the sharp conditions of the uniqueness for inverse nodal problems abstract in this paper we provide the sharp conditions of the uniqueness for inverse nodal sturm liouville problems defined on interval 0 1 with separated boundary conditions we prove that the potential and boundary parameters can be uniquely determined by a dense nodal subset contained on a 1 a 2 0 1 with 1 2 a 1 a 2 through two cases of a 1 0 and a 1 0 where in the latter case the nodal subset also need to be paired note that the dense nodal subset was required to be twin for both cases in the previous works

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

half inverse problem for diffusion operators on the finite interval abstract the potential function q x in the regular and singular sturm liouville problem can be uniquely determined from two spectra inverse problem for diffusion operator given at the finite interval eigenvalues normal numbers also on two spectra are solved half inverse spectral problem for a sturm liouville operator consists in reconstruction of this operator by its spectrum and half of the potential in this study by using the hochstadt and lieberman s method we show that if q x is prescribed on 2 then only one spectrum is sufficient to determine q x on the interval 0 2 for diffusion operator

enumerating representations in finite wreath products abstract let be a group finite or infinite h a finite group and let rn denote the sequence h sn of symmetric wreath products as well as certain variants of it including in particular h an and wn the weyl group of type dn we compute the exponential generating function for the number hom rn of representations in rn and for some refinements of this sequence under very mild finiteness assumptions on always met for instance if is finitely generated this generalizes in a uniform way the connection between the problem of counting finite index subgroups in a group and the enumeration of actions on finite sets on the one hand and the recent results of chigira concerning solutions of the equation xm 1 in the groups h sn h an and wn on the other we also study the asymptotics of the function hom g h sn for arbitrary finite groups g and h

uniqueness results for one dimensional schrodinger operators with purely discrete spectra we provide an abstract framework for singular one dimensional schrodinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely as an example we apply our findings to prove new uniqueness results for perturbed quantum mechanical harmonic oscillators in addition we also show how to establish a hochstadt lieberman type result for these operators our approach is based on the singular weyl titchmarsh kodaira theory which is extended to cover the present situation

enumerating representations in finite wreath products ii explicit formulae abstract we present several contributions to the enumerative theory of wreath product representations developed in a previous paper by the first named author adv in math 153 2000 118 154 theorem 3 1 of the present paper establishes an explicit formula for one of the key ingredients in the description of the corresponding generating functions given in muller 2000 the exterior function building on theorem 1 in muller 2000 and the latter result we derive explicit formulae for the exponential generating function of the series hom r n in the case where is dihedral or a finite abelian group and the representation sequence r n is any of h sn or h an with a fixed finite group h or the sequence hom g wn moreover we verify a conjecture concerning the asymptotic behaviour of the sequence hom g wn for finite groups g made in muller 2000 in the case when g is dihedral or abelian

uniqueness theorems in inverse spectral theory for one dimensional schrodinger operators new unique characterization results for the potential v x in connection with schrodinger operators on r and on the half line 0 are proven in terms of appropriate krein spectral shift functions particular results obtained include a generalization of a well known uniqueness theorem of borg and marchenko for schrodinger operators on the half line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for schrodinger operators with confining potentials on the entire real line

incomplete inverse spectral problems for dirac bessel operators we investigate the incomplete inverse spectral problems for dirac bessel operators defined on 0 1 we show that when the potential is known on the subinterval a 1 0 1 the potential on the whole interval can be uniquely determined in terms of appropriate partial information on the spectral data including eigenvalues and norming constants moreover these results can be applied to solve the incomplete inverse spectral problems for a class of bessel operators we investigate the incomplete inverse spectral problems for dirac bessel operators defined on 0 1 we show that when the potential is known on the subinterval a 1 0 1 the potential on the whole interval can be uniquely determined in terms of appropriate partial information on the spectral data including eigenvalues and norming constants moreover these results can be applied to solve the incomplete inverse spectral problems for a class of bessel operators

inverse spectral problems and closed exponential systems consider the inverse eigenvalue problem of the schr odinger operator de fined on a finite interval we give optimal and almost optimal conditions for a set of eigenvalues to determine the schr odinger operator these conditions are simple closedness properties of the exponential system corresponding to the known eigenvalues the statements contain nearly all former results of this topic we give also conditions for recovering the weyl titchmarsh m function from its values m en

