fredholm integral equation with potential kernel and its structure resolvent the purpose of this paper is to establish and solve the fredholm integral equation of the second kind with potential kernel from the semi symmetric hertz problem for two different elastic materials in three dimensions the kernel of the problem is represented in the non homogeneous wave equation also the structure resolvent of the problem is obtained

goursat functions for an infinite plate with a generalized curvilinear hole in ζ plane we used a rational mapping function with complex constants to derive exact and close expressions of goursat functions for the first and second fundamental problems plane elasticity problems of an infinite plate weakened by a hole having arbitrary shape notable the area outside the hole is conformally mapped outside a unit circle by means of the rational mapping the complex variable method has been applied in the procedure and it transforms the fundamental problems to the integro differential equations that can be solved by using cauchy type integrals the hole takes different famous shapes which can be found throughout the nature like tunnels caves excavation in soil or rock etc many previous works have been considered as special cases of this work also many new cases can be derived from the problem

toeplitz matrix method and nonlinear integral equation of hammerstein type the existence and uniqueness solution of the nonlinear integral equation of hammerstein type with discontinuous kernel are discussed the normality and continuity of the integral operator are proved toeplitz matrix method is used as a numerical method to obtain a nonlinear system of algebraic equations also many important theorems related to the existence and uniqueness of the produced algebraic system are derived finally numerical examples when the kernel takes a logarithmic and carleman forms are discussed and the estimate error in each case is calculated

on the stability of the classical collocation method for cauchy type singular integral equations abstract the classical collocation method for cauchy type singular integral equations of the second kind on an interval using gaussian quadrature rules with respect to jacobi nodes is investigated in general the nodes and the weights of the quadrature rules can only be evaluated approximately in the present paper the question is answered how precise our knowledge about them must be in order to ensure that the corresponding algebraic system of equations is solvable yet and its solution is close to the solution of the exact system

the perturbed galerkin method for cauchy singular integral equations with constant coefficients we have considered the numerical solution of cauchy singular integral equations with constant coefficients via a polynomial based galerkin method l 2 and uniform convergence results are proved under holder continuity conditions on the kernel and right hand side and taking into account the quadrature errors produced when inner products are replaced by numerical quadratures these results generalize those of miel who considered only the case of equations of the first kind with index 0

spectral properties of schrodinger operators with irregular magnetic potentials for a spin 12 particle the two dimensional schrodinger operatorh a for a spin 12 particle is considered the magnetic fieldbgenerated byadoes not grow in some directions and stabilizes to a positively homogeneous function it is shown that the spectrum h a consists of disc h a and 0 the latter being an isolated eigenvalue of infinite multiplicity the former accumulating to only the principal term of the asymptotics of disc h a and of h a v wherebandvdo not grow in some directions is computed

compactness of the solution operator to in weighted l2 spaces abstract in this paper we discuss compactness of the canonical solution operator to on weigthed l 2 spaces on c n for this purpose we apply ideas which were used for the witten laplacian in the real case and various methods of spectral theory of these operators we also point out connections to the theory of dirac and pauli operators

a new numerical approximation for volterra integral equations combining two quadrature rules abstract this paper describes a numerical approximation to the solution of volterra integral equations of the second kind this algorithm combines trapezoidal and simpson rules we prove the convergence of the method numerical examples are provided to illustrate the accuracy of the method

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

the eigenvalue distribution of degenerate dirichlet forms we obtain a positive lower bound to the spectrum of certain second order elliptic operators h on l2 where rn has a lipschitz boundary and the coefficients of h become singular as one approaches the boundary we also find a general formula for the order of the asymptotic eigenvalue distribution of h in some situations where the classical limit formula of weyl courant titchmarsh and others is not applicable

nonclassical eigenvalue asymptotics abstract the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 and for x y all of which are non weyl because of infinite phase space volumes are computed along the way a general inequality on quantum partition functions computed in a kind of born oppenheimer approximation is proved

maximal generalized solution of eikonal equation abstract we study the dirichlet problem for the eikonal equation 1 2 u x 2 a x 0 in u x x on without continuity assumptions on the map a we find a class of maps a contained in the space l for which the problem admits a maximal generalized solution providing a generalization of the notion of viscosity solution

