one dimensional schrodinger operators with singular potentials a schwartz distributional formulation using an extension of the hormander product of distributions we obtain an intrinsic formulation of one dimensional schrodinger operators with singular potentials this formulation is entirely defined in terms of standard schwartz distributions and does not require as some previous approaches the use of more general distributions or generalized functions we determine in the new formulation the action and domain of the schrodinger operators with arbitrary singular boundary potentials we also consider the inverse problem and obtain a general procedure for constructing the singular pseudo potential that imposes a specific set of local boundary conditions this procedure is used to determine the boundary operators for the complete four parameter family of one dimensional schrodinger operators with a point interaction finally the and potentials are studied in detail and the corresponding schrodinger operators are shown to coincide with the norm resolvent limit of specific sequences of schrodinger operators with regular potentials

an example of unitary equivalence between self adjoint extensions and their parameters abstract the spectral problem for self adjoint extensions is studied using the machinery of boundary triplets for a class of symmetric operators having weyl functions of a special type we calculate explicitly the spectral projections in the form of operator valued integrals this allows one to give a constructive proof of the fact that in certain intervals the resulting self adjoint extensions are unitarily equivalent to the parameterizing boundary operator acting in a smaller space and one is able to provide an explicit form for the associated unitary transform applications to differential operators on metric graphs and to direct sums are discussed

a new class of solvable models in quantum mechanics describing point interactions on the line the authors provide a detailed analysis of properties of the schrodinger operator in l2 r which formally can be written as d2 dx2 sigma y in gamma nu y delta y where delta is the derivative of dirac s delta function and y contained in implied by r is discrete this model allows for an explicit calculation of spectral properties special emphasis is given to the periodic case y z nu nu j j in z where the spectrum and the density of states are explicitly computed also the spectrum for a half crystal is given the authors study in detail spectral consequences when various defects and impurities are added to the periodic case

spectral theory of semibounded sturm liouville operators with local interactions on a discrete set we study the hamiltonians hx q with type point interactions at the centers xk on the positive half line in terms of energy forms we establish analogs of some classical results on operators hq d2 dx2 q with locally integrable potentials q lloc1 r in particular we prove that the hamiltonian hx q is self adjoint if it is lower semibounded this result completes the previous results of brasche perturbation of schrodinger hamiltonians by measures selfadjointness and semiboundedness j math phys 26 621 1985 on lower semiboundedness also we prove the analog of molchanov s discreteness criteria birman s result on stability of a continuous spectrum and investigate discreteness of a negative spectrum in the recent paper kostenko a and malamud m 1 d schrodinger operators with local point interactions on a discrete set j differ equations 249 253 2010 it was shown that the spectral properties of hx hx 0 correlate with the corresponding spectral properties of a certain clas

bonded dissimilar strips with a crack perpendicular to the functionally graded interface the mode i crack problem for two dissimilar infinite strips bonded through a functionally graded interfacial zone is investigated within the framework of plane elasticity the graded interfacial zone is treated as a nonhomogeneous continuum having the continuously varying elastic modulus between the dissimilar homogeneous strips a crack is assumed to be embedded in one of the strips perpendicular to the nominal interface with the aid of the stiffness matrix approach a set of homogeneous conditions relevant to the proposed crack problem is readily satisfied subsequent application of the mixed conditions on the cracked plane then leads to a singular integral equation of the first kind which is solved numerically in consequence the variations of stress intensity factors are provided as functions of geometric and material parameters of the bonded structure the effect of the material nonhomogeneity in the interfacial zone is further addressed by measuring the degree of correspondence with the results that are obtained based on the use of the homogenized interfacial elastic property

spectral theory of schrodinger operators with infinitely many point interactions and radial positive definite functions abstract a number of results on radial positive definite functions on r n related to schoenberg s integral representation theorem are obtained they are applied to the study of spectral properties of self adjoint realizations of two and three dimensional schrodinger operators with countably many point interactions in particular we find conditions on the configuration of point interactions such that any self adjoint realization has purely absolutely continuous non negative spectrum we also apply some results on schrodinger operators to obtain new results on completely monotone functions

characterization of one dimensional point interactions for the schrodinger operator by means of boundary conditions we find a new representation of all boundary conditions corresponding to the so called point interactions we show that there is one to one correspondence between one dimensional point interactions and boundary conditions of the form with satisfying 2 2 1