hochstadt lieberman theorem for dirac operator with eigenparameter dependent boundary conditions abstract we consider an inverse problem for the dirac operator with the spectral parameter in the boundary conditions in this work using hochstadt and lieberman s method hochstadt and lieberman 1978 1 we show that if the potential q x is prescribed on the half interval 1 2 1 then a single spectrum of the dirac operator with eigenparameter dependent boundary conditions on the whole interval 0 1 suffices to determine q x on another half interval 0 1 2 and the boundary condition at x 0

a survey on inverse and generalized inverse eigenvalue problems for jacobi matrices in this paper we give a survey of well known works of constructing an specific tridiagonal matrix using prescribed data then we focus on two well known methods namely m functions and lanczos algorithm we describe the first method for a generalized inverse eigenvalue problem and the second one for a classic inverse eigenvalue problem we show that these two methods are linked together in both method we prove that the solution is unique

inverse spectral problems for sturm liouville operators with singular potentials part iii reconstruction by three spectra abstract we solve the inverse spectral problem of recovering the singular potential from w 12 0 1 of a sturm liouville operator by its spectra on the three intervals 0 1 0 a and a 1 for some a 0 1 necessary and sufficient conditions on the spectral data are derived and uniqueness of the solution is analyzed

inverse spectral results for schrodinger operators on the unit interval with partial information given on the potentials we pursue the analysis of the schrodinger operator on the unit interval in inverse spectral theory initiated in the work of amour and raoux inverse spectral results for schrodinger operators on the unit interval with potentials in lp spaces inverse probl 23 2367 2007 while the potentials in the work of amour and raoux belong to l1 with their difference in lp 1 p we consider here potentials in wk 1 spaces having their difference in wk p where 1 p k 0 1 2 it is proved that two potentials in wk 1 0 1 being equal on a 1 are also equal on 0 1 if their difference belongs to wk p 0 a and if the number of their common eigenvalues is sufficiently high naturally this number decreases as the parameter a decreases and as the parameters k and p increase

weyl titchmarsh functions of vector valued sturm liouville operators on the unit interval the matrix valued weyl titchmarsh functions m of vector valued sturm liouville operators on the unit interval with the dirichlet boundary conditions are considered the collection of the eigenvalues i e poles of m and the residues of m is called the spectral data of the operator the complete characterization of spectral data or equivalently n n weyl titchmarsh functions corresponding to n n self adjoint square integrable matrix valued potentials is given if all n eigenvalues of the averaged potential are distinct

the uniqueness for inverse discrete transmission eigenvalue problems abstract the inverse problem for the discrete analog of the transmission eigenvalue problem with a spherically symmetric index of refraction is considered some uniqueness results are provided which imply that can be completely determined if only partial entries are given on together with partial transmission eigenvalues furthermore some similar results are presented for a related schrodinger equation

inverse spectral analysis with partial information on the potential ii the case of discrete spectrum we discuss results where the discrete spectrum or partial information on the discrete spectrum and partial information on the potential q of a one dimensional schrodinger operator h d 2 dx 2 q determine the potential completely included are theorems for finite intervals and for the whole line in particular we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of h on a finite interval and knowledge of q over a corresponding fraction of the interval the methods employed rest on weyl m function techniques and densities of zeros of a class of entire functions

the class of 1 3 groups with homocyclic regulator quotient of exponent p4 has bounded representation type abstract the class of almost completely decomposable groups with a critical typeset of type 1 3 and a homocyclic regulator quotient of exponent p 4 is shown to be of bounded representation type there are only nine near isomorphism types of indecomposables all of rank 6

on boundary conditions of an inverse strum liouville problem it is known that if the potential function q x in a sturm liouville problem is prescribed over the interval 0 1 2 and the boundary condition at the point zero is fixed then a single spectrum suffices to determine q x on the interval 1 2 1 uniquely explicit examples are given that show the necessity of specifying the boundary condition at zero even if the boundary condition at one is given

inverse spectral problems for the sturm liouville operator on a d star graph abstract in this work we consider inverse spectral problems for the sturm liouville differential operator on a d star type graph with standard matching conditions in the internal vertex where the integer d 2 by using the yurko s method yurko 2008 27 yurko 2009 28 we show that 1 if the potential function q j x on a fixed edge e j is prescribed on the interval 2 then the reciprocal of d of the spectrum suffices to determine q j x on the whole interval 0 2 the 2 over d of the spectrum suffices to determine q j x on a fixed edge e j

on the location of the weyl circles the paper deals with explicit estimates concerning certain circles in the complex plane which were associated with sturm liouville problems by h weyl by the use of riccati equations instead of linear integral equations improvements are obtained for results of everitt and halvorsen concerning the behaviour of the titchmarsh weyl m coefficient

