elliptic boundary value problems with λ dependent boundary conditions abstract in this paper second order elliptic boundary value problems on bounded domains r n with boundary conditions on depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework for a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure in the special case of rational nevanlinna or riesz herglotz functions on the boundary the solution operator is obtained in an explicit form in the product hilbert space l 2 l 2 m which is a natural generalization of known results on linear elliptic boundary value problems and rational boundary value problems for ordinary second order differential equations

locally definite normal operators in krein spaces abstract we introduce the spectral points of two sided positive type of bounded normal operators in krein spaces it is shown that a normal operator has a local spectral function on sets which are of two sided positive type in addition we prove that the riesz dunford spectral subspace corresponding to a spectral set which is only of positive type is uniformly positive the restriction of the operator to this subspace is then normal in a hilbert space

towards a matrix theory for unbounded operator matrices sur l espace produit r l 2 r on considere l operateur 7b a a 0 ou d d dx est defini sur d a w 1 r 0 est la mesure de dirac en 0 c l 2 r est un operateur de r dans l 2 r et 7b a a un domaine d 7b a r w 1 r on introduit une fonction caracteristique non commutative donnant une facon efficace pour calculer le spectre de 7b a on obtient des resultats sur la generation de semi groupes par des matrices operateur

on the negative squares of indefinite sturm liouville operators abstract the number of negative squares of all self adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a krein space is characterized in terms of the behaviour of an abstract titchmarsh weyl function near 0 and these results are applied to a large class of symmetric and self adjoint indefinite sturm liouville operators with indefinite weight functions

non symmetric perturbations of self adjoint operators abstract we investigate the effect of non symmetric relatively bounded perturbations on the spectrum of self adjoint operators in particular we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates these results extend and improve classical perturbation results by kato and by gohberg kre n further we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator in the latter case we can even allow for perturbations with relative bound 1 the generality of our results is illustrated by several applications massive and massless dirac operators point coupled periodic systems and two channel hamiltonians with dissipation

a wider class of stable gyroscopic systems abstract it is well known that in a physically interesting sense n n gyroscopic systems of the form l 2 i b c where c 0 and b is indefinite and invertible are stable whenever b ki k 1 c for some real k 0 it is shown that stability is retained under a considerably weaker condition formulated in terms of the spectral radius of b 1 i 1 c

the similarity problem for j nonnegative sturm liouville operators abstract sufficient conditions for the similarity of the operator a 1 r x d 2 d x 2 q x with an indefinite weight r x sgn x r x are obtained these conditions are formulated in terms of titchmarsh weyl m coefficients sufficient conditions for the regularity of the critical points 0 and of j nonnegative sturm liouville operators are also obtained this result is exploited to prove the regularity of 0 for various classes of sturm liouville operators this implies the similarity of the considered operators to self adjoint ones in particular in the case r x sgn x and q l 1 r 1 x d x we prove that a is similar to a self adjoint operator if and only if a is j nonnegative the latter condition on q is sharp i e we construct q 1 l 1 r 1 x d x such that a is j nonnegative with the singular critical point 0 hence a is not similar to a self adjoint operator for periodic and infinite zone potentials we show that j positivity is sufficient for the similarity of a to a self adjoint operator in the case q 0 we prove the regularity of the critical point 0 for a wide class of weights r this yields new results for forward backward diffusion equations

eigencurves for two parameter sturm liouville equations this paper concerns two parameter sturm liouville problems of the form p x y q x y lambda r x mu y quad a leqslant x leqslant b with self adjoint boundary conditions at a and b the set of lambda mu in bf r 2 for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions our focus is on these graphs which are termed eigencurves in the literature although eigencurves have been used in a variety of ways for about a century they seem comparatively underdeveloped in their own right our plan is to give motivation for the topic elementary properties of eigencurves illustrations on a simple example first studied by richardson in 1918 and since then by several authors and some natural questions which may whet the reader s appetite some of these questions lead to new types of inverse sturm liouville problems

spectral inclusion for unbounded block operator matrices abstract in this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators a admitting a block operator matrix representation for diagonally dominant and off diagonally dominant block operator matrices we show that the recently introduced quadratic numerical range w 2 a contains the eigenvalues of a and that the approximate point spectrum of a is contained in the closure of w 2 a this provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non convex set w 2 a several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems both in the non self adjoint case and in the self adjoint case applications to dirac operators and to two channel hamiltonians are given

scattering matrices and weyl functions for a scattering system a a 0 consisting of self adjoint extensions a and a 0 of a symmetric operator a with finite deficiency indices the scattering matrix s and a spectral shift function are calculated in terms of the weyl function associated with a boundary triplet for a and a simple proof of the krein birman formula is given the results are applied to singular sturm liouville operators with scalar and matrix potentials to dirac operators and to schrodinger operators with point interactions

on frames for krein spaces abstract a definition of frames for krein spaces is proposed which extends the notion of j orthonormal bases of krein spaces a j frame for a krein space h is in particular a frame for h in the hilbert space sense but it is also compatible with the indefinite inner product meaning that it determines a pair of maximal uniformly j definite subspaces an analogue to the maximal dual pair associated to a j orthonormal basis also each j frame induces an indefinite reconstruction formula for the vectors in h which resembles the one given by a j orthonormal basis