minima of quasisuperminimizers abstract we show that unlike minima of superharmonic functions which are again superharmonic the same property fails for q quasisuperminimizers more precisely if u i is a q i quasisuperminimizer i 1 2 where 1 q 1 q 2 then u min u 1 u 2 is a q quasisuperminimizer but there is an increase in the optimal quasisuperminimizing constant q we provide the first examples of this phenomenon i e that q q 2 in addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to kinnunen and martio moreover our lower and upper bounds turn out to be quite close we also study a similar phenomenon in pasting lemmas for quasisuperminimizers where q q 1 q 2 turns out to be optimal and provide results on exact quasiminimizing constants of piecewise linear functions on the real line which can serve as approximations of more general quasiminimizers

an exactly solvable periodic schrodinger operator the authors explicitly determine the band spectrum of a periodically strongly singular schrodinger operator in l2 r associated with the differential expression d2 dx2 s2 1 4 cos2 x s 2 1 4 sin2 x x epsilon r x epsilon pi z 2 0 s s 1 the corresponding density of states is calculated analytically

sharp exponents and a wiener type condition for boundary regularity of quasiminimizers abstract we obtain a sufficient condition for boundary regularity of quasiminimizers of the p energy integral in terms of a wiener type sum the exponent in the sum is independent of the dimension and is explicitly expressed in terms of p and the quasiminimizing constant we also show by an example that the exponent is sharp in a certain sense

one dimensional schrodinger operators with δ interactions on cantor type sets we introduce a novel approach for defining a interaction on a subset of the real line of lebesgue measure zero which is based on sturm liouville differential expression with measure coefficients this enables us to establish basic spectral properties e g self adjointness lower semiboundedness and spectral asymptotics of hamiltonians with interactions concentrated on sets of complicated structures

pt invariant point interactions in one dimension by using wu and yu s pseudo potential we construct point interactions in one dimension that are complex but conform to space time reflection invariance the resulting point interactions are equivalent to those obtained by albeverio fei and kurasov as self adjoint extensions of the kinetic energy operator

on the spectral theory of gesztesy seba realizations of 1 d dirac operators with point interactions on a discrete set abstract we investigate spectral properties of gesztesy seba realizations d x and d x of the 1 d dirac differential expression d with point interactions on a discrete set x x n n 1 r here n n 1 and n n 1 r the gesztesy seba realizations d x and d x are the relativistic counterparts of the corresponding schrodinger operators h x and h x with and interactions respectively we define the minimal operator d x as the direct sum of the minimal dirac operators on the intervals x n 1 x n then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator d x in the case d x inf x i x j i j 0 it turns out that the boundary operators b x and b x parameterizing the realizations d x and d x are jacobi matrices these matrices substantially differ from the ones appearing in spectral theory of schrodinger operators with point interactions we show that certain spectral properties of the operators d x and d x correlate with the corresponding spectral properties of the jacobi matrices b x and b x respectively using this connection we investigate spectral properties self adjointness discreteness absolutely continuous and singular spectra of gesztesy seba realizations moreover we investigate the non relativistic limit as the velocity of light c most of our results are new even in the case d x 0

modeling method for a crack problem of functionally graded materials with arbitrary properties piecewise exponential model abstract a new model piecewise exponential model pe model is developed to investigate the crack problem of the functionally graded materials fgms with arbitrary properties in the pe model the functionally graded material is divided into some nonhomogeneous layers along the gradient direction of the properties with each layer s properties varying exponentially by this way the real material properties can be approached by a series of exponential functions since the real material properties are used on both surfaces of each nonhomogeneous layer the nature of continuously varying properties of fgms can be approached accurately the influences of the local nonhomogeneity on the crack tip fields can be fully considered by using the new model the fracture problem of a functionally graded strip with arbitrary properties and a crack vertical to the free surfaces is studied the integral transform method the theory of residues and the theory of singular integral equation are applied some representative samples with different kinds of nonhomogeneous properties are analyzed and the corresponding stress intensity factors sifs are presented it is shown that the pe mode is effective for investigating the crack problems of the fgms with arbitrary properties

spectral enclosures for non self adjoint extensions of symmetric operators abstract the spectral properties of non self adjoint extensions a b of a symmetric operator in a hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding weyl functions these extensions are given in terms of abstract boundary conditions involving an in general non symmetric boundary operator b in the abstract part of this paper sufficient conditions for sectoriality and m sectoriality as well as sufficient conditions for a b to have a non empty resolvent set are provided in terms of the parameter b and the weyl function special attention is paid to weyl functions that decay along the negative real line or inside some sector in the complex plane and spectral enclosures for a b are proved in this situation the abstract results are applied to elliptic differential operators with local and non local robin boundary conditions on unbounded domains to schrodinger operators with potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line and to quantum graphs with non self adjoint vertex couplings