remarks on a new inverse nodal problem in a recent paper x f yang proved a uniqueness theorem on inverse nodal problems that links to inverse spectral theory on one hand and reduces the redundancy of the classical inverse nodal problems on the other hand in this note we improve yang s theorem by weakening its conditions and simplifying its proof we also discuss variants of yang s theorem

the infinite period limit of the inverse formalism for periodic potentials soit l operateur de hill h p d 2 dx 2 v p x quand v p x est une fonction p periodique de x on montre que les formules pour la solution du probleme inverse ont une limite si on prend v p comme une repetition periodique d une fonction a support compact v x et que l on fait tendre p vers l infini

inverse nodal and inverse spectral problems for discontinuous boundary value problems inverse nodal and inverse spectral problems are studied for second order differential operators on a finite interval with discontinuity conditions inside the interval uniqueness theorems are proved and a constructive procedure for the solution is provided

almost completely decomposable groups and unbounded representation type abstract almost completely decomposable groups with a regulating regulator and a p primary regulator quotient are studied it is shown that there are indecomposable such groups of arbitrarily large rank provided that the critical typeset contains some basic configuration and the exponent of the regulator quotient is sufficiently large

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

the inverse interior transmission eigenvalue problem with mixed spectral data abstract the inverse spectral problem for the interior transmission eigenvalue problem with the unit time is studied by given spectral data the authors show that the refractive index on the whole interval can be uniquely determined by parts of its transmission eigenvalues together with the partial information on the refractive index in particular we pose and solve a new type of inverse spectral problems involving the interior transmission eigenvalue problem with complex transmission eigenvalues except for at most finite real eigenvalues

a nodal inverse problem for second order sturm liouville operators with indefinite weights in this paper we study an inverse problem for weighted second order sturm liouville equations we show that the zeros of any subsequence of eigenfunctions or a dense set of nodes are enough to determine the weight we impose weaker hypotheses for positive weights and we generalize the proof to include indefinite weights moreover the parameters in the boundary conditions can be determined numerically by using a shooting method

on wohlfahrt series and wreath products abstract suppose that a group a contains only a finite number of subgroups of index d for each positive integer d let g s n be the wreath product of a finite group g with the symmetric group s n on 1 n for each positive integer n let k n be a subgroup of g s n containing the commutator subgroup of g s n if the sequence k n 0 satisfies a certain compatible condition then the exponential generating function n 0 hom a k n x n g n n of the sequence hom a k n 0 takes the form of a sum of exponential functions

rank one perturbations at infinite coupling abstract we discuss rank one perturbations a a r a 0 self adjoint let d x be the spectral measure defined by a z 1 d x x z we prove there is a measure d which is the weak limit of 1 2 d x as if is cyclic for a then a the strong resolvent limit of a is unitarily equivalent to multiplication by x on l 2 r d this generalizes results known for boundary condition dependence of sturm liouville operators on half lines to the abstract rank one case

inverse scattering with fixed energy and an inverse eigenvalue problem on the half line recently a g ramm 1999 has shown that a subset of phase shifts 1 l 0 1 determines the potential if the indices of the known shifts satisfy the miintz condition l 0 l l 1 l we prove the necessity of this condition in some classes of potentials the problem is reduced to an inverse eigenvalue problem for the half line schrodinger operators

uniqueness of solution to inverse dirac spectral problems associated with incomplete spectral data in this paper we study the inverse dirac spectral problem defined on 0 a with the boundary condition at a involving a nevanlinna function f when f is known a priori we prove that the potentials and the other boundary condition are uniquely determined in terms of appropriate partial information on the spectral data consisting of eigenvalues and normalizing constants

inverse problems dense nodal subset on an interior subinterval abstract the inverse nodal problem for the sturm liouville problems defined on interval 0 1 with separated boundary conditions is considered we prove that a twin dense subset of the nodal set in interior subinterval a 1 a 2 0 1 uniquely determines the potential on 0 1 and the boundary conditions through two cases of 1 2 a 1 a 2 and 1 2 a 1 a 2 note that for the latter we need additional spectral information which is associated with the derivatives of eigenfunctions at some known nodal points

an inverse problem for the heat equation abstract let u t u xx q x u 0 x 1 t 0 u 0 t 0 u 1 t a t u x 0 0 where a t is a given function vanishing for t t a t 0 t 0 a t dt u x 0 t b 0 t for all t 0 does this information determine q x uniquely do the measurements of the flux u x 1 t b t give more information about q x than b 0 t does these questions are answered in this note

