trace expansions for elliptic cone operators with stationary domains we analyze the behavior of the trace of the resolvent of an ellip tic cone differential operator as the spectral parameter tends to infinity the resolvent splits into two components one associated with the minimal exten sion of the operator and another of finite rank depending on the particular choice of domain we give a full asymptotic expansion of the first compo nent and expand the component of finite rank in the case where the domain is stationary the results make use and develop further our previous investi gations on the analytic and geometric structure of the resolvent the analysis of nonstationary domains considerably more intricate is pursued elsewhere

functional determinant of the massive laplace operator and the multiplicative anomaly after a brief survey of zeta function regularization issues and of the related multiplicative anomaly illustrated with a couple of basic examples namely the harmonic oscillator and quantum field theory at finite temperature an application of these methods to the computation of functional determinants corresponding to massive laplacians on spheres in arbitrary dimensions is presented explicit formulas are provided for the laplace operator on spheres in dimensions and for vector and tensor laplacians on the unitary sphere s4

bicubic b spline surfaces constrained by the biharmonic pde abstract bicubic b spline surface constrained by the biharmonic pde is presented in this paper by representing the biharmonic pde in the form of the bilinear b spline bases we find the regular vector valued coefficients and discover that bicubic b spline surface can satisfy the biharmonic pde when the control points of the boundaries for open or closed surfaces are given the inner control points can be fully determined for each case of the surfaces open in both directions closed in one direction and closed in both directions a linear system for solving inner control points is established some examples show the effectiveness of the given method

a mathematical model for the partition function in this work we get the formula 5 that it allows the explicit computation of the renormalized partition function associated with a physical system whose states belong to a differentiable manifold of finite dimension and whose energy function is invariant under the action of a compact lie group furthermore we extend the formula above to the infinite dimensional case using regularization technique

a pragmatic approach to the problem of the self adjoint extension of hamilton operators with the aharonov bohm potential we consider the problem of the self adjoint extension of hamilton operators for charged quantum particles in the pure aharonov bohm potential infinitely thin solenoid we present a pragmatic approach to the problem based on the orthogonalization of the radial solutions for different quantum numbers then we discuss a model of a scalar particle with a magnetic moment which allows us to explain why the self adjoint extension contains arbitrary parameters and give a physical interpretation

fundamental solution of the heat and schrodinger equations with point interaction abstract we find explicit formulae for the fundamental solution for the heat and time dependent schrodinger equations with point interaction in dimension n 3 we discuss in detail the small time asymptotics of these fundamental solutions the case n 2 has the peculiarity of involving logarithmic terms in the expansion

spectral analysis of a flat plasma sheet model the spectral analysis of the electromagnetic field on the background of an infinitely thin flat plasma layer is carried out this model loosely imitates a single base plane from graphite and it is of interest for theoretical studies of fullerenes by making use of the hertz potentials the solutions to maxwell equations with the appropriate matching conditions at the plasma layer are derived and on this basis the spectrum of electromagnetic oscillations is determined the model is naturally split into the te sector and tm sector both the sectors have positive continuous spectra but the tm modes have in addition a bound state namely the surface plasmon this analysis relies on the consideration of the scattering problem in the te and tm sectors the spectral zeta function and integrated heat kernel are constructed for different branches of the spectrum in an explicit form as a preliminary the rigorous procedure of integration over the continuous spectra is formulated by introducing the spectral density in terms of the scattering phase shifts the asymptotic expansion of the integrated heat kernel at small values of the evolution parameter is derived by making use of the technique of integral equations developed earlier by the same authors the local heat kernel green s function or fundamental solution is also constructed as a by product a new method is demonstrated for deriving the fundamental solution to the heat conduction equation or to the schrodinger equation on an infinite line with the like source in particular for the heat conduction equation on an infinite line with the source a nontrivial counterpart is found namely a spectral problem with point interaction which possesses the same integrated heat kernel while the local heat kernels fundamental solutions in these spectral problems are different

derivation of the time dependent propagator for the three dimensional schrodinger equation with one point interaction an explicit formula for the time dependent propagator of the schrodinger equation with one point interaction in three dimensions is given the derivation is based on the inverse laplace transformation applied to the corresponding resolvent