on the spectrum of the dirichlet laplacian for horn shaped regions in rn with infinite volume abstract an inequality for trace et d is proven where d is the dirichlet laplacian for horn shaped regions d in rn the results of rozenbljum and simon for the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 0 are easily recovered an example of a horn shaped region in r2 where that asymptotics is exponential is given

invariant formulation for the minimum induced drag conditions of nonplanar wing systems under the hypotheses of linear potential flow and rigid wake aligned with the freestream a configuration invariant analytical formulation for the induced drag minimization of single wing nonplanar systems is presented following a variational approach the resulting euler lagrange integral equation in the unknown circulation distribution is obtained the kernel presents a singularity of the first order and an efficient computational method ideal for the early conceptual phases of the design is proposed munk s theorem on the normalwash and its relation with the geometry of the wing under optimal conditions is naturally obtained with the present method moreover munk s constant of proportionality not provided in his original work is demonstrated to be the ratio between the freestream velocity and the optimal aerodynamic efficiency the augmented munk s minimum induced drag theorem is then formulated additional induced drag theorems are demonstrated following the derivations of this invariant proced

euclidean limit of l2 spectral properties of the pauli hamiltonians on constant curvature riemann surfaces we realize the pauli hamiltonians with constant magnetic field 0 on a simply connected riemann surface m of constant scalar curvature as second order differential operators acting on differential 1 forms of m we also study the asymptotic behaviour of some aspects of their l2 spectral properties when the euclidean limit is taken more exactly we show that the l2 eigenprojector kernels on the plane i e 0 corresponding to the landau levels 8 l l 0 1 can be recovered from the l2 eigenprojector kernels of of the curved riemann surfaces m 0 in the limit

trace properties of the dirichlet laplacian on obtient des bornes superieures et inferieures pour tr exp ht en termes de la fonction r x

nonclassical eigenvalue asymptotics for operators of schrodinger type on considere des operateurs de la forme a v x sur r n ou la metrique ij x 0 et le potentiel v x 0 on etudie la fonction de denombrement des valeurs propres par un calcul pseudo differentiel avec des symboles a valeur d operateurs

a simple separable hamiltonian having bound states in the continuum the author discusses a simple separable two freedom hamiltonian h describing the motion of a point particle in a plane region v 0 between two infinite parallel hard walls v infinity and with a rectangular finite potential well v v0 the quantal spectrum of h is discrete below the continuum limit ionisation energy different from the classical escape energy e 0 by the zero point energy of the transverse motion and has infinitely many bound states embedded in the continuum the author shows that this spectrum is fragile against perturbations of h if the hard walls are made diverging the bound states in the continuum disappear while if they are converging the continuum disappears and the spectrum is pure point

spectral properties of the canonical solution operator to we study boundedness compactness and schatten class membership of the canonical solution operator to restricted to 0 1 forms with holomorphic coefficients on l2 d where is a measure with the property that the monomials form an orthogonal family in l2 d the characterizations are formulated in terms of moment properties of our results generalize the results of the first author to several variables contain some known results for several variables and also cover new ground

on the numerical solution of cauchy type singular integral equations by the collocation method the collocation method for the numerical solution of fredholm integral equations of the second kind is applied properly modified to the numerical solution of cauchy type singular integral equations of the first or the second kind but with constant coefficients this direct method of numerical solution of cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the fredholm integral equation of the second kind equivalent to the cauchy type singular integral equation as well as with another method based also on the regularization procedure for the numerical solution of the same class of equations finally the convergence of the method is discussed

numerical solution of nonlinear volterra fredholm hammerstein integral equations via tau collocation method with convergence analysis in this paper we consider the nonlinear volterra fredholm hammerstein integral equations the approximate solution for the nonlinear volterra fredholm hammerstein integral equations is obtained by using the tau collocation method to do this the nonlinear volterra fredholm hammerstein integral equations is transformed into a system of nonlinear algebraic equations in matrix form thus by solving this system unknown coefficients are obtained the spectral rate of convergence for the proposed method is established in the l 2 norm the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the tau collocation method is of high accuracy more convenient and efficient for solving nonlinear volterra fredholm hammerstein integral equations

infiniteness of zero modes for the pauli operator with singular magnetic field abstract we establish that the pauli operator describing a spin 1 2 two dimensional quantum system with a singular magnetic field has under certain conditions an infinite dimensional space of zero modes possibly both spin up and spin down moreover there is a spectral gap separating the zero eigenvalue from the rest of the spectrum in particular infiniteness takes place if the field has infinite flux which settles this previously unknown case of aharonov casher theorem