local solvability and stability of inverse problems for sturm liouville operators with a discontinuity abstract partial inverse problems are studied for sturm liouville operators with a discontinuity the main results of the paper are local solvability and stability of the considered inverse problems our approach is based on a constructive algorithm for solving the inverse problems the key role in our method is played by the riesz basis property of a special vector functional system in a hilbert space in addition we obtain a new uniqueness theorem for recovering the potential on a part of the interval by using a fractional part of the spectrum

inverse spectral problem with partial information on the potential the case of the whole real line the schrodinger operator is considered on the real axis we discuss the inverse spectral problem where discrete spectrum and the potential on the positive half axis determine the potential completely we do not impose any restrictions on the growth of the potential but only assume that the operator is bounded from below has discrete spectrum and the potential obeys under these assertions we prove that the potential for x 0 and the spectrum of the problem uniquely determine the potential on the whole real axis also we study the uniqueness under slightly different conditions on the potential the method employed uses weyl m function techniques and asymptotic behavior of the herglotz functions

determination of a differential pencil from interior spectral data abstract in this paper inverse spectra problems for a differential pencil are studied by using the approach similar to those in hochstadt and lieberman 1978 14 and ramm 2000 26 we prove that 1 if p x or q x is full given on the interval 0 then a set of values of eigenfunctions at the mid point of the interval 0 and one spectrum suffice to determine q x or p x on the interval 0 and all parameters in the boundary conditions 2 if p x or q x is full given on the interval 0 then some information on eigenfunctions at some internal point b 2 and parts of two spectra suffice to determine q x or p x on the interval 0 and all parameters in the boundary conditions

asymptotic behavior for differences of eigenvalues of two sturm liouville problems with smooth potentials abstract the asymptotics for the differences of the eigenvalues of two sturm liouville problems defined on 0 with the same boundary conditions and different smooth potentials is considered under the assumptions of that both problems with a suite of boundary conditions have the same one full spectrum and both potential functions and their derivatives are the same at the endpoint x the asymptotic expressions associated with other boundary conditions are provided

uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions abstract inverse spectral problems are considered for differential pencils with boundary conditions depending polynomially on the spectral parameter and with a finite number of transmission conditions we give formulations of the associated inverse problems such as titchmarsh weyl theorem hochstadt lieberman theorem and mochizuki trooshin theorem and prove corresponding uniqueness theorems the obtained results are generalizations of the similar results for the classical sturm liouville operator on a finite interval

generating functions for permutation representations abstract we discuss the categorical approach to representations in wreath products and generalize the wohlfahrt formula of the exponential generating function for the number of permutation representations

on the uniqueness of inverse spectral problems associated with incomplete spectral data abstract the inverse spectral problem for a sturm liouville problem which consists of a sturm liouville equation defined on the interval 0 a and two separated boundary conditions at the endpoints 0 and a one of which is involved function f is considered in this paper when f is the type of herglotz functions and is known as a priori we prove that the potential on 0 a and the other boundary condition can be uniquely determined in terms of appropriate partial information on the spectrum and partial information on the set of normalizing constants

fuchsian groups coverings of riemann surfaces subgroup growth random quotients and random walks abstract fuchsian groups acting as isometries of the hyperbolic plane occur naturally in geometry combinatorial group theory and other contexts we use character theoretic and probabilistic methods to study the spaces of homomorphisms from fuchsian groups to symmetric groups we obtain a wide variety of applications ranging from counting branched coverings of riemann surfaces to subgroup growth and random finite quotients of fuchsian groups as well as random walks on symmetric groups in particular we show that in some sense almost all homomorphisms from a fuchsian group to alternating groups an are surjective and this implies higman s conjecture that every fuchsian group surjects onto all large enough alternating groups as a very special case we obtain a random hurwitz generation of an namely random generation by two elements of orders 2 and 3 whose product has order 7 we also establish the analogue of higman s conjecture for symmetric groups we apply these results to branched coverings of riemann surfaces showing that under some assumptions on the ramification types their monodromy group is almost always sn or an another application concerns subgroup growth we show that a fuchsian group has n o 1 index n subgroups where is the measure of and derive similar estimates for so called eisenstein numbers of coverings of riemann surfaces a final application concerns random walks on alternating and symmetric groups we give necessary and sufficient conditions for a collection of almost homogeneous conjugacy classes in an to have product equal to an almost uniformly pointwise our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups

