modified numerov s method for inverse sturm liouville problems in this paper we propose a new modified numerov s method for recovering from eigenvalues a symmetric potential of a sturm liouville operator with dirichlet boundary conditions we use interpolation to refine the mesh sufficiently for numerov s method to be effective even without the asymptotic correction technique of andrew and paine accuracy and stability of the method are investigated convergence of the method is established our method is extended to deal with natural boundary conditions numerical experiments confirm its competitiveness

generalized point interactions in one dimensional quantum mechanics there is a four parameter family of point interactions in one dimensional quantum mechanics they represent all possible self adjoint extensions of the kinetic energy operator if time reversal invariance is imposed the number of parameters is reduced to three one of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known we present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so called interaction we emphasize that there is little resemblance between the interaction and what its name suggests

asymptotic correction of numerov s eigenvalue estimates with natural boundary conditions abstract asymptotic correction at negligible extra cost greatly improves the accuracy of higher eigenvalues computed by finite difference or finite element methods and generally increases the accuracy of the lower ones as well this paper gives a brief overview of the technique and describes how its previous use with numerov s method may be extended to problems with natural boundary conditions numerical results indicate that it is just as successful as with dirichlet boundary conditions

reconstruction techniques for classical inverse sturm liouville problems this paper gives constructive algorithms for the classical inverse sturm liouville problem it is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator two methods of solving this latter problem are then provided and numerical examples are presented

potential approximations to δ an inverse klauder phenomenon with norm resolvent convergence we show that there is a family schrodinger operators with scaled potentials which approximates the interaction hamiltonian in the norm resolvent sense this approximation based on a formal scheme proposed by cheon and shigehara has nontrivial convergence properties which are in several respects opposite to those of the klauder phenomenon

correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

spectral asymptotics for schrodinger operators with periodic point interactions abstract spectrum of the second order differential operator with periodic point interactions in l 2 r is investigated classes of unitary equivalent operators of this type are described spectral asymptotics for the whole family of periodic operators are calculated it is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely it is proven that the spectrum of the operators with anomalous spectral asymptotics when the ratio between the lengths of the bands and gaps tends to zero at infinity can be approximated by standard periodic weighted operators with step wise density functions it is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal generalized resolvent of the periodic schrodinger operator with certain energy dependent boundary conditions the operator acting in an extended hilbert space such that its resolvent restricted to l 2 r coincides with the formal resolvent is constructed explicitly

matrix methods for computing eigenvalues of sturm liouville problems of order four abstract this paper examines and develops matrix methods to approximate the eigenvalues of a fourth order sturm liouville problem subjected to a kind of fixed boundary conditions furthermore it extends the matrix methods for a kind of general boundary conditions the idea of the methods comes from finite difference and numerov s methods as well as boundary value methods for second order regular sturm liouville problems moreover the determination of the correction term formulas of the matrix methods is investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q 0 are known in this case finally some numerical examples are illustrated

distribution theory for discontinuous test functions and differential operators with generalized coefficients investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions such a distribution theory for test functions having discontinuity at one point is developed a four parameter family of schrodinger operators formed by the operators with singular potential singular metrics and singular gauge field is considered it is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point approximation by operators with smooth coefficients is discussed c 1996 academic press inc

the absence of the absolutely continuous spectrum for δ wannier stark ladders a modification of the kronig penney model consisting of equidistantly spaced interactions is considered we prove that an absolutely continuous spectrum of such a system disappears under the influence of an external electric field the result extends to periodic lattices of nonidentical interactions and potentials which are lower unbounded and up to a bounded term asymptotically decreasing with bounded first two derivatives

the generalized star product and the factorization of scattering matrices on graphs in this article we continue our analysis of schrodinger operators on arbitrary graphs given as certain laplace operators in the present article we give the proof of the composition rule for the scattering matrices this composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs we perform a detailed analysis of the generalized star product for arbitrary unitary matrices the relation to the theory of transfer matrices is also discussed

