the klein gordon equation of generalized hulthen potential in complex quantum mechanics we have investigated the reality of exact bound states of complex and or pt symmetric non hermitian exponential type generalized hulthen potential the klein gordon equation has been solved by using the nikiforov uvarov method which is based on solving the second order linear differential equations by reduction to a generalized equation of hypergeometric type in many cases of interest negative and positive energy states have been discussed for different types of complex potentials

equivalence of unstable anharmonic oscillators and double wells we give a direct and general proof of the equality of the energy levels of a pair of different quantum problems the double well and the unstable anharmonic oscillator defined by complex translation the resonances of the two problems are also equal and are the analytic continuations of the anharmonic oscillator energy levels

green s functions of two dimensional anisotropic plates containing an elliptic hole abstract for a two dimensional anisotropic plate the green s function satisfying traction free boundary conditions around an elliptic hole is developed using stroh s formalism a combination of this function and the boundary element method shows that it is the most effective approach for solving hole problems the generality of the present green s function is shown by the broader meaning of the following words two dimensional includes not only in plane but also antiplane problems and the problems where in plane and anti plane deformations couple each other anisotropic which need not have any material symmetry restrictions means that it covers the solutions given in the literature which only deal with orthotropic or monoclinic materials elliptic includes the special case where the minor axis of the ellipse tends to zero i e the case of a griffith crack the accuracy of the numerical method presented is then verified by comparison with exact or accepted solutions of several examples such as an infinite or a finite plate with an elliptic hole or a crack under in plane or anti plane loading the materials used are isotropic orthotropic or laminated composites finally problems where the hole boundary is not traction free are solved such as rigid inclusions and pin loaded holes

pt symmetric pseudo perturbation recipe an imaginary cubic oscillator with spikes the pseudo perturbative shifted l expansion technique pslet is shown to be applicable in the non hermitian symmetric context the construction of bound states for several symmetric potentials is presented with special attention paid to v r ir3 ir oscillators

symmetrically regularized eckart poschl teller and hulthen potentials parity times time reversal symmetry of complex hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for hermiticity perhaps an equally important role is played by the related strengthened analyticity assumptions in a constructive illustration we complexify a few potentials solvable only in s wave then we continue their domain from the semi axis to the whole axis and obtain new exactly solvable models their energies turn out to be real as expected the new one dimensional spectra themselves differ quite significantly from their s wave predecessors

algebraic and scattering aspects of a symmetric solvable potential we study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions we determine the transmission and reflection coefficients for the symmetric case and also formulate the problem in terms of an su 1 1 potential group which allows unified treatment of the discrete and the continuous spectra in a natural way we find that bound and scattering states of the symmetric problem supply a basis for the unitary irreducible representations of the su 1 1 potential group and this gives a straightforward group theoretical interpretation of the fact that the complex invariant potential has a real energy spectrum

matching method and exact solvability of discrete symmetric square wells discrete symmetric square wells are studied their wavefunctions are found to be proportional to classical tshebyshev polynomials of complex argument the compact secular equations for energies are derived giving the real spectra in certain intervals of non hermiticity strengths z it is amusing to note that although the known square well re emerges in the usual continuum limit a twice as rich upside down symmetric spectrum is exhibited by all its present discretized predecessors

uniqueness theorems on meromorphic functions sharing one value in this paper we study with a weighted sharing method the uniqueness problem of f n z k and g n z k sharing one value and obtain some results which extend the theorems given by m fang s bhoosnurmath and s dyavanal et al

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

the cal pt symmetric rosen morse ii potential effects of the asymptotically non vanishing imaginary potential component bound and scattering solutions of the symmetric rosen morse ii potential are investigated the energy eigenvalues and the corresponding wavefunctions are written in a closed analytic form and it is shown that this potential always supports at least one bound state it is found that with increasing non hermiticity the real bound state energy spectrum does not turn into complex conjugate pairs i e the spontaneous breakdown of symmetry does not occur rather the energy eigenvalues remain real and shift to positive values closed expression is found for the pseudo norm of the bound states and its sign is found to follow the 1 n rule similarly to the known scattering examples the reflection coefficients exhibit a handedness effect while the transmission coefficient picks up a complex phase factor when the direction of the incoming wave is reversed it is argued that the unusual findings might be caused by the asymptotically non vanishing though finite imaginary potential component comparison with the real rosen morse ii potential is also made