a necessary and sufficient condition for strong convergence in sobolev spaces abstract we present an improved version of a necessary and sufficient condition for strong convergence in the sobolev space w 1 p and provide a nontrivial application to a class of fully nonlinear partial differential equations involving the range of the map w 1 p u u p l 1

dirichlet neumann bracketing for horn shaped regions we use dirichlet neumann bracketing to obtain sharp upper and lower bounds for the spectral counting function of the dirichlet laplacian for a horn shaped region in rm the first and second term and an estimate for the remainder in the asymptotic expansion of the spectral counting function are obtained for a region in r2 given by x1 x2 x1er x2e x1 x2 1 2 12 212

integral equation of mixed type and integrals of orthogonal polynomials a method is used to obtain the general solution of the fredholm volterra integral equation of the first kind fviefk in the space l2 0 a c 0 t 0 x a 0 t t the kernel of the fredholm integal term is considered in a generalized discontinuous form which belongs to c 0 a 0 a while the kernel of volterra integral term is a positive continuous function in the class c 0 t many interesting cases of integrals of orthogonal polynomials and spectral relationships are obtained and established from this work

on the number of negative eigenvalues for a schrodinger operator with magnetic field we consider the schrodinger operator with magnetic field h sum limits j 1 n left frac 1 i frac partial partial x j a j right 2 vin mathbb r n under certain conditions on the magnetic fieldb curla we generalize the fefferman phong estimates bull a m s 9 129 206 1983 on the number of negative eigenvalues for v to the operatorh upper and lower bounds are established our estimates incorporate the contribution from the magnetic field the conditions onb in particular are satisfied if the magnetic potentialsaj x are polynomials

eigenvalue asymptotics of the neumann laplacian of regions and manifolds with cusps abstract we study the eigenvalue asymptotics of a neumann laplacian n in unbounded regions of r2 with cusps at infinity a typical example is x y r 2 x 1 y x 2 and prove that n e n n e h v e 2 vol where hv is the canonical one dimensional schrodinger operator associated to the problem we establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a dirichlet laplacian d for a class of cusp type regions of infinite volume

the convergence of a collocation method for a class of cauchy singular integral equations in several recent papers we have studied the numerical solution of cauchy singular integral equations of the first kind by a polynomial collocation method 4 61 this technique has proved to be quite efftcient for many problems even when the kernel but not the right hand side is discontinuous 4 61 the l convergence and stability was established in s and 6 in 121 ioakimidis extended this method to solve cauchy singular integral equations of the second kind with constant coefficients although some discussion of convergence was given it appears at least to this author that the proof is incomplete and the conditions on the kernel and the right hand side are somewhat more restrictive than necessary in this paper we will show that the convergence analysis given in 5 6 can be generalized to establish the l convergence of collocation under rather mild restrictions on the data although a variety of numerical techniques such as galerkin s method and direct quadrature have been used extensively to solve cauchy singular equations they seem to be generally suited for problems with smooth kernels for problems with discontinuous kernels collocation methods appear to be a reasonable compromise between time consuming galerkin methods 2 and direct quadrature methods which require the evaluation of the kernel at pairs of points where the kernel may become unbounded 3 131 it is well known that for fredholm equations collocation is related to product integration quadrature methods 11 since such methods seem not to have been developed for the solution of cauchy singular equations collocation appears to be a good way of dealing with fairly general integral equations of this type 4 71

generalizations of the classical weyl and colin de verdiere s formulas and the orbit method the classical weyl formula expresses the leading term of the asymptotics of the counting function n h of the spectrum of a self adjoint operator h in an invariant form one can hear the volume of the subset of the cotangent bundle where the symbol of the operator h is less than in particular it is applicable to schrodinger operators with electric potentials growing at infinity the weyl formula is formulated in an invariant form however it gives for magnetic schrodinger operators with magnetic tensors growing at infinity for these operators colin de verdiere s formula is known but the form of the latter is not invariant in this article we suggest an invariant generalization of both weyl s and colin de verdiere s formulas for wide classes of schrodinger operators with polynomial electric and magnetic fields the construction is based on the orbit method due to kirillov and it allows one to hear the geometry of coadjoint orbits