strong unique continuation for higher order elliptic equations with gevrey coefficients article history we address the strong unique continuation problem for higher order elliptic partial differential equations in 2d with gevrey co efficients we provide a quantitative estimate of unique continu ation observability estimate and prove that the solutions satisfy the strong unique continuation property for ranges of the gevrey exponent strictly including non analytic gevrey classes as an ap plication we obtain a new upper bound on the hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients published by elsevier inc

convergence of numerov s method for inverse sturm liouville problems in this paper we discuss the convergence of numerov s method in andrew 2005 2006 8 9 for computing sturm liouville potentials from the given eigenvalues by using the asymptotic estimate for the eigenvalue of the sturm liouville problem and the error in the finite difference eigenvalue convergence of numerov s method for symmetric potentials is proved based on the method of symmetric extension we establish a convergence result of numerov s method for the nonsymmetric potential from two spectra numerical examples are reported to confirm the theoretically predicted convergence

multiple eigenvalue sensitivity analysis abstract this paper surveys the development of the sensitivity analysis of multiple eigenvalues and the associated invariant subspaces of two kinds of matrix eigenvalue problems in which the matrices are analytically dependent on several parameters

four parameter point interaction in 1d quantum systems the author constructs a four parameter point interaction for a nonrelativistic particle moving on a line as the limit of a short range interaction with a range tending toward zero for particular choices of the parameters one can obtain a delta interaction or the so called delta interaction the hamiltonian corresponding to the four parameter point interaction is shown to correspond to the four parameter self adjoint hamiltonian of the free particle moving on the line with the origin excluded

boundary value method for inverse sturm liouville problems in this paper we present a method to recover symmetric and non symmetric potential functions of inverse sturm liouville problems from the knowledge of eigenvalues the linear multistep method coupled with suitable boundary conditions known as boundary value method bvm is the main tool to approximate the eigenvalues in each iteration step of the used newton method the bvm was extended to work for neumann neumann boundary conditions moreover a suitable approximation for the asymptotic correction of the eigenvalues is given numerical results demonstrate that the method is able to give good results for both symmetric and non symmetric potentials

exponentially fitted numerov methods the error in the estimate of the kth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh length h is o k 6h 4 it is shown that the error can be reduced to o k 3h 4 by using one of the three versions of the exponentially fitted numerov method numerical examples demonstrate the usefulness of this approach even for low values of k

the inverse sturm liouville problem ii soit q q 0 2 l r 2 0 1 le probleme de sturm liouville y q x y y 0 x 1 y 0 cos y 0 sin 0 y 1 cos y 1 sin 0 a un spectre discret de valeurs propres simples 0 q 1 q avec des fonctions propres h 0 x q h 1 x q on fixe p dans 0 2 l r 2 0 1 et soit m p q 0 2 l r 2 0 1 n q n p n 0 l espace de tous les points q de meme spectre que p on etudie m p comme une sous variete de 0 2 l r 2 0 1

strong unique continuation for products of elliptic operators of second order we study strong unique continuation for products of elliptic operators the main tools are carleman inequalities for second order elliptic operators we obtain strong unique continuation assuming either gevrey regularity of the coefficients or some pointwise conditions on the coefficients

recovery of the m function from spectral data for generalized sturm liouville problems the sturm liouville problem y qy y y 0 cos y 0 sin y y 1 h g is studied where h and g are real polynomials generalized norming constants p n k associated with eigenvalues n are defined and formulae are given for recovering the m function from these constants this leads to a uniqueness theorem for the associated inverse problem