uniqueness and value sharing of entire functions abstract in this paper we study the uniqueness problems on entire functions sharing one value with the same multiplicities we generalize and unify some previous results of fang and hua m l fang x h hua entire functions that share one value j nanjing univ math biquarterly 13 1 1996 44 48 yang and hua c c yang x h hua uniqueness and value sharing of meromorphic functions ann acad sci fenn math 22 2 1997 395 406 and fang m l fang uniqueness and value sharing of entire functions comput math appl 44 2002 828 831

green s functions for the biharmonic equation bonded elastic media a comprehensive formulation of green s functions for biharmonic equations with respect to perfectly bonded bimaterial linear isotropic elastic media is presented this is accomplished by extending mindlin s nuclei of strain method which employs the method of images the analysis involves properly scaling the nuclei grouping the nuclei into different classes and introducing a matrix structure for each class the strength vectors of the nuclei required at the object and image points to represent the solution are obtained by matrix vector multiplications involving two invariant matrices the reflection matrix and the transmission matrix algebraic expressions for the elements of the reflection and transmission matrices which depend on the elastic properties of the materials are presented

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

a new pt symmetric complex hamiltonian with a real spectrum we construct an isospectrum system in terms of a real and a complex potential to show that the underlying pt symmetric complex hamiltonian possesses a real spectrum which is shared by its real partner

complex square well a new exactly solvable quantum mechanical model recently a class ofpt invariant quantum mechanical models described by the non hermitian hamiltonianhdp2 x2 ix was studied it was found that the energy levels for this theory are real for all 0 here the limit as 1is examined it is shown that in this limit the theory becomes exactly solvable a generalization of this hamiltonian h d p 2 x 2m ix md 1 2 3 is also studied and thispt symmetric hamiltonian becomes exactly solvable in the large limit as well in effect what is obtained in each case is a complex analogue of the hamiltonian for the square well potential expansions about the large limit are obtained

the second main theorem for small functions and related problems we shall establish the following three results in more general forms 1 the second main theorem for small functions let f be a meromorphic function on the complex plane c let a1 aq be distinct meromorphic functions on c assume that ai are small with respect to f i e t r a 0 corollary 1 here as usual in nevanlinna theory the terms t r f and n r ai f denote for the characteristic function and the truncated counting function respectively 2 application to functional equations let kc be the field of meromorphic functions on c for a function r 0 r put k c a kc t r a o r which is a subfield of kc then the following holds let f x y k c x y be a polynomial in two variables over k c assume that the curve f x y 0 over k c has genus greater than one if 1 2 kc satisfy the functional equation f 1 2 0 then both 1 and 2 are contained in k c corollary 2 3 height inequality for curves over function fields let k be a function field of one variable over c let x be a smooth projective curve over k let d x be a reduced divisor let l be an ample line bundle on x and let e 0 then we have hk kx d p n 1 k s d p dk p ehk l p oe 1 for all p x k d theorem 5 here the notations are introduced in v1 v3 see also section 9 our proof uses ahlfors theory of covering surfaces and the geometry of the moduli space of q pointed stable curves of genus 0

bender wu branch points in the cubic oscillator the analytic continuation of the resonances of the cubic anharmonic oscillator to complex values of the coupling constant is studied with semiclassical and numerical methods bender wu branch points at which level crossing occurs are calculated and labelled by a process of analytic continuation the different resonances are the values that a single analytic function takes on different sheets of a riemann surface whose topology is described

uniqueness theorems for entire functions concerning fixed points this paper is devoted to studying the uniqueness problem of entire functions sharing one value or fixed points we improve some results given by fang and extend some results given by fang and qiu and by lin and yi

non hermitian matrix description of the calpcalt symmetric anharmonic oscillators the symmetric differential schrodinger equation h e with the operator h h x p2 ax4 i x3 cx2 i x h x on l2 is studied at a 0 it is rearranged as a linear algebraic diagonalization with rigorous proof our non variational construction of bound states offers an infinite dimensional analogue to the recent finite dimensional quasi exact solution available at the less common a 0

large order perturbation theory for a non hermitian pt symmetric hamiltonian a precise calculation of the ground state energy of the complex pt symmetric hamiltonian h p2 14x2 i x3 is performed using high order rayleigh schrodinger perturbation theory the energy spectrum of this hamiltonian has recently been shown to be real using numerical methods here we present convincing numerical evidence that the rayleigh schrodinger perturbation series is borel summable and show that pade summation provides excellent agreement with the real energy spectrum pade analysis provides strong numerical evidence that the once subtracted ground state energy considered as a function of 2 is a stieltjes function the analyticity properties of this stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable hamiltonian h p2 14 x2 ex3

fixed points and uniqueness of meromorphic functions in the paper we study the uniqueness and the shared fixed points of meromorphic functions and prove two main theorems which improve the results of fang and fang and qiu