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

semi transparent boundary conditions in the worldline formalism the interaction of a quantum field with a background containing a dirac delta function with support on a surface of codimension 1 represents a particular kind of matching conditions on that surface for the field in this paper we show that the worldline formalism can be applied to this model we obtain the asymptotic expansion of the heat kernel corresponding to a scalar field on in the presence of an arbitrary regular potential and subject to this kind of matching conditions on a flat surface we also consider two such surfaces and compute their casimir attraction due to the vacuum fluctuations of a massive scalar field weakly coupled to the corresponding dirac deltas

zeta regularized determinants for conic manifolds we study relative zeta regularized determinants of laplace type operators on compact conic manifolds we establish gluing formulae for relative zeta regularized determinants for arbitrary self adjoint extensions of the laplace beltrami operator we express the relative determinants for these as a ratio of the determinants of certain finite matrices for the self adjoint extensions corresponding to dirichlet and neumann conditions the formula is particularly simple and elegant

levinson s theorem in one dimension heuristics for partial waves in three dimensions levinson s theorem asserts that delta 0 pi nb where delta p is the phase shift at wavenumber p and nb the number of bound states delta infinity 0 by convention the corresponding theorem in one dimension calls first for a systematic parametrisation of the transmission amplitude t p cos theta ei tau and of the left right incidence reflection amplitudes rl r p i sin theta exp i tau or i rho where the phase angles tau theta rho are functions of p if the potential is not everywhere zero and excluding throughout the exceptional case where it has a zero energy bound state heuristic arguments show that rl r 0 1 that theta 0 1 2 pi 1 2 pi when nb is odd even and that tau 0 pi nb 1 2 by convention tau infinity 0 theta infinity thus mod tau 0 mod cannot be less than 1 2 pi no matter how weak the potential the transition to the limit of zero potential is non uniform in the special case of reflection symmetric potentials rho 0 one can subdivide nb nb e nb o and define even and odd parity phase shifts e 1 2 tau theta and delta 1 2 tau theta then e 0 pi nb e 1 2 delta 0 pi nb o the appendix shows how e 0 is obtainable by suitably adapting the familiar s wave argument which exploits the analyticity properties of the jost solutions

hermitian symplectic geometry and the factorization of the scattering matrix on graphs hermitian symplectic spaces provide a natural framework for the extension theory of symmetric operators here we show that hermitian symplectic spaces may also be used to describe the solution to the factorization problem for the scattering matrix on a graph i e we derive a formula for the scattering matrix of a graph in terms of the scattering matrices of its subgraphs the solution of this problem is shown to be given by the intersection of a lagrange plane and a coisotropic subspace which in an appropriate hermitian symplectic space forms a new lagrange plane the scattering matrix is given by a distinguished basis to the lagrange plane using our construction we are also able to give a simple proof of the unitarity of the scattering matrix as well as provide a characterization of the discrete eigenvalues embedded in the continuous spectrum

solving the triharmonic equation over multi patch planar domains using isogeometric analysis abstract we present a framework for solving the triharmonic equation over bilinearly parameterized planar multi patch domains by means of isogeometric analysis our approach is based on the construction of a globally c 2 smooth isogeometric spline space which is used as discretization space the generated c 2 smooth space consists of three different types of isogeometric functions called patch edge and vertex functions all functions are entirely local with a small support and numerical examples indicate that they are well conditioned the construction of the functions is simple and works uniformly for all multi patch configurations while the patch and edge functions are given by a closed form representation the vertex functions are obtained by computing the null space of a small system of linear equations several examples demonstrate the potential of our approach for solving the triharmonic equation

fundamental measure density functional theory for nonadditive hard core mixtures the one dimensional case we treat a one dimensional binary mixture of hard core particles that possess nonadditive diameters for this model a density functional theory is constructed following similar principles as an earlier extension of rosenfeld s fundamental measure theory to three dimensional nonadditive hard sphere mixtures the theory applies to arbitrary positive and moderate negative nonadditivity and reduces to percus exact functional in the additive case bulk direct correlation functions are obtained as functional derivatives of the excess free energy functional results for the partial pair correlation functions in bulk as calculated via the ornstein zernike route and using the direct correlation functions as input show very good agreement with results from our monte carlo computer simulations of the mixture