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

unique continuation for some evolution equations this work is concerned with unique continuation results for some evolution equations essentially of parabolic type to be more precise let l be an evolution operator acting on functions defined on some connected open set b of r r x r l is said to have the unique continuation property if every solution u of lu 0 which vanishes one some nonempty open set u i of fl vanishes in the horizontal component of o in 0 i e in x t e 0 3x x t e co the first general result of this type was established by mizohata 6 in the case where l is a second order parabolic operator with smooth coefficients however mizohata s proof is somewhat delicate it is based in particular on an extended class of calderon zygmund operators and does not lead obviously to a weakening of the smoothness of the coefficients on the other hand it is important for the applications to deal with nonsmooth coefficients since l is often obtained as the linearization of some nonlinear operator at a nonnecessarily smooth solution we refer to lions 4 and kernevez and lions 3 for applications to control theory and to 1 l for applications to transversality techniques let us now describe the content of this work in the first section we prove a unique continuation theorem when l is a second order parabolic equation our proof is simple and based on the derivation of a carleman estimate which is reminiscent of the classical carleman estimates for second order elliptic operators see 121 this carleman inequality allows the weakening of the 118 0022 0396187 3 00

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

boundary conditions in path integrals from point interactions for the path integral of the one dimensional dirac particle a proposal is given of how to implement point interactions and boundary conditions in the path integral the starting point is a path integral formulation of a dirac particle in one dimension the implementation of a point interaction yields by means of a perturbation expansion the corresponding green function for a relativistic particle in the non relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen by a proper combination of the various possibilities of implementing the point interaction the whole range of the four parameter family of boundary conditions for point interactions can be exploited in the relativistic case as well as in the non relativistic limit in addition making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line in particular dirichlet and neumann boundary conditions emerge the method is illustrated with some examples

on quantitative uniqueness for elliptic equations we address the question of quantitative uniqueness for the equation delta u v u with either periodic or dirichlet boundary conditions in a disk we construct solutions u corresponding to potential functions v such that u vanishes of order mathrm const vert v vert l infty 2 3 the example also shows sharpness of recently obtained bounds in the case of a parabolic equation u t delta u v u

accurate numerical bounds for the spectral points of singular sturm liouville problems over 0 x the eigenvalues of singular sturm liouville problems defined over the semi infinite positive real axis are examined on a truncated interval 0 x l as functions of the boundary point l as a basic theoretical result it is shown that the eigenvalues of the truncated interval problems satisfying dirichlet and neumann boundary conditions provide respectively upper and lower bounds to the eigenvalues of the original problem moreover the unperturbed system in a perturbation problem where l remains sufficiently small admits analytical solutions in terms of the bessel functions of the first kind applications to the schrodinger equations of diatomic molecules and a harmonic oscillator confirm the practical implementation of this approach in calculating highly accurate numerical eigenvalue enclosures it is worth mentioning that this study is therefore a completion of the paper j comput appl math 115 2000 535 where similar problems on the whole real axis x were treated along the same lines

distribution theory for schrodinger s integral equation much of the literature on point interactions in quantum mechanics has focused on the differential form of schrodinger s equation this paper in contrast investigates the integral form of schrodinger s equation while both forms are known to be equivalent for smooth potentials this is not true for distributional potentials here we assume that the potential is given by a distribution defined on the space of discontinuous test functions first by using schrodinger s integral equation we confirm a seminal result by kurasov which was originally obtained in the context of schrodinger s differential equation this hints at a possible deeper connection between both forms of the equation we also sketch a generalisation of kurasov s j math anal appl 201 1 297 323 1996 result to hypersurfaces second we derive a new closed form solution to schrodinger s integral equation with a delta prime potential this potential has attracted considerable attention including some controversy interestingly the derived propagator satisfies boundary conditions that were previously derived using schrodinger s differential equation third we derive boundary conditions for super singular potentials given by higher order derivatives of the delta potential these boundary conditions cannot be incorporated into the normal framework of self adjoint extensions we show that the boundary conditions depend on the energy of the solution and that probability is conserved this paper thereby confirms several seminal results and derives some new ones in sum it shows that schrodinger s integral equation is a viable tool for studying singular interactions in quantum mechanics