left definite sturm liouville problems abstract left definite regular self adjoint sturm liouville problems with either separated or coupled boundary conditions are studied we give an elementary proof of the existence of eigenvalues for these problems for any fixed equation we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions some of our results here yield an algorithm for numerically computing the eigenvalues of a left definite problem with an arbitrary coupled boundary condition our inequalities imply that the well known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case moreover we study the continuous and differentiable dependence of the eigenvalues of the general left definite problem on all the parameters in its differential equation and boundary condition

the slopes determined by n points in the plane let m12 m13 mn 1 n be the slopes of the nlines con necting n points in general position in the plane the ideal in of all algebraic relations among the mij defines a configuration space called the slope variety of the complete graph we prove that in is reduced and cohen macaulay give an explicit grobner basis for it and compute its hilbert series combinatorially we proceed chiefly by studying the associated stanley reisner simplicial com plex which has an intricate recursive structure in addition we are able to answer many questions about the geometry of the slope variety by translating them into purely combinatorial problems concerning enumeration of trees

the inverse spectral problem for indefinite strings motivated by the study of certain nonlinear wave equations in particular the camassa holm equation we introduce a new class of generalized indefinite strings associated with differential equations of the form begin aligned u z u omega z 2 u upsilon end aligned on an interval 0 l where omega is a real valued distribution in h 1 mathrm loc 0 l upsilon is a non negative borel measure on 0 l and z is a complex spectral parameter apart from developing basic spectral theory for these kinds of spectral problems our main result is an indefinite analogue of m g krein s celebrated solution of the inverse spectral problem for inhomogeneous vibrating strings

eigenvalues of schrodinger operators with definite and indefinite weights in this paper computational algorithms are presented to compute the eigenvalues of schrodinger operators with definite and indefinite weights the algorithms are based on titchmarsh weyl s theory and can be applied to solve a wide class of problems in quantum mechanics explicit examples are presented some of which recover the known exact solutions finally some analytical bounds have been reviewed for definite and indefinite singular problems weyl s spectral bisection algorithm has been proposed to solve definite problems as algebraic problems open problems have been presented based on the results of the presented algorithms

on the spectral theory of singular indefinite sturm liouville operators we consider a singular sturm liouville differential expression with an indefinite weight function and we show that the corresponding self adjoint differential operator in a krein space locally has the same spectral properties as a definitizable operator

eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations abstract udy the perturbation theory of structured matrices under structured rank one perturbations and then focus on several classes of complex matrices generic jordan structures of perturbed matrices are identified it is shown that the perturbation behavior of the jordan structures in the case of singular j hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in 18 28 30 31 thus a generic structured perturbation would not be generic if considered as an unstructured perturbation in other settings of structured matrices the generic perturbation behavior of the jordan structures within the confines imposed by the structure follows the pattern of that of unstructured perturbations

inverse spectral problems for left definite sturm liouville equations with indefinite weight abstract inverse sturm liouville problems with indefinite weight are considered theorems analogous to those of both hochstadt and gelfand levitan are proved as a sample result under suitable symmetry conditions we show that half of one spectrum suffices to determine the potential function uniquely

elliptic eigenvalue problems with eigenparameter dependent boundary conditions abstract let a be a uniformly elliptic second order linear operator on a smooth bounded domain r n we study the eigenvalue problem au u subject to boundary conditions b0u b1u on where bj are linear boundary operators the problem is recast in the form a u u in a hilbert or krein space and results are given on the location and type of the spectrum full and half range completeness and regularity of critical points

boundary triples and weyl m functions for powers of the jacobi differential operator abstract the abstract theory of boundary triples is applied to the classical jacobi differential operator and its powers in order to obtain the weyl m function for several self adjoint extensions with interesting boundary conditions separated periodic and those that yield the friedrichs extension these matrix valued nevanlinna herglotz m functions are to the best knowledge of the author the first explicit examples to stem from singular higher order differential equations the creation of the boundary triples involves taking pieces determined in 26 of the principal and non principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space these maps act like quasi derivatives which are usually not well defined for all functions in the maximal domain of singular expressions however well defined regularizations of quasi derivatives are produced by putting the pieces of the non principal solutions through a modified gram schmidt process

indefinite sturm liouville problems and half range completeness where the function h changes sign on i such problems occur in certain physical models particularly in transport theory and statistical physics see 2 9 11 131 and references therein as for the standard eigenvalue problem it is true under quite general conditions and not difficult to prove that there is a set of eigenfunctions un which is complete in an appropriate hilbert space of functions this is known as full range completeness for the physical problems the question of interest is generally that of half range completeness to formulate this we set

sturm liouville operators with indefinite weight functions and eigenvalue depending boundary conditions we consider a class of boundary value problems for sturm liouville operators with indenite weight functions the spectral parameter appears nonlinearly in the boundary condition in the form of a function which has the property that 7 is a generalized nevanlinna function we construct linearizations of these boundary value problems and study their spectral properties