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

on the existence of compressive solitary waves in compacting media necessary conditions for the existence of compressive solitary wave solutions of a partial differential equation derived by scott and stevenson 1986 which describes the two phase fluid flow in a medium compacting under gravity are derived it is shown that for compressive solitary wave solutions to exist which satisfy certain boundary conditions it is necessary that n m 1 where n and m are the exponents in power laws relating the permeability of the medium and the viscosity of the solid matrix respectively to the voidage the effect of the value of the exponents n and m on the shape of the solitary wave is investigated by using existing analytical solutions and new numerical solutions

complex calogero model with real energies recently the symmetry of many single particle non hermitian hamiltonians has been conjectured to be sufficient to keep their spectrum real we demonstrate this and how the similar concept of a weakened hermiticity can be extended to some exactly solvable two and three particle models

meromorphic functions that share fixed points abstract in this paper we prove the following result let f z and g z be two nonconstant meromorphic entire functions n 11 n 6 a positive integer if f n z f z and g n z g z have the same fixed points then either f z c 1 e cz 2 g z c 2 e cz 2 where c 1 c 2 and c are three constants satisfying 4 c 1 c 2 n 1 c 2 1 or f z tg z for a constant t such that t n 1 1

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

exact transient solutions of buried dynamic point forces for elastic bimaterials the exact transient closed form solutions for applying time dependent point forces at a depth below the interface of dissimilar solids are obtained in this study the problem is solved by application of laplace transform methods the inverse transforms are evaluated by means of cagniard s method the equations of wave fronts for reflected and refracted waves induced from the bimaterial interface by the incident waves are determined in this study the corresponding static solutions of stresses and displacements are reduced from the obtained transient solutions numerical results of stress fields during the transient process are obtained for different cases and compared with the corresponding static values

a family of complex potentials with real spectrum we consider a two parameter non hermitian quantum mechanical hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions in contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane our model is integrable along a line parallel to the real axis

explicit wavefunctions for shape invariant potentials by operator techniques the authors obtain explicit expressions for the wavefunctions of all the known shape invariant potentials by using the operator method

uniqueness and value sharing of entire functions abstract in this paper we study the uniqueness of entire functions and prove the following theorem let f z and g z be two nonconstant entire functions n k two positive integers with n 2 k 4 if f n z k and g n z k share 1 with counting the multiplicity then either f z c 1 e cz g z c 2 e cz where c 1 c 2 and c are three constants satisfying 1 k c 1 c 2 n nc 2 k 1 or f z tg z for a constant t such that t n 1

uniform asymptotic and jwkb expansions for anharmonic oscillators we show explicitly the relation between the uniform asymptotic and the jeffreys wentzel kramers brillouin jwkb wavefunctions and between the matching of uniform asymptotic expansions and the complete jwkb connection formulae written in terms of stokes multipliers and loop integrals as an application we give a unified derivation of the asymptotic behaviour of the imaginary part of the resonances in anharmonic oscillators and via dispersion relations the corresponding asymptotic behaviour of the rayleigh schrodinger perturbation theory coefficients

uniqueness of meromorphic functions and differential polynomials in this paper we study the uniqueness theorem of meromorphic functions concerning differential polynomials and obtain two theorems which improve and generalize the related results of fang s s bhoosnurmath and r s dyavanal and xiao yu zhang and wei chuan lin

poschl teller paradoxes in the well known poschl teller trigonometric potential well a symmetric regularization x x ie of the impenetrable end point barriers is performed this leads to the four different solvable generalizations of the model as a byproduct the scheme clarifies certain apparent paradoxes encountered in the classically forbidden coupling regime

spiked and symmetrized decadic potentials supporting elementary n plets of bound states we show that the potential wells v x x10 ax8 bx6 cx4 dx2 f x2 with a central spike possess arbitrary finite multiplets of elementary exact bound states their strong asymptotic growth implies an ambiguity in their symmetrically generalized quantization via complex boundary conditions but the three eligible methods coincide at our exceptional solutions

eigenvalue bounds for a class of singular potentials in n dimensions the eigenvalue bounds obtained earlier hall r l and saad n 1998 j phys a math gen 31 963 for smooth transformations of the form are extended to n dimensions in particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator potential and is valid for all discrete eigenvalues arbitrary angular momentum l and spatial dimension n

uniqueness of meromorphic functions sharing one value in this paper we study the uniqueness of meromorphic resp entire functions sharing one value and obtain some results which improve and generalize the related results due to fang bhoosnurmath dyavanal yang hua fang hua bai han etc

uniqueness of meromorphic functions sharing a small function and its applications in this paper we shall study the uniqueness problems of meromorphic functions sharing a small function our results improve or generalize many previous results on value sharing of meromorphic functions