small potential corrections for the discrete eigenvalues of the sturm liouville problem the inverse sturm liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem y qy y holds on 0 fory 0 y 0 and a set of given eigenvalues since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding sturm liouville eigenvalues asymptotic corrections for the s serve to get better estimates forq let k 1 k n be the first eigenvalues of let k be the corresponding discrete eigenvalues obtained by the finite element method for and let k k for the special caseq 0 then starting from an asymptotic correction technique proposed by paine de hoog and anderssen new estimates for the errors bar lambda k lambda k of the corrected discrete eigenvalues bar lambda k lambda k mu k k 2 are obtained and confirm and improve the knowno kh 2 h n 1 behaviour the estimates are based on new sobolev inequalities and on fourier analysis and it is shown that bar lambda k lambda k leqq c 1 kh 2 for 4 c 2 k n 1 2 wherec 1 andc 2 are constants depending onq which tend to 0 for vanishingq

numerical solution of the inverse spectral problem for bessel operators we consider some inverse spectral problems associated with the singular sturm liouville equation u q x 1 x 2 u lu0

on the existence of resonances in the transmission probability for interactions arising from derivatives of dirac s delta function the scattering properties of regularizing finite range potentials constructed in the form of squeezed rectangles which approximate the first and second derivatives of the dirac delta function x are studied in the zero range limit particularly for a countable set of interaction strength values a non zero transmission through the point potential x defined as the weak limit in the standard sense of distributions of a special dipole like sequence of rectangles is shown to exist when the rectangles are squeezed to zero width a tripole sequence of rectangles which gives in the weak limit the distribution x is demonstrated to exhibit the total transmission for a countable sequence of the rectangle s width that tends to zero however this tripole sequence does not admit a well defined point interaction in the zero range limit making sense only for a finite range of the regularizing rectangular like potentials

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

bvms for computing sturm liouville symmetric potentials abstract the paper deals with the numerical solution of inverse sturm liouville problems with unknown potential symmetric over the interval 0 the proposed method is based on the use of a family of boundary value methods obtained as a generalization of the numerov scheme aimed to the computation of an approximation of the potential belonging to a suitable function space of finite dimension the accuracy and stability properties of the resulting procedure for particular choices of such function space are investigated the reported numerical experiments put into evidence the competitiveness of the new method

resonant tunnelling through short range singular potentials a three parameter family of point interactions constructed from sequences of symmetric barrier well barrier and well barrier well rectangles is studied in the limit when the rectangles are squeezed to zero width but the barrier height and the well depth become infinite the zero range limit the limiting generalized potentials are referred to as the second derivative of dirac s delta function x with a renormalized coupling constant 0 or simply as like point interactions as a result a whole family of self adjoint extensions of the one dimensional schrodinger operator is shown to exist which results in full and partial resonant tunnelling through this class of singular potentials the resonant tunnelling occurs for countable sets of interaction strength values in the space which are the roots of several transcendental equations the comparison with the previous results for like point interactions is also discussed

classifying the general family of 1d point interactions a scattering approach we propose a classification scheme for the complete family of 1d point interactions to do so we first review the solutions of the wavefunctions and green functions of the problem second we derive the exact time dependent propagators in such a way that we can write the expressions for the k s in a very compact form as they should such expressions do reduce to the known results in the literature according to the potential parameter values then we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets finally we show that the physical features associated with the scattering process can be used to establish a classification procedure moreover these physical characteristics are directly related to the potential parameters leading to the many formulae for the k s as an application we present numerical calculations for gaussian wave packets

the inverse sturn liouville problem iii on considere la theorie spectrale inverse du probleme de sturm liouville y q x y y 0 x 1 avec les conditions aux limites y 0 0 by 1 y 1 0 q l r 2 0 1 et b r

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

a convergence theorem for newton like methods in banach spaces on the basis of the results obtained in a series of papers 25 28 a convergence theorem for newton s method in banach spaces is given which improves the theorems of kantorovich 4 lancaster 8 and ostrowski 10 the error bounds obtained improve the recent results of potra 17

finite difference methods for half inverse sturm liouville problems abstract we investigate finite difference solution of the hochstadt lieberman problem for a sturm liouville operator defined on 0 given the value of the potential q on c where 0

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m