the very unusual properties of the resolvent heat kernel and zeta function for the operator d2 dr2 1 4r2 in this paper we analyze the resolvent the heat kernel and the spectral zeta function of the operator d2 dr2 1 4r2 over the finite interval the structural properties of these spectral functions depend strongly on the chosen self adjoint realization of the operator a choice being made necessary because of the singular potential present only for the friedrichs realization standard properties are reproduced for all other realizations highly nonstandard properties are observed in particular for k n we find terms like logt k in the small t asymptotic expansion of the heat kernel furthermore the zeta function has s 0 as a logarithmic branch point

remarks on inverse scattering in one dimension this paper answers the following questions 1 what are the consequences in the matrix marchenko inversion scheme if a given s matrix lacks forward analyticity and 2 in particular does the condition known as the miracle depend on forward analyticity and if not what properties of s does it depend on the answers are 1 if the input s matrix lacks forward analyticity then the output s matrix has it anyway and 2 integrability of krl r is sufficient for the miracle to occur it is also found that the matrix marchenko procedure simultaneously constructs the potentials for two scattering problems which differ only by the signs of their reflection coefficients

spectral functions for the schrodinger operator on r with a singular potential in this article we analyze the spectral zeta function the heat kernel and the resolvent of the operator d2 dr2 r2 r2 over the interval 0 for 1 4 depending on the self adjoint extension chosen nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed in particular for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at s k k n0 do occur this implies that the small t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk ln t l 1 for k l n0 the corresponding statements for the resolvent are also shown furthermore we evaluate the zeta determinant of the operator for all values of and any self adjoint extension

the heat equation with singular coefficients the small time asymptotics of the kernel ofe th is defined and derived for h frac d 2 dx 2 frac kappa x 2 on 1 lemmas on singular asymptotics in the sense of distributions are formulated and used the results are applied to derive an index formula on 1

spectral functions of non essentially self adjoint operators fil falomir horacio alberto consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet la plata instituto de fisica la plata universidad nacional de la plata facultad de ciencias exactas instituto de fisica la plata argentina

deriving the rosenfeld functional from the virial expansion in this article we replace the semiheuristic derivation of the rosenfeld functional of hard convex particles with the systematic calculation of mayer clusters it is shown that each cluster integral further decomposes into diagrams of intersection patterns that we classify by their loop number this extends the virial expansion of the free energy by an expansion in the loop order with the rosenfeld functional as its leading contribution rosenfeld s weight functions then follow from the derivation of the intersection probability by generalizing the equation of blaschke santalo and chern it is found that the 0 loop order can be derived exactly and reproduces the rosenfeld functional we further discuss the influence of particle dimensions topologies and geometries on the mathematical structure of the calculation

density functional theory for nearest neighbor exclusion lattice gases in two and three dimensions to speak about fundamental measure theory obliges us to mention dimensional crossover this feature inherent to the systems themselves was incorporated in the theory almost from the beginning although at first it was thought to be a consistency check for the theory it rapidly became its fundamental pillar thus becoming the only density functional theory which possesses such a property it is straightforward that dimensional crossover connects for instance the parallel hard cube system three dimensional with that of squares two dimensional and rods one dimensional we show here that there are many more connections which can be established in this way through them we deduce from the functional for parallel hard hyper cubes in the simple hyper cubic lattice the corresponding functionals for the nearest neighbor exclusion lattice gases in the square triangular simple cubic face centered cubic and body centered cubic lattices as an application the bulk phase diagram for all these systems is obtained

an exactly solvable model for depletion phenomena abstract the equilibrium properties of a one dimensional mixture of hard rods of two different lengths are obtained these are used to test the free volume approximation which has in the past been applied in the study of hard sphere mixtures in three dimensions three different versions of the free volume approximation are tested in the most primitive of these the smaller spheres or rods are assumed not to interact with each other in higher levels of the approximation the interactions between the smaller molecules are partly taken into account the results show the ranges of compositions and densities over which the approximations are accurate and they identify one form of the approximation as the most reliable

