation it is by now well recognized how the quantum time evolution for time dependent hamiltonians that are quadratic both in position and momentum variables generates squeezing on the other hand squeezed states and their high n generalizations are known to provide a simple classical mechanical description of general quantum time evolution in this paper these ideas are combined to show the relevance of generalized squeezed states in the semiclassical limit of the evolution of time dependent wave packets for general time dependent hamiltonians including the multidimensional case

on the spectrum and weakly effective operator for dirichlet laplacian in thin deformed tubes abstract we study the laplacian in deformed thin bounded or unbounded tubes in r 3 i e tubular regions along a curve r s whose cross sections are multiplied by an appropriate deformation function h s 0 one of the main requirements on h s is that it has a single point of global maximum we find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero it is shown that such behaviors are not influenced by some geometric features of the tube such as curvature torsion and twisting and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator

diffusion processes in thin tubes and their limits on graphs the present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary respectively reflecting at the boundary and corresponding processes in the limit where the thin tubular domains are shrinking to graphs the methods we use are probabilistic ones for shrinking we use big potentials respectively reflection on the boundary of tubes we show that there exists a unique limit process and we characterize the limit process by a second order differential generator acting on functions defined on the limit graph with kirchhoff boundary conditions at the vertices

graph c algebras branching systems and the perron frobenius operator in this paper we show how to produce a large number of representations of a graph c algebra in the space of the bounded linear operators in l2 x these representations are very concrete and in the case of graphs that satisfy condition l we use our techniques to realize the associated graph c algebra as a subalgebra of the bounded operators in l2 r we also show how to describe the perron frobenius operator of ergodic theory in terms of the representations we associate to a graph

variational problems on multiply connected thin strips i basic estimates and convergence of the laplacian spectrum let m be a planar embedded graph whose arcs meet transversally at the vertices let be a strip shaped domain around m of width except in a neighborhood of the singular points assume that the boundary of is smooth we define comparison operators between functions on and on m and we derive energy estimates for the compared functions we define a laplace operator on m which is in a certain sense the limit of the laplace operator on with neumann boundary conditions in particular we show that the p th eigenvalue of the laplacian on converges to the p th eigenvalue of the laplacian on m as tends to 0 a similar result holds for the magnetic schrodinger operator

squeezing by tuning the oscillator frequency the author examines the possibility of generating squeezed states out of coherent states by external changes of the oscillator frequency using the evolution operator method developed by cheng and fung he investigates the evolution of a coherent state of a time dependent harmonic oscillator as well as its squeezing and coherence property two specific models of frequency variation are explored namely one whose frequency is exponentially decreasing in time and the other with a periodic frequency his analyses indicates that the wavefunction of the time dependent oscillator starts as a coherent state at t o and evolves as a squeezed state at a later time it is also shown that squeezing cannot be generated by an adiabatic change of the oscillator frequency because the variances of q and p turn out to be adiabatic invariants whereas a sudden change can produce squeezing