convergence analysis of hybrid functions method for two dimensional nonlinear volterra fredholm integral equations abstract an interesting and computationally efficient scheme is proposed for solving a class of two dimensional nonlinear volterra fredholm integral equations the collocation method in conjunction with two dimensional hybrid functions is utilized to convert these types of equations to a nonlinear system of equations that can be solved numerically by the newton s method the major advantages of this method are simplicity of performing and favorable convergence rate for an approximate solution the convergence analysis of the hybrid functions and the error bound of the presented method are discussed in detail moreover some numerical experiments are provided to confirm our theoretical analysis

refined asymptotics for eigenvalues on domains of infinite measure in this work we study the asymptotic distribution of eigenvalues in one dimensional open sets the method of proof is rather elementary based on the dirichlet lattice points problem which enable us to consider sets with infinite measure also we derive some estimates for the spectral counting function of the laplace operator on unbounded two dimensional domains

on the weighted galerkin method of numerical solution of cauchy type singular integral equations the well known weighted galerkin method for the direct numerical solution of cauchy type singular integral equations of the first and the second kind based on the approximation of the unknown function by a finite series of appropriate orthogonal polynomials is shown to be identical to the corresponding method based on the reduction of the cauchy type singular integral equation to an equivalent fredholm integral equation of the second kind and the application to the latter of the weighted galerkin method this result permits the direct transfer of the whole set of convergence and rate of convergence results for the weighted galerkin method of numerical solution of fredholm integral equations with regular kernels to the case of cauchy type singular integral equations an application to the case where chebyshev polynomials are used is also made and the available convergence results for the direct weighted galerkin method are rederived with essentially no analysis

fredholm volterra integral equation and generalized potential kernel a method is used to solve the fredholm volterra integral equation of the first kind in the space l 2 w xc 0 t w x y email protected x 2 y 2

formule de weyl pour les groupes de lie nilpotents pour tout groupe de lie g reel nilpotent simplement connexe on definit les classes de symboles sur le dual de son algebre de lie pour toute representation unitaire irreductible de g et pour tout symbole p on definit l operateur de symbole de weyl p agissant sur l espace de la representation on etablit alors une formule donnant le comportement asymptotique des valeurs propres de l operateur en fonction des valeurs prises par son symbole de weyl sur l orbite co adjointe associee a la representation par la methode de kirillov dans le cas ou le symbole est elliptique dans la direction du cone asymptote a l orbite

the galerkin method for singular integral equations revisited abstract in an earlier work the author has obtained error bounds for the galerkin method for solving cauchy singular integral equations discovering that the usually neglected constants contain the riemann zeta function when evaluated in the supremum norm the aim of this investigation is twofold to show that the occurrence of the riemann zeta function in the error bound for the chebyshev norm is sharp and secondly to use this result to obtain a class of forcing functions for which the method does not yield an approximate solution differing from the analytical one by at most a prescribed error tolerance these counterexamples indicate that in practical situations for functions exhibiting a behavior similar to the one presented here galerkin s method might not lead to an acceptable solution

the trace of the heat kernel in lipschitz domains we establish the existence of an asymptotic expansion as t 0 for the trace of the heat kernel for the neumann laplacian in a bounded lip schitz domain the proof of an asymptotic expansion for the heat kernel for the dirichlet laplacian is also sketched the treatment of the dirichlet lapla cian extends work of brossard and carmona who obtained the same result in c1 domains a classical question in analysis asks what is the relationship between the spectrum of the laplacian on a domain and the geometry of the domain this question has been studied by many authors including 1 3 7 11 12 14 15 17 19 one approach to this problem proceeds by considering the trace of the heat semigroup on d tr 0 i g t x x dx te jd where g is the heat kernel on d and 0 ai a2 a3 is the sequence of eigenvalues for a on d one attempts to give an asymptotic expansion for tx g near t 0 whose coefficients are quantities associated with the geometry of d thus one can deduce geometrical information about d from knowledge of the spectrum in this paper we will address this question under minimal smoothness as sumptions on the boundary of d in particular we will obtain asymptotic ex pansions for tx g when the order of smoothness of the boundary corresponds exactly to the order of accuracy of the asymptotic expansion in order to state our main theorem we need one definition by a lipschitz domain we mean a bounded connected open set whose boundary may be locally described as the graph of a lipschitz function our main result is main theorem suppose that d c r is a lipschitz domain then the heat kernel for the laplacian with dirichlet boundary conditions on 3d satisfies tx g t 4ntyn 2 d n x dd o txl2