dimensional crossover of the fundamental measure functional for parallel hard cubes we present a regularization of the recently proposed fundamental measure functional for a mixture of parallel hard cubes the regularized functional is shown to have correct dimensional crossovers to any smaller dimension thus allowing its use to study highly inhomogeneous phases such as the solid phase furthermore it is shown how the functional of the slightly more general model of parallel hard parallelepipeds can be obtained using the zero dimensional functional as a generating functional extensions to the multicomponent system a restricted orientation model and mixtures or prallel hard cylinders are also given s0031 9007 97 03105 0

deriving fundamental measure theory from the virial series consistency with the zero dimensional limit fundamental measure theory fmt for hard particles has great potential for predicting the phase behavior of colloidal and nanometric shapes the modern versions of fmt are usually derived from the zero dimensional limit a system of at most one particle confined in a collection of cavities in the limit that all cavities shrink to the size of the particle in phys rev e 85 041150 2012 a derivation from an approximated and resummed virial expansion was presented whose result was not fully consistent with the fmt from the zero dimensional limit here we improve on this derivation and obtain exactly the same fmt functional as was obtained earlier from the zero dimensional limit as a result further improvements of fmt based on the virial expansion can now be formulated some of which we suggest in the outlook

adjoints of elliptic cone operators we study the adjointness problem for the closed extensions of a general b elliptic operator a x diff m b m e 0 initially defined as an unbounded operator a c c m e x l 2 b m e x l 2 b m e inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the case where a is a symmetric semibounded operator is of particular interest and we give a complete description of the domain of the friedrichs extension of such an operator

heat kernel expansion on noncompact domains and a generalized zeta function regularization procedure heat kernel expansion and zeta function regularization are discussed for laplace type operators with discrete spectrum in noncompact domains since a general theory is lacking the heat kernel expansion is investigated by means of several examples it is pointed out that for a class of exponential analytic interactions generically the noncompactness of the domain gives rise to logarithmic terms in the heat kernel expansion then a meromorphic continuation of the associated zeta function is investigated a simple model is considered for which the analytic continuation of the zeta function is not regular at the origin displaying a pole of higher order for a physically meaningful evaluation of the related functional determinant a generalized zeta function regularization procedure is proposed

the resolvent expansion for second order regular singular operators abstract the paper gives a scheme for computing the asymptotics of tr e tl as t 0 where l is an elliptic operator of the form l d 2 x 2 a x and a x is a family of operators satisfying appropriate ellipticity and smoothness conditions a principle example is the laplace operator for a manifold with an asymptotically conic singularity the expansion has the usual terms away from the singularity appropriately regularized at x 0 plus singular contributions determined by the function of a 0 1 4 1 2 applications to index theorems are given in a subsequent paper

explicit derivation of the propagator for a dirac delta potential an explicit derivation of the propagator is given for a particle in a dirac delta potential lambda delta x by making a direct application of a variational technique followed by a systematic use of laplace transforms although the solution of the corresponding schrodinger equation has been known for years the solution of the propagator has been obtained only recently

regular singular sturm liouville operators and their zeta determinants abstract we consider sturm liouville operators on the line segment 0 1 with general regular singular potentials and separated boundary conditions we establish existence and a formula for the associated zeta determinant in terms of the wronski determinant of a fundamental system of solutions adapted to the boundary conditions this generalizes the earlier work of the first author treating general regular singular potentials but only the dirichlet boundary conditions at the singular end and the recent results by kirsten loya park for general separated boundary conditions but only special regular singular potentials

the spectral determinant of the isotropic quantum harmonic oscillator in arbitrary dimensions we show that the spectral determinant of the isotropic quantum harmonic oscillator converges exponentially to one as the space dimension grows to infinity we determine the precise asymptotic behaviour for large dimension and obtain estimates valid for all cases with the same asymptotic behaviour in the large as a consequence we provide an alternative proof of a conjecture posed by bar and schopka concerning the convergence of the determinant of the dirac operator on s n determining the exact asymptotic behaviour for this case and thus improving the estimate on the rate of convergence given in the work of moller math ann 343 35 51 2009