common eigenvalue problem and periodic schrodinger operators abstract let a be a subset of the family of all self adjoint extensions of a symmetric operator a0 with equal deficiency indices in a hilbert space assuming that a0 has a purely residual spectrum we describe the set of eigenvalues common to all self adjoint extensions from a this abstract result is used to show that the one dimensional periodic schrodinger operator with local point interactions is absolutely continuous

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

remarks on quasi sub minima on considere la notion de minimum d integrales variationnelles en liaison avec les solutions d equations elliptiques et de systemes sous forme divergence

a partially insulated interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate under heat flux supply in this paper thermal response of an orthotropic functionally graded coating substrate structure with a partially insulated interface crack under heat flux supply is considered it is assumed that there exists thermal resistance to heat conduction through the crack region the mixed boundary value problems are reduced to a system of singular integral equations and solved numerically higher order asymptotic terms for the singular integral kernels are considered to improve the accuracy and the convergence efficiency of the numerical integrals numerical results are presented to show the effects of the orthotropy parameters thermo elastic nonhomogeneity parameters and dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors sifs

boundary relations and their weyl families the concepts of boundary relations and the corresponding weyl families are introduced let s be a closed symmetric linear operator or more generally a closed symmetric relation in a hilbert space h let h be an auxiliary hilbert space let graphics and let jh be defined analogously a unitary relation g from the krein space h 2 j h to the kre in space h 2 j h is called a boundary relation for the adjoint s if ker gamma s the corresponding weyl family m lambda is de ned as the family of images of the defect subspaces n over cap lambda lambda is an element of c r under gamma here gamma need not be surjective and is even allowed to be multi valued while this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex hilbert space h and the class of unitary relations gamma h 2 j h h 2 j h it also generalizes the notion of so called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems moreover these new notions yield for instance the following realization theorem every h valued maximal dissipative for lambda is an element of c holomorphic family of linear relations is the weyl family of a boundary relation which is unique up to unitary equivalence if certain minimality conditions are satisfied further connections between analytic and spectral theoretical properties of weyl families and geometric properties of boundary relations are investigated and some applications are given

dynamic investigation of a functionally graded layered structure with a crack crossing the interface abstract the dynamic response of a functionally graded layered structure with a crack crossing the interface is analyzed the in plane impact loading condition is considered by using the laplace and fourier integral transforms singular integral equation method and residue theory the present problem is reduced to a singular integral equation in the laplace transform domain the influences of young s modulus ratio thickness ratio and crack length and location on the dynamic stress intensity factors dsifs are investigated particularly the dsifs corresponding to different crack locations are shown in the case when the crack center moves from one layer to another layer through the interface the peak and static values and overshoot characteristics of the dsifs are analyzed it is found that these values typically exhibit kinking behavior when the crack tips arrive at the interface this study is different from previous other investigations in the following respects 1 the dynamic response of a crack crossing the interface of a functionally graded structure is studied analytically which has hardly been done in the past and 2 the present model can be reduced to some important problems such as a functionally graded coating substrate structure with a crack in the graded coating or homogeneous substrate or one intersecting the interface

boundary regularity for quasiminima in gg giaquinta giusti introduced the notion of quasiminima a concept which is very useful in unifying the c 0 regularity theory of elliptic equations in divergence form in particular their treatment provides regularity results for minima of nondifferentiable variational integrals they also show that weak solutions of nonlinear elliptic equations in divergence form under natural and general structural conditions are quasiminima as are solutions to a wide class of obstacle problems because of the great interplay between quasiminima and the holder theory of partial differential equations it is important to develop the subject fully indeed significant progress has already been made recently for example harnack inequalities have been established in dbt for sub and super quasiminima which run parallel to the results obtained by trudinger t for weak sub and supersolutions of quasilinear equations in divergence form in particular the results in dbt imply that quasiminima are holder continuous a result first established by giaquinta giusti gg

a regularity classification of boundary points for p harmonic functions and quasiminimizers a regularity classification of boundary points for p harmonic funktions and quasiminimizers