on the asymptotics of the heat equation and bounds on traces associated with the dirichlet laplacian we use the feynman kac formula and a decomposition of the brownian bridge to obtain pointwise estimates on the diagonal elements of the heat kernel we also compute bounds on trace et d where d is the dirichlet laplacian for i a bounded region d in rn with smooth boundary or ii an unbounded region d in r2 with finite area

spectral asymptotics for the schrodinger operator with potential which steadies at infinity we consider the discrete spectrum of the selfadjoint schrodinger operatora h h 2 v defined inl2 m with potentialv which steadies at infinity i e v x g x f 1 o 1 as x for 0 and some homogeneous functionsg andf of order zero let h 0 be the total multiplicity of the eigenvalues ofa h smaller thanm m being the minimum value ofg over the unit spheresm 1 hence m coincides with the lower bound of the essential spectrum ofa h we study the asymptotic behaviour of 1 as 0 or of h ash 0 the number 0 being fixed we find that these asymptotics depend essentially on the structure of the submanifold ofsm 1 where the functiong takes the valuem and generically are nonclassical i e even as a first approximation 2 m h differs from the volume of the set x 2m h2 2 v x m

on the spectral counting function for the dirichlet laplacian abstract we obtain upper and lower bounds for the spectral counting function associated to the dirichlet laplacian for an open set d in euclidean space satisfying a geometric and a potential theoretic condition

dixmier traces and residues on weak operator ideals we develop the theory of modulated operators in general principal ideals of compact operators for laplacian modulated operators we establish connes trace formula in its local euclidean model and a global version thereof it expresses dixmier traces in terms of the vector valued wodzicki residue we demonstrate the applicability of our main results in the context of log classical pseudo differential operators studied by lesch and a class of operators naturally appearing in noncommutative geometry c 2020 elsevier inc all rights reserved

on the numerical solutions of fredholm volterra integral equation toeplitz matrix method and the product nystrom method are described for mixed fredholm volterra singular integral equation of the second kind the results are compared with the exact solution of the integral equation the error of each method is calculated

eigenvalue asymptotics and exponential decay of eigenfunctions for schrodinger operators with magnetic fields we consider the schr6dinger operator with magnetic field h va x 2 v x in rj assuming that v 0 and icurl a i v 1 is locally in certain reverse holder class we study the eigenvalue asymptotics and exponential decay of eigenfunctions introduction consider the schrodinger operator with magnetic field 0 1 h h a v jv a x v x inlr n 3 where i 1 v rn r is the electric scalar potential and a rn rn is the magnetic vector potential under some natural conditions on a and v h admits a self adjoint realization which is still denoted by h in l2 ir1n in this paper we shall study its eigenvalue asymptotics and exponential decay of its eigenfunctions let n a h denote the number of eigenvalues counting multiplicity of h smaller than a or the dimension of the spectral projection for h corresponding to x a in the case a x 0 the following phase space volume estimate due to cwickel lieb rosenbljum sil theorem 9 3 p 95 is classical 0 2 n a a v cnl i x e r n x r n 1h12 v x l where cn is a constant depending on n on the other hand there exist some simple degenerate potentials v x i e potentials which do not tend to infinity in all direction as lxi oo e g v x v xi x2 xn x2x n xn for which the right hand side of 0 2 is infinite and nevertheless a v x has a discrete spectrum furthermore in the case where a x 0 and v x 0 the phase space volume of the set where the symbol of h is less than a n i x e nr x r n e ijaj x l2 l j 1 received by the editors may 19 1995 1991 mathematics subject classification primary 35p20 35j10

landau levels on the hyperbolic plane in the presence of aharonov bohm fields abstract we consider the magnetic schrodinger operators on the poincare upper half plane with constant gaussian curvature 1 we assume the magnetic field is given by the sum of a constant field and the dirac measures placed on some lattice we give a sufficient condition for each landau level to be an infinitely degenerated eigenvalue we also prove the lowest landau level is not an eigenvalue if the above condition fails in particular the infinite degeneracy of the lowest landau level is equivalent to the infiniteness of the zero modes of the two dimensional pauli operator