cluster density functional theory for lattice models based on the theory of mobius functions rosenfeld s fundamental measure theory for lattice models is given a rigorous formulation in terms of the theory of mobius functions of partially ordered sets the free energy density functional is expressed as an expansion in a finite set of lattice clusters this set is endowed with a partial order so that the coefficients of the cluster expansion are connected to its mobius function because of this it is rigorously proven that a unique such expansion exists for any lattice model the low density analysis of the free energy functional motivates a redefinition of the basic clusters zero dimensional cavities which guarantees a correct zero density limit of the pair and triplet direct correlation functions this new definition extends rosenfeld s theory to lattice models with any kind of short range interaction repulsive or attractive hard or soft one or multicomponent finally a proof is given that these functionals have a consistent dimensional reduction i e the functional for dimension d can be obtained from that for dimension d d d if the latter is evaluated at a density profile confined to a d dimensional subset

resolvents of elliptic cone operators we prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions this is achieved by the construction of a suitable parametrix and reduction to the boundary special attention is devoted to the clarification of the analytic structure of the resolvent

regularized products and determinants zeta regularized products are used to define determinants of operators in infinite dimensional spaces this article provides a general theory of regularized products and determinants which delivers a better approach to their existence and explicit determination

representation of piecewise biharmonic surfaces using biquadratic b splines piecewise biquadratic b spline surface satisfying biharmonic condition is studied in this paper by applying biharmonic pde biquadratic b spline surface is fully determined by boundary control vertices a linear system for solving inner control vertices is established its coefficient matrix is block tridiagonal and the proof of non singularity of coefficient matrix is presented under the condition that all knots do not coincide with each other a few examples are given to show the effectiveness of piecewise biharmonic biquadratic b spline surface an interpolation method on given boundary points is presented if boundary curves are open

isogeometric analysis for surface pdes with extended loop subdivision abstract we investigate the isogeometric analysis for surface pdes based on the extended loop subdivision approach the basis functions consisting of quartic box splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly and construct the solution space for dependent variables as well which is consistent with the concept of isogeometric analysis the subdivision process is equivalent to the h refinement of nurbs based isogeometric analysis the performance of the proposed method is evaluated by solving various surface pdes such as surface laplace beltrami harmonic biharmonic triharmonic equations which are defined on the limit surfaces of extended loop subdivision for different initial control meshes numerical experiments show that the proposed method has desirable performance in terms of the accuracy convergence and computational cost for solving the above surface pdes defined on both open and closed surfaces the proposed approach is proved to be second order accuracy in the sense of l 2 norm with theoretical and or numerical results which is also outperformed over the standard linear finite element by several numerical comparisons

density functional for the widom rowlinson model we present a density functional theory for the m component widom rowlinson model for a mixture of spherical particles where the unlike species interact with a hard core potential and the interactions between like species vanish the functional is exact for small densities and in the zero dimensional limit it predicts the fluid structure in good agreement with simulations and yields a continuous demixing phase transition for m 2 in the limit of large m the widom rowlinson model reduces to effective hard spheres in the mixed phase and the asakura oosawa colloid ideal polymer model in the demixed phase within the present theory both cases are captured correctly for intermediate m we find a first order demixing phase transition with a rapidly broadening density discontinuity upon increasing m

integral equations for heat kernel in compound media by making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the laplace operator a2 in the case of compound media in each of the media the parameter a2 acquires a certain constant value at the interface of the media the conditions are imposed which demand the continuity of the temperature and the heat flows the integration in the equations is spread out only over the interface of the media as a result the dimension of the initial problem is reduced by 1 the perturbation series for the integral equations derived are nothing else as the multiple scattering expansions for the relevant heat kernels thus a rigorous derivation of these expansions is given in the one dimensional case the integral equations at hand are solved explicitly abel equations and the exact expressions for the regarding heat kernels are obtained for diverse matching conditions derivation of the asymptotic expansion of the integrated heat kernel for a co