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

stress analysis in plane orthotropic material by the boundary element method abstract the boundary element method is a more suitable technique than the finite element method for problems having large stress gradients one such problem is the stresses around mechanical fasteners to solve a problem by the boundary element method requires a solution of an integral equation the integrand of the integral equation is a product of a known green s function and an unknown function unlike isotropic material the plane orthotropic material can have three forms of green s function depending upon the relationship of the four material constants to solve the integral equations numerically the unknown function is approximated by a piecewise continuous linear function the boundary is approximated by a sum of straight line segments the result of the two approximations is integrals over straight line segments the integrand of which is a product of linear polynomials and one of the three green s functions these integrals arc evaluated analytically by exploiting the common features in the three forms of green s function a very efficient algorithm can be designed numerical results arc presented for a circular hole in an infinite medium and in a coupon the results show good correlation with analytical results for all kinds of orthotropic materials

a boundary element method for determining the effect of holes on the stress distribution around a crack a boundary integral procedure is obtained for examining the effect of a finite number of holes on the crack tip stress intensity factors for a plane crack in a homogeneous anisotropic elastic material numerical results for specific examples involving a transversely isotropic material are given

entire or meromorphic functions sharing one value in this paper we deal with the uniqueness problems on entire or meromorphic functions concerning differential polynomials that share one value with the same multiplicities moreover we greatly generalize some results obtained by fang lin and yi fang and fang

shape invariant potentials with pt symmetry suitable complexification of the well known hyperbolic rosen morse oscillator in one dimension is shown to give the fourth and last exactly solvable model which combines the shape and invariance

generalized swanson models and their solutions we analyse a class of non hermitian quadratic hamiltonians which are of the form where are real constants with and and are generalized creation and annihilation operators respectively thus these hamiltonians may be classified as generalized swanson models it is shown that the eigenenergies are real for a certain range of values of the parameters a similarity transformation mapping the non hermitian hamiltonian h to a hermitian one h is also obtained it is shown that h and h share identical energies as explicit examples the solutions of a couple of models based on the trigonometric rosen morse i and the hyperbolic rosen morse ii type potentials are obtained we also study the case where the non hermitian hamiltonian is symmetric

singularities interfaces and cracks in dissimilar anisotropic media for a non pathological bimaterial in which an interface crack displays no oscillatory behaviour it is observed that apart possibly from the stress intensity factors the structure of the near tip field in each of the two blocks is independent of the elastic moduli of the other block collinear interface cracks are analysed under this non oscillatory condition and a simple rule is formulated that allows one to construct the complete solutions from mode iii solutions in an isotropic homogeneous medium the general interfacial crack tip field is found to consist of a two dimensional oscillatory singularity and a one dimensional square root singularity a complex and a real stress intensity factors are proposed to scale the two singularities respectively owing to anisotropy a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields however defined does not recover the classical stress intensity factors as the bimaterial degenerates to be non pathological collinear crack problems are also formulated in this context and a strikingly simple mathematical structure is identified interactive solutions for singularity interface and singularity interface crack are obtained the general results are specialized to decoupled antiplane and in plane deformations for this important case it is found that if a material pair is non pathological for one set of relative orientations of the interface and the two solids it is non pathological for any set of orientations for bonded orthotropic materials an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized a complex variable representation is presented for a class of degenerate orthotropic materials throughout the paper the equivalence of the lekhnitskii and stroh formalisms is emphasized the article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids

symmetric effective mass schrodinger equations we outline a general method of obtaining exact solutions of symmetric schrodinger equations with a position dependent effective mass using this method exact solutions of some symmetric potentials have been obtained we have also discussed the construction of isospectral symmetric potentials

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

uniqueness of meromorphic functions sharing one small function in this paper we study the uniqueness problem of nonlinear differential polynomials of meromorphic functions that share one small function a z our results extend the corresponding result obtained by zhang et al 2008

on the completeness of the quasinormal modes of the poeschl teller potential the completeness of the quasinormal modes of the wave equation with poschl teller potential is investigated a main result is that after a large enough time t 0 the solutions of this equation corresponding to c data with compact support can be expanded uniformly in time with respect to the quasinormal modes thereby leading to absolutely convergent series explicit estimates for t 0 depending on both the support of the data and the point of observation are given for the particular case of an early time and zero distance between the support of the data and observational point it is shown that the corresponding series is not absolutely convergent and hence that there is no associated sum which is independent of the order of summation