hybrid legendre polynomials and block pulse functions approach for nonlinear volterra fredholm integro differential equations this paper introduces an approach for obtaining the numerical solution of the nonlinear volterra fredholm integro differential nvfid equations using hybrid legendre polynomials and block pulse functions these hybrid functions and their operational matrices are used for representing matrix form of these equations the main characteristic of this approach is that it reduces nvfid equations to a system of algebraic equations which greatly simplifying the problem numerical examples illustrate the validity and applicability of the proposed method

on the numerical treatment of the singular integral equation of the second kind here a numerical treatment for solving the integral equation of the second kind with cauchy kernel is presented the singular term has been removed and the solution in the legendre polynomial form has been used to obtain a system of linear algebraic equation this system is solved numerically

numerical solution for system of cauchy type singular integral equations with its error analysis in complex plane abstract in this paper the problem of finding numerical solution for a system of cauchy type singular integral equations of first kind with index zero is considered the analytic solution of such system is known but it is of limited use as it is a nontrivial task to use it practically due to the presence of singularity in the known solution itself therefore a residual based galerkin method is proposed with legendre polynomials as basis functions to find its numerical solution the proposed method converts the system of cauchy type singular integral equations into a system of linear algebraic equations which can be solved easily further hadamard conditions of well posedness are established for system of cauchy singular integral equations as well as for system of linear algebraic equations which is obtained as a result of approximation of system of singular integral equations with cauchy kernel the theoretical error bound is derived which can be used to obtain any desired accuracy in the approximate solution of system of cauchy singular integral equations the derived theoretical error bound is also validated with the help of numerical examples

fredholm volterra integral equation of the first kind and contact problem regular and singular asymptotic methods are applied to one and two dimensional fredholm volterra integral equation f vie of the first kind that arise in the treatment of various two dimensional axisymmetric and three dimensional problems with mixed boundary conditions in the mechanics of continuous media the solution of the integral equation is obtained in the space l 2 w xc 0 t 0

further convergence results for the weighted galerkin method of numerical solution of cauchy type singular integral equations the convergence of the weighted lalerkin method based on chcbvshen or jacobi polvnoinials for the direct numerical solution of one dimensional real cauchv tvpe singular integral equations of the first and of the second kind on a finite interval is proved under sufficientlv weak continuitv assumptions for the kernels and the right side functions of thc integral equations

numerical methods of singular integral equation for solving some problem in fluid dynamic consider a contact problem of two different layers under certain conditions the aim of this work is obtaining the stress potential function between the two layers and obtaining the solution numerically using two different method toeplitz matrix method and product nystrom method numerical results are considered and the error in each case is computed

adventures of the coupled yang mills oscillators i semiclassical expansion we study the quantum mechanical motion in the x2y2 potentials with n 2 3 which arise in the spatially homogeneous limit of the yang mills ym equations these systems show strong stochasticity in the classical limit 0 and exhibit a quantum mechanical confinement feature we calculate the partition function z t going beyond the thomas fermi tf approximation by means of the semiclassical expansion using the wigner kirkwood wk method we derive a novel compact form of the differential equation for the wk function after separating the motion in the channels of the equipotential surface from the motion in the central region we show that the leading higher order corrections to the tf term vanish up to eighth order in if we treat the quantum motion in the hyperbolic channels correctly by adiabatic separation of the degrees of freedom finally we obtain an asymptotic expansion of the partition function in terms of the parameter g24t3

classical and non classical eigenvalue asymptotics for magnetic schrodinger operators abstract in this paper we consider schrodinger operators with magnetic fields and study which of the magnetic field and the scalar potential contributes to the leading term of the asymptotic distribution of eigenvalues by studying the asymptotic behavior of the trace of the heat kernels we see that if the norm of the magnetic field diverges at infinity faster than the scalar potential only the magnetic field contributes to the leading asymptotic and that only the scalar potential does if the opposite under mild conditions

the partition function and level density for yang mills higgs quantum mechanics we calculate the partition function z t and the asymptotic integrated level density n e for yang mills higgs quantum mechanics for two and three dimensions n 2 3 due to the infinite volume of the phase space on energy shell for n 2 it is not possible to completely disentangle the coupled oscillators x2 y2 model from the higgs sector the situation is different for n 3 for which is finite the transition from order to chaos in these systems is expressed by the corresponding transitions in z t and n e analogous to the transitions in adjacent level spacing distribution from poisson distribution to wigner dyson distribution we also discuss a related system with quartic coupled oscillators and two dimensional quartic free oscillators for which contrary to ymhqm both coupling constants are dimensionless