the trace formula for schrodinger operators on the line this paper discusses certain aspects of the spectral and inverse spectral problems for the schrodinger operator l q frac d 2 dx 2 q x for q x c the space of bounded continuous functions the trace formula of the title is the relation q 0 lambda 0 sum limits j 1 infty lambda 2j lambda 2j 1 2 mu j with appropriate choices of j j 0 and j j 1 which is a familiar relation in the theory of hill s equation we characterize the set subseteq c of potentials for which this holds further extensions of the theory of hill s equation are also obtained from the spectrum l q a torust q is constructed which is in general infinite dimensjional everyq x can be mapped to a continuous path ont q described by the auxiliary spectrum j j 1 under certain geometrical conditions on l q this path is the orbit of ac 1 vector field ont q and the mapping extends to one from the hull q overline q x xi xi in mathbb r to the closure of this orbit in particular q is compact these results have applications in the theory of schrodinger operators with ergodic potentials

absolute summability of the trace relation for certain schrodinger operators a recently established general trace formula for one dimensional schrodinger operators is systematically studied in the context of short range potentials potentials which approach different spatial asymptotes sufficiently fast and appropriate impurity defect interactions in one dimensional solids we prove the absolute summability of the trace formula and establish its connections with scattering quantities such as reflection coefficients in each case

inverse spectral problems for the sturm liouville operator with discontinuity abstract in this work we consider the sturm liouville operator on a finite interval 0 1 with discontinuous conditions at 1 2 we prove that if the potential is known a priori on a subinterval b 1 with b 1 2 then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions for the case b 1 2 parts of either one or two spectra can uniquely determine the potential and a part of parameters

uniqueness of the interior transmission problem with partial information on the potential and eigenvalues abstract the inverse spectral problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b a uniqueness theorem for the potential q of the derived sturm liouville problem b q is presented from the data involving fractions of the eigenvalues of the problem b q on a finite interval and knowledge of q over a corresponding fraction of the interval the methods employed rest on weyl function techniques and properties of zeros of a class of entire functions

rank one perturbations with infinitesimal coupling we consider a positive self adjoint operator a and formal rank one perturbations b a where h 2 a but h 1 a with h s a the usual scale of spaces we show that b can be defined for such and what are essentially negative infinitesimal values of in a sense we will make precise every rank one perturbation is one of three forms i h 1 a r ii h 1 or iii the new type we consider here

1 2 groups with p3 regulator quotient abstract for a prime p and a poset 1 2 1 2 3 of types p reduced almost completely decomposable groups with critical typeset 1 2 and a p power as regulating index are called 1 2 groups the number of near isomorphism types of indecomposable 1 2 groups depends on the exponent p k of the regulator quotient it is shown that indecomposable 1 2 groups with a regulator quotient of exponent p 3 have rank 4 and if the types i and the prime p are fixed then there are precisely four near isomorphism types of indecomposable groups it is unknown for which exponent p k 0 of the regulator quotient exist infinitely many near isomorphism types of indecomposable 1 2 groups

the inverse spectral problem for differential pencils by mixed spectral data abstract an inverse problem for differential pencils of second order is studied we show that the potentials on the whole interval can be uniquely determined by partial information on potentials and parts of two spectra

inverse eigenvalue problems in this article we consider inverse eigenvalue problems for the schrodinger operator on a finite interval we extend and strengthen previously known uniqueness theorems a partially known potential is identified by some sets of eigenvalues and norming constants

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

almost periodicity of some random potentials in this paper we study some random potentials with purely absolutely continuous spectrum in one dimension there are no examples of random potentials with absolutely continuous spectrum that are not almost periodic though it is known from kotani 3 that such potentials are deterministic hence one could conjecture that any random potential with purely absolutely continuous spectrum is almost periodic we show this result for random potentials with infinitely many bands in their spectrum such that the gaps behave well in the finite bands case concini and johnson l discussed the same problem a precise definition of the behavior of the gaps could be found in condition 2 3 we make use of the results of levitan 7 kotani 3 4 and moser 9 for doing this we also discuss the behavior of the lyapunov exponent asymptotically and its relation to the smoothness of the potentials

inverse problem for a quadratic pencil of sturm liouville operator abstract the purpose of this paper is to solve the inverse problem for a quadratic pencil of sturm liouville operator on a finite interval by using the hochstadt s method it is shown that the potential function can be given from two spectras as uniquely later by generalizing this result the hochstadt s theorem which is related to the structure of difference for the potential functions is obtained