1 d schrodinger operators with local point interactions on a discrete set abstract spectral properties of 1 d schrodinger operators h x d 2 d x 2 x n x n x x n with local point interactions on a discrete set x x n n 1 are well studied when d inf n k n x n x k 0 our paper is devoted to the case d 0 we consider h x in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding weyl functions we show that the spectral properties of h x like self adjointness discreteness and lower semiboundedness correlate with the corresponding spectral properties of certain classes of jacobi matrices based on this connection we obtain necessary and sufficient conditions for the operators h x to be self adjoint lower semibounded and discrete in the case d 0 the operators with type interactions are investigated too the obtained results demonstrate that in the case d 0 as distinguished from the case d 0 the spectral properties of the operators with and type interactions are substantially different

δ interaction as a limit of a thin neumann waveguide with transversal window abstract we consider a waveguide like domain consisting of two thin straight tubular domains connected through a tiny window the perpendicular size of this waveguide is of order e under the assumption that the window is appropriately scaled we prove that the neumann laplacian on this domain converges in a kind of norm resolvent sense as e 0 to a one dimensional schrodinger operator corresponding to a interaction of a non negative strength we estimate the rate of this convergence also we prove the convergence of spectra

thermoelastic problem of steady state heat flows disturbed by a crack at an arbitrary angle to the graded interfacial zone in bonded materials plane thermoelasticity solutions are presented for the problem of a crack in bonded materials with a graded interfacial zone the interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermoelastic moduli between dissimilar homogeneous half planes the crack is assumed to exist in one of the half planes at an arbitrary angle to the graded interfacial zone disturbing uniform steady state heat flows the fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic thermoelasticity equations formulation of the current nonisothermal crack problem lends itself to the derivation of two sets of cauchy type singular integral equations for heat conduction and thermal stress analyses the heat flux intensity factors and the thermal stress intensity factors are defined and evaluated in order to quantify the singular characters of temperature gradients and thermal stresses respectively in the near tip region numerical results include the variations of such crack tip field intensity factors versus the crack orientation angle for various combinations of material and geometric parameters of the dissimilar media bonded through the thermoelastically graded interfacial zone the dependence of the near tip thermoelastic singular field on the degree of crack surface partial insulation is also addressed

spherical schrodinger operators with δ type interactions we investigate spectral properties of spherical schrodinger operators also known as bessel operators with point interactions concentrated on a discrete set we obtain necessary and sufficient conditions for these hamiltonians to be self adjoint lower semibounded and also we investigate their spectra we also extend the classical bargmann estimate to such hamiltonians in certain cases we express the number of negative eigenvalues explicitly by means of point interactions and the corresponding intensities we apply our results to schrodinger operators in l2 rn with a singular interaction supported by an infinite family of concentric spheres

one dimensional schrodinger operators with interactions singular on a discrete set on etudie le caractere auto adjoint des operateurs de schrodinger sur y u j i x j y j ou on suppose y j x j 1

on the unitary equivalence of absolutely continuous parts of self adjoint extensions the classical weyl von neumann theorem states that for any self adjoint operator a0 in a separable hilbert space h there exists a non unique hilbert schmidt operator c c such that the perturbed operator a0 c has purely point spectrum we are interesting whether this result remains valid for non additive perturbations by considering the set exta of self adjoint extensions of a given densely defined symmetric operator a in h and some fixed a0 a0 exta we show that the ac parts a ac and a0ac of a a exta and a0 are unitarily equivalent provided that the resolvent difference ka a i 1 a0 i 1 is compact and the weyl function m of the pair a a0 admits weak boundary limits m t w limy 0m t iy for a e t r this result generalizes the classical kato rosenblum theorem moreover it demonstrates that for such pairs a a0 the weyl von neumann theorem is in general not true in the class exta

sturm liouville boundary value problems with operator potentials and unitary equivalence consider the minimal sturm liouville operator a amin generated by the differential expression a d2dt2 t in the hilbert space l2 r h where t t 0 in h we investigate the absolutely continuous parts of different self adjoint realizations of a in particular we show that dirichlet and neumann realizations ad and an are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the krein realization moreover if inf ess t inf t then the part a acea ad of any self adjoint realization a of a is unitarily equivalent to ad in addition we prove that the absolutely continuous part a ac of any realization a is unitarily equivalent to ad provided that the resolvent difference a i 1 ad i 1 is compact the abstract results are applied to elliptic differential expressions in the half space