spectral properties of schrodinger operators with magnetic fields for a spin 12 particle abstract we discuss the spectral properties of schrodinger operators with magnetic fields especially for a spin 1 2 particle we are interested in the essential spectrum and the discrete spectrum we classified the spectrum by the asymptotic behavior of the magnetic field at infinity our main tools are the supersymmetry and the complex analysis in several variables we show that 0 is an eigenvalue with infinite multiplicity under suitable conditions

hybrid function method for solving fredholm and volterra integral equations of the second kind numerical solutions of fredholm and volterra integral equations of the second kind via hybrid functions are proposed in this paper based upon some useful properties of hybrid functions integration of the cross product a special product matrix and a related coefficient matrix with optimal order are applied to solve these integral equations the main characteristic of this technique is to convert an integral equation into an algebraic hence the solution procedures are either reduced or simplified accordingly the advantages of hybrid functions are that the values of n and m are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function for the solutions of integral equations we propose that the available optimal values of n and m can minimize the relative errors of the numerical solutions the high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples the hybrid function method is superior to other piecewise constant orthogonal functions w f blyth r l may p widyaningsih volterra integral equations solved in fredholm form using walsh functions anziam j 45 e 2004 c269 c282 m h reihani z abadi rationalized haar functions method for solving fredholm and volterra integral equations j comp appl math 200 2007 12 20 for these problems

analysis of zero modes for dirac operators with magnetic links abstract in this paper we provide a means to approximate dirac operators with magnetic fields supported on links in s 3 and r 3 by dirac operators with smooth magnetic fields then we proceed to prove that under certain assumptions the spectral flow of paths along these operators is the same in both the smooth and the singular case we recently characterized the spectral flow of such paths in the singular case this allows us to show the existence of new smooth compactly supported magnetic fields in r 3 for which the associated dirac operator has a non trivial kernel using clifford analysis we also obtain criteria on the magnetic link for the non existence of zero modes

on a symptotic methods for fredholm volterra integral equation of the second kind in contact problems a method is used to obtain the general solution of fredholm volterra integral equation of the second kind in the space l2 c 0 t 0 t t is the domain of integrations the kernel of the fredholm integral term belong to c and has a singular term and a smooth term the kernel of volterra integral term is a positive continuous in the class c 0 t while is the domain of integration with respect to the fredholm integral term besides the separation method the method of orthogonal polynomials has been used to obtain the solution of the fredholm integral equation the principal singular part of the kernel which corresponds to the selected domain of parameter variation is isolated the unknown and known functions are expanded in a chebyshev polynomial and an infinite algebraic system is obtained

galerkin s method for operator equations with nonnegative index with application to cauchy singular integral equations abstract we consider solving operator equation 1 hu ku f where h and k are bounded linear operators between two real hilbert spaces h1 and h2 operator h is assumed to have a finite dimensional nullspace n h and a bounded right inverse h1 h2 h1 and k is compact it follows that dim n h k dim n h so that to obtain uniqueness the m additional conditions 2 u k 1 bk k 1 2 h dim n h m are imposed where the k k 1m are an orthonormal basis for n h to solve 1 and 2 these equations are converted to an equivalent equation of the second kind to which galerkin s method is applied using the basis 1 2 m m 1 n it was shown that this method is equivalent to the method of weighted residuals when h 1 h the adjoint of h the results are applied to obtain convergence proofs of some numerical methods for solving several classes of cauchy singular integral equations whose kernels are only square integrable

convergence and stability of a collocation method for the generalized airfoil equation we establish the mean square convergence and numerical stability of a collocation method for solving a class of cauchy singular integral equations the method uses the jacobi polynomials p 1 2 1 2 n as basis elements and the zeros of the polynomial p 1 2 1 2 n 1 as collocation points uniform convergence is shown to hold for the first iterate convergence is proved via vainikko s embedding method and stability is demonstrated by generalizing the approach of atkinson for integral equations of the second kind

galerkin solution of a singular integral equation with constant coefficients galerkin methods are used to approximate the singular integral equation email protected x email protected 1 email protected t email protected 1 1k t x f t dt f x 10 are constants in this case f is decomposed as f x 1 x a 1 x bu x where email protected 0 0 is a small enough constant

spectral properties of pauli operators on the poincare upper half plane we investigate the essential spectrum of the pauli operators and the dirac and the schrodinger operators with magnetic fields on the poincare upper half plane the magnetic fields under consideration are asymptotically constant which may be equal to zero or diverge at infinity moreover the aharonov casher type result is also considered