on the resolvent and spectral functions of a second order differential operator with a regular singularity we consider the resolvent of a second order differential operator with a regular singularity admitting a family of self adjoint extensions we find that the asymptotic expansion for the resolvent in the general case presents unusual powers of which depend on the singularity the consequences for the pole structure of the function and for the small t asymptotic expansion of the heat kernel are also discussed

the exotic heat trace asymptotics of a regular singular operator revisited we discuss the exotic properties of the heat trace asymptotics for a regular singular operator with general boundary conditions at the singular end as observed by falomir muschietti pisani and seeley unusual poles of the functions for some regular singular differential operators j phys a 36 39 9991 10010 2003 10 1088 0305 4470 36 39 302 as well as by kirsten loya and park the very unusual properties of the resolvent heat kernel and zeta function for the operator d2 dr2 1 4r2 j math phys 47 4 043506 2006 10 1063 1 2189194 we explain how their results alternatively follow from the general heat kernel construction by mooers heat kernel asymptotics on manifolds with conic singularities j anal math 78 1 36 1999 10 1007 bf02791127 a natural question that has not been addressed yet as the latter work did not elaborate explicitly on the singular structure of the heat trace expansion beyond the statement of non polyhomogeneity of the heat kernel

spectral functions for the flat plasma sheet model the present work is based on bordag m et al 2005 j phys a math gen 38 11027 where the spectral analysis of the electromagnetic field on the background of an infinitely thin flat plasma layer is carried out the solutions to maxwell equations with the appropriate matching conditions at the plasma layer are derived and the spectrum of electromagnetic oscillations is determined the spectral zeta function and the integrated heat kernel are constructed for different branches of the spectrum in an explicit form the asymptotic expansion of the integrated heat kernel at small values of the evolution parameter is derived the local heat kernels are considered also

exercises in exact quantization the formalism of exact 1d quantization is reviewed in detail and applied to the spectral study of three concrete schrodinger hamiltonians d2 dq2 v q on the half line q 0 with a dirichlet or neumann condition at q 0 emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters we first discuss the homogeneous potential v q qn as n versus its solvable n limit an infinite square well useful distinctions are established between regular and singular behaviours of spectral quantities various identities among the square well spectral functions are unravelled as limits of finite n properties the second model is the quartic anharmonic oscillator the zero energy spectral determinants det d2 dq2 q4 vq2 are explicitly analysed in detail revealing many special values algebraic identities between taylor coefficients and functional equations of a quartic type coupled to asymptotic v properties of airy type the third study addresses the potentials v q qn vqn 2 1 of even degree their zero energy spectral determinants prove computable in closed form and the generalized eigenvalue problems with v as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case corresponding to n 2 these results partly reflect the presence of quasi exactly solvable potentials in the family above

on self adjoint boundary conditions for singular sturm liouville operators bounded from below abstract we extend the classical boundary values 0 1 g a w u a 0 g a lim x a g x u a 0 x g 1 a p g a w u a 0 g a lim x a g x g a u a 0 x u a 0 x for regular sturm liouville operators associated with differential expressions of the type r x 1 d d x p x d d x q x for a e x a b r to the case where is singular on a b r and the associated minimal operator t m i n is bounded from below here u a 0 and u a 0 denote suitably normalized principal and nonprincipal solutions of u 0 u for appropriate 0 r respectively our approach to deriving the analog of 0 1 in the singular context employing principal and nonprincipal solutions of u 0 u is closely related to a seminal 1992 paper by niessen and zettl 58 we also recall the well known fact that the analog of the boundary values in 0 1 characterizes all self adjoint extensions of t m i n in the singular case in a manner familiar from the regular case we briefly discuss the singular weyl titchmarsh kodaira m function and finally illustrate the theory in some detail with the examples of the bessel legendre and kummer resp laguerre operators

explicit zeta functions for bosonic and fermionic fields on a non commutative toroidal spacetime explicit formulae for the zeta functions s corresponding to bosonic 2 and to fermionic 3 quantum fields living on a non commutative partially toroidal spacetime are derived formulae for the most general case of the zeta function associated with a quadratic linear constant form in z are obtained they provide the analytical continuation of the zeta functions in relation to the whole complex s plane in terms of series of bessel functions of fast exponential convergence thus being extended chowla selberg formulae as is well known this is the most convenient expression that can be found for the analytical continuation of a zeta function in particular the residua of the poles and their finite parts are explicitly given an important novelty is the fact that simple poles show up at s 0 as well as in other places simple or double depending on the number of compactified non compactified and non commutative dimensions of the spacetime where they had never appeared before this poses a challenge to the zeta function regularization procedure