pt symmetric potentials and the so 2 2 algebra starting from a differential realization of the generators of the so 2 2 algebra we connect the eigenvalue equation of the casimir invariant either with the hypergeometric equation or the schr odinger equation in the latter case we consider problems for whichso 2 2 appears as a potentia la lgebra connecting states with the same energy in different potentials we analyse the role of the two so 2 1 subalgebras and point out their importance for pt symmetric problems where the doubling of bound states is known to occur we present two mechanisms for this and illustrate them with the example of the scarf and the p oschl teller ii potentials we also analyse scattering states transmission

new class of conditionally exactly solvable potentials in quantum mechanics motivated by an idea of dutra 1993 we obtain a new class of one dimensional conditionally exactly solvable potentials for which the entire spectra can be obtained in an algebraic manner provided one of the potential parameters is assigned a fixed negative value it is shown that using shape invariant potentials as input one may generate different classes of such potentials even in more than one dimension we also illustrate that wkb and supersymmetry inspired wkb methods provide very good approximations for these potentials with the latter doing comparatively better

new exactly solvable isospectral partners for symmetric potentials we examine in detail the possibility of applying the darboux transformation to non hermitian hamiltonians in particular we propose a simple method of constructing exactly solvable symmetric potentials by applying darboux transformation to higher states of an exactly solvable symmetric potential it is shown that the resulting hamiltonian and the original one are pseudo supersymmetric partners we also discuss application of the darboux transformation to hamiltonians with spontaneously broken symmetry

on the multiple values and uniqueness of meromorphic functions on annuli the purpose of this article is to deal with the multiple values and uniqueness of meromorphic functions on annuli we prove a general theorem on the multiple values and uniqueness of meromorphic functions on annuli from which an analog of nevanlinna s famous five value theorem is proposed

image singularities of green s functions for isotropic elastic bimaterials subjected to concentrated forces and dislocations abstract green s functions for isotropic materials in the two dimensional problem for elastic bimaterials with perfectly bonded interface are reexamined in the present study although the green s function for an isotropic elastic bimaterial subjected to a line force or a line dislocation has been discussed by many authors the physical meaning and the structure of the solution are not clear in this investigation the green s function for an elastic bimaterial is shown to consist of eight green s functions for a homogeneous infinite plane one of the novel features is that green s functions for bimaterials can be expressed directly by knowing green s functions for the infinite plane if the applied load is located in material 1 the solution for the half plane of material 1 is constructed with the help of five green s functions corresponding to the infinite plane however the solution for the half plane of material 2 only consists of three green s functions for the infinite plane one of the five green s functions of material 1 and all the three green s functions of material 2 have their singularities located in the half plane where the load is applied and the other four image singularities of material 1 are located outside the half plane at the same distance from the interface as that of the applied load the nature and magnitude of the image singularities for both materials are presented explicitly from the principle of superposition and classified according to different loads it is known that for the problem of anisotropic bimaterials subjected to concentrated forces and dislocations the image singularities are simply concentrated forces and dislocations with the stress singularity of order o 1 r however higher orders o 1 r2 and o 1 r3 of stress singularities are found to exist in this study for isotropic bimaterials the highest order of the stress singularity is o 1 r3 for the image singularities of material 1 and is o 1 r2 for material 2 using the present solution green s functions associated with the problems of elastic half plane with free and rigidly fixed boundaries for homogeneous isotropic elastic solid are obtained as special cases

variational and perturbative schemes for a spiked harmonic oscillator a variational analysis of the spiked harmonic oscillator hamiltonian operator d2 dx2 x2 l l 1 x2 x where is a real positive parameter is reported in this work the formalism makes use of the functional space spanned by the solutions of the schrodinger equation for the linear harmonic oscillator hamiltonian supplemented by a dirichlet boundary condition and a standard procedure for diagonalizing symmetric matrices the eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system valid for positive and relatively large values of the coupling parameter additionally a large coupling perturbative expansion is carried out and the contributions up to fourth order to the ground state energy are explicitly evaluated numerical results are compared for the special case 5 2

green functions of piezoelectric material with an elliptic hole or inclusion abstract in this paper the green functions of infinite two dimensional piezoelectric material containing an elliptic hole or an elliptic piezoelectric inclusion are obtained with the extended stroh formalism the results can be used to investigate the mechanical electric coupling behaviours of infinite piezoelectric material with hole or inclusion acted by a set of generalized concentrated forces and more importantly as kernels of boundary integral equations in bem analyses for finite size piezoelectric materials with general forces and boundary conditions