refined asymptotic expansion for the partition function of unbounded quantum billiards this paper presents a refined asymptotic expansion for the partition function t tr et of quantum billiards in the unbounded regions 0 x 0 yx 1 0 and 0 ye x 1 r2 where is the dirichlet laplacian simon ann phys 146 209 1983 j funct anal 53 84 1983 determined the leading divergence of the trace of the heat kernel for the first class of systems standard techniques are combined for the evaluation of for bounded region billiards with results by van den berg j funct anal 71 279 1987 for horn shaped regions using an optimized way of dividing the region into narrow and wide parts to determine the first three terms in the asymptotic expansion of results are also stated for bounded regions with cusps that can be obtained by the same method as an application the spectral staircase of the strongly chaotic billiard system defined in the region 0 xy 2 x 0 which has been discussed in connection with the riemann function and the search for quantum chaos is c

eigenvalue problems for the schrodinger operator with the magnetic field on a compact riemannian manifold abstract we consider the eigenvalue problem of the schrodinger operator with the magnetic field on a compact riemannian manifold first we discuss the least eigenvalue we give a representation of the least eigenvalue by the variational formula and give a relation to the least eigenvalue of the schrodinger operator without the magnetic field second we discuss the asymptotic distribution of eigenvalues by obtaining the asymptotic expansion of the kernel of semigroup here we use the theory of asymptotic expansion for wiener functionals

viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape from shading equations with discontinuous intensities the shape from shading sfs equation relating u y r the unknown angular height of a surface to i y r the known synthetic aperture radar sar intensity data from the surface is i frac u r 2 sqrt 1 u r 2 u y 2 where y and r are axial and radial cylindrical coordinates unlike the more common eikonal sfs equation which relates surface height in cartesian coordinates to optical photographic intensity data the above radar equation can be transformed into hamilton jacobi cauchy form ur g i uy 0 we explore the case where i is a discontinuous function which occurs commonly in radar data by considering sequences of continuous intensity functions that converge to i we obtain corresponding sequences of viscosity solutions we prove that these sequences must converge we also establish conditions that guarantee that these sequences converge to a common limit which we define as the solution to the radar equation finally we establish and demonstrate that when this common limit exists monot

numerical solution of fredholm volterra integral equation in one dimension with time dependent in this paper we consider fredholm volterra integral equation in one dimension with time dependent the kernel of the fredholm term is assumed to be singular while the kernel of volterra term is continuous the existence of the solution is proved and a numerical method is presented to get an approximate to such solution

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

localization in infinite billiards a comparison between quantum and classical ergodicity consider the non compact billiard in the first quandrant bounded by the positive x semiaxis the positive y semiaxis and the graph of f x x 1 1 2 although the schnirelman theorem holds the quantum average of the position xis finite on any eigenstate while classical ergodicity entails that the classical time average of xis unbounded

the essential spectrum of schrodinger operators with asymptotically constant magnetic fields on the poincare upper half plane we study the essential spectrum of the magnetic schrodinger operators on the poincare upper half plane and establish a hyperbolic analog of iwatsuka s result j math kyoto univ 23 3 475 480 1983 on the stability of the essential spectrum under perturbations from constant magnetic fields

on viscosity solutions of hamilton jacobi equations we consider the dirichlet problem for hamilton jacobi equations and prove existence uniqueness and continuous dependence on boundary data of lipschitz continuous maximal viscosity solutions

numerical solution of nonlinear volterrafredholmhammerstein integral equations in two dimensional spaces based on block pulse functions in this paper the numerical technique based on block pulse functions bpfs has been developed to approximate the solutions of nonlinear volterrafredholmhammerstein integral equations in two dimensional spaces these functions are orthogonal and have compact support on 0 1 the proposed method reduces the integral equations to a system of nonlinear algebraic equations that can be easily solved by any numerical method also the convergence of the proposed approach is discussed furthermore in order to show the accuracy and reliability of the above mentioned algorithm the new approach is verified through some numerical examples a numerical technique based on block pulse functions is proposed two dimensional nonlinear volterrafredholmhammerstein integral equations are firstly numerically solved the proposed method is to transform the volterrafredholm integral equations into a system of algebraic equations the main characteristic behind the algorithm is operator theory