explicit bezier control net of a pde surface abstract the pde under study here is a general fourth order linear elliptic partial differential equation having prescribed the boundary control points we provide the explicit expression of the whole control net of the associated pde bezier surface in other words we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points the set of scalar coefficients of these combinations works like a mould for pde surfaces thus once this mould has been computed for a given degree real time manipulation of the resulting surfaces becomes possible by modifying the prescribed information

fundamental measure theory and dimensional interpolation for the hard spheres fluid abstract this paper presents a short review of the recent advances in the development of density functional for hard spheres the fundamental measure theory as developed originally by y rosenfeld and in its more recent versions is compared with other density functional approximations the results for dimensional crossover from the hard spheres fluid in three dimensions to hard rods in one dimension and to zero dimensional cavities are discussed new results on the fluid crystal coexistence and on the crossover to the hard disc fluid in two dimensions are presented

deficiency indices and singular boundary conditions in quantum mechanics we consider schrodinger operators h in l2 rn n n with countably infinitely many local singularities of the potential which are separated from each other by a positive distance it is proved that due to locality each singularity yields a separate contribution to the deficiency index of h in the special case where the singularities are pointlike and the potential exhibits certain symmetries near these points we give an explicit construction of self adjoint boundary conditions

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

unusual poles of the ζ functions for some regular singular differential operators we consider the resolvent of a system of first order differential operators with a regular singularity admitting a family of self adjoint extensions we find that the asymptotic expansion for the resolvent in the general case presents powers of which depend on the singularity and can take even irrational values the consequences for the pole structure of the corresponding and functions are also discussed

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

zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

theory of asymmetric nonadditive binary hard sphere mixtures it is shown that the formal procedure of integrating out the degrees of freedom of the small spheres in a binary hard sphere mixture works equally well for nonadditive as it does for additive mixtures for highly asymmetric mixtures small size ratios the resulting effective hamiltonian of the one component fluid of big spheres which consists of an infinite number of many body interactions should be accurately approximated by truncating after the term describing the effective pair interaction using a density functional treatment developed originally for additive hard sphere mixtures the zero one and two body contribution to the effective hamiltonian are determined it is demonstrated that even small degrees of positive or negative nonadditivity have significant effect on the shape of the depletion potential the second virial coefficient b2 corresponding to the effective pair interaction between two big spheres is found to be a sensitive measure of the effects of nonadditivity the variation of b2 with the density of the small spheres shows significantly different behavior for additive slightly positive and slightly negative nonadditive mixtures possible repercussions of these results for the phase behavior of binary hard sphere mixtures are discussed and it is suggested that measurements of b2 might provide a means of determining the degree of nonadditivity in real colloidal mixtures

connection conditions and the spectral family under singular potentials to describe a quantum system whose potential is divergent at one point one must provide proper connection conditions for the wavefunctions at the singularity generalizing the scheme used for point interactions in one dimension we present a set of connection conditions which are well defined even if the wavefunctions and or their derivatives are divergent at the singularity our generalized scheme covers the entire u 2 family of quantizations self adjoint hamiltonians admitted for the singular system we use this scheme to examine the spectra of the coulomb potential v x e2 x and the harmonic oscillator with square inverse potential v x m 2 2 x2 g x2 and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra we further show that for any parity invariant singular potential v x v x the spectrum is determined solely by the eigenvalues of the characteristic matrix u u 2

a third order partial differential equation for isotropic boundary based triangular bezier surface generation abstract we approach surface design by solving a linear third order partial differential equation pde we present an explicit polynomial solution method for triangular bezier pde surface generation characterized by a boundary configuration the third order pde comes from a symmetric operator defined here to overcome the anisotropy drawback of any operator over triangular bezier surfaces

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

the application of cubic trigonometric b spline to the numerical solution of the hyperbolic problems in this paper a collocation finite difference scheme based on new cubic trigonometric b spline is developed and analyzed for the numerical solution of a one dimensional hyperbolic equation wave equation with non local conservation condition the usual finite difference scheme is used to discretize the time derivative while a cubic trigonometric b spline is utilized as an interpolation function in the space dimension the scheme is shown to be unconditionally stable using the von neumann fourier method the accuracy of the proposed scheme is tested by using it for several test problems the numerical results are found to be in good agreement with known exact solutions and with existing schemes in literature

determinant of the laplacian on a non compact three dimensional hyperbolic manifold with finite volume the functional determinant of laplace type operators on a three dimensional non compact hyperbolic manifold with invariant fundamental domain of finite volume is expressed via the selberg zeta function related to the picard group

peel or coat spheres by convolution repeatedly a convolution transformation is presented that maps the four fundamental measures minkowski functionals of a three dimensional sphere to those of a sphere with a different radius it is shown that the set of all these transformations parametrized by the induced change in radius forms an abelian commutative group and hence constitutes a flexible framework for the manipulation of spheres the corresponding one dimensional case is laid out and the relationship to fundamental measure density functional theory is discussed briefly

heat kernel expansion for semitransparent boundaries we study the heat kernel for an operator of laplace type with a function potential concentrated on a closed surface we derive the general form of the small t asymptotics and explicitly calculate several first heat kernel coefficients

pde triangular bezier surfaces harmonic biharmonic and isotropic surfaces we approach surface design by solving second order and fourth order partial differential equations pdes we present many methods for designing triangular bezier pde surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces moreover we introduce and study a second order and a fourth order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular bezier surfaces

zeta determinant for laplace operators on riemann caps the goal of this paper is to compute the zeta function determinant for the massive laplacian on riemann caps or spherical suspensions these manifolds are defined as compact and boundaryless d dimensional manifolds deformed by a singular riemannian structure the deformed spheres considered previously in the literature belong to this class after presenting the geometry and discussing the spectrum of the laplacian we illustrate a method to compute its zeta regularized determinant the special case of the deformed sphere is recovered as a limit of our general formulas

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

explicit time dependent schrodinger propagators the authors compute explicitly the time dependent schrodinger and heat propagator for the potentials lambda 2m m 1 cosh2 lambda x a delta function potential several cases of periodic delta function potentials and a delta function potential in a square well they also discuss the semiclassical approximation for the delta function potential

construction of triharmonic bezier surfaces from boundary conditions abstract the surface of partial differential equation pde surface is a surface that satisfies the pde with boundary conditions which can be applied in surface modeling and construction in this paper the construction of tensor product bezier surfaces of triharmonic equation from different boundary conditions is presented the internal control points of the resulting triharmonic bezier surface can be obtained uniquely by the given boundary condition some representative examples show the effectiveness of the presented method

effective computation of traces determinants and ζ functions for sturm liouville operators abstract the principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents and resolvent differences fredholm determinants functions and function regularized determinants associated with linear operators in a hilbert space in particular we detail the connection between fredholm and function regularized determinants concrete applications of our formalism to general i e three coefficient regular sturm liouville operators on bounded intervals with various separated and coupled boundary conditions and schrodinger operators on a half line are provided and further illustrated with an array of examples

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points

generating function density functional theory free energy functionals and direct correlation functions for hard spheres abstract a recent density functional theory for the inhomogeneous hard sphere fluid expresses the free energy in terms of a set of derivatives with respect to the particle radius of a simple generating function in the present paper this generating function approach gfa is reformulated in terms of general conditions on the coefficients of the differential operator involved in the theory this gives more flexibility to the theory and allows for ready derivation of new functionals that are tested in the calculation of density profiles of hard spheres near a hard wall comparison with the results of the previous formulation of the gfa shows a better performance of the new functionals and also gives a much deeper understanding of the structure of the theory and its connections with the fundamental measures theory of rosenfeld a simple modification of the gfa pair direct correlation functions yields explicit analytic expressions with very good agreement with simulation results