on symmetries in the theory of finite rank singular perturbations for a nonnegative self adjoint operator a0 acting on a hilbert space h singular perturbations of the form a0 v v 1nbij j i are studied under some additional requirements of symmetry imposed on the initial operator a0 and the singular elements j a concept of symmetry is defined by means of a one parameter family of unitary operators u that is motivated by results due to r s phillips the abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials v and the corresponding self adjoint realizations of a0 v the results are applied for the investigation of singular perturbations of the schrodinger operator in l2 r3 and for the study of a fractional p adic schrodinger type operator with point interactions

the order of neutrality for linear operators on inner product spaces abstract let a be a symmetric linear operator defined on all of a possibly degenerate indefinite inner product space h let n be the set of all subspaces of h which are a invariant neutral in the sense of the indefinite scalar product and finite dimensional it is shown that members of n which are maximal with respect to inclusion all have the same dimension this is called the order of neutrality of a and admits immediate application to self adjoint operators on a pontrjagin space

hyponormal and strongly hyponormal matrices in inner product spaces complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied based on earlier works on normal matrices the notions of hyponormal and strongly hyponormal matrices are introduced a full characterization of such matrices is given and it is shown how those matrices are related to different concepts of normal matrices in degenerate inner product spaces finally the existence of invariant semidefinite subspaces for strongly hyponormal matrices is discussed

elliptic problems involving an indefinite weight we consider a selfadjoint elliptic eigenvalue problem which is derived formally from a variational problem of the form lu a o x u in q b1u 0 on f j 1 m where l is a linear elliptic operator of order 2m defined in a bounded open set q c rn n 2 with boundary f the bj are linear differential operators defined on f and w is a real valued function assuming both positive and negative values for our problem we prove the completeness of the eigenvectors and associated vectors in two function spaces which arise naturally in such an indefinite problem we also establish some results concerning the eigenvalues of the problem which complement the known results and investigate the structure of the principal subspaces

on finite rank perturbations of definitizable operators it was shown by p jonas and h langer that a selfadjoint definitizable operator a in a krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator b is selfadjoint and the resolvent set b is nonempty it is the aim of this note to prove a more general variant of this perturbation result where the assumption on b is dropped as an application a class of singular ordinary differential operators with indefinite weight functions is studied

geometry of graph varieties first published in transactions of the american mathematical society in volume 355 2003 4151 4169 published by the american mathematical society

spectral points of type π and π of self adjoint operators in krein spaces via approximative eigensequences we introduce the notion of spectral points of type and for self adjoint operators in krein spaces they are stable under compact perturbations for real spectral points of type and which are not in the interior of the spectrum we prove that the growth of the resolvent in some neighbourhood of them is of finite order there exists a local spectral function with singularities it turns out that all spectral subspaces corresponding to sufficiently small neighbourhoods of points of type or type are pontryagin spaces

an analytic family of cal p cal t symmetric hamiltonians with real eigenvalues we study nonlinear perturbative expansions for symmetric local schrodinger operators the schrodinger operator is a sum of the harmonic oscillator hamiltonian and a local symmetric potential depending in general nonlinearly on the perturbation parameter a specific class of models having real spectrum for any value of the parameter is proposed

spectral bounds for indefinite singular sturm liouville operators with uniformly locally integrable potentials abstract the non real spectrum of a singular indefinite sturm liouville operator a 1 r d d x p d d x q with a sign changing weight function r consists under suitable additional assumptions on the real coefficients 1 p q r l loc 1 r of isolated eigenvalues with finite algebraic multiplicity which are symmetric with respect to the real line in this paper bounds on the absolute values and the imaginary parts of the non real eigenvalues of a are proved for uniformly locally integrable potentials q and potentials q l s r for some s 1 the bounds depend on the negative part of q on the norm of 1 p and in an implicit way on the sign changes and zeros of the weight function

p mathcal t symmetric non self adjoint operators diagonalizable and non diagonalizable with a real discrete spectrum consider in the operator family h0 a 1a1 a dad d 2 is the quantum harmonic oscillator with rational frequencies w is a symmetric bounded potential and g is a real coupling constant we show that if g being an explicitly determined constant the spectrum of h g is real and discrete moreover we show that the operator has a real discrete spectrum but is not diagonalizable

accumulation of complex eigenvalues of indefinite sturm liouville operators spectral properties of singular sturm liouville operators of the form with the indefinite weight x sgn x on are studied for a class of potentials with lim x v x 0 the accumulation of complex and real eigenvalues of a to zero is investigated and explicit eigenvalue problems are solved numerically

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

self adjoint analytic operator functions and their local spectral function for a self adjoint analytic operator function a which satisfies on some interval of the real axis the virozub matsaev condition a local spectral function q on the values of which are non negative operators is introduced and studied in the particular case that a i a with a self adjoint operator a it coincides with the orthogonal spectral function of a an essential tool is a linearization of a by means of a self adjoint operator in some krein space and the local spectral function of this linearization the main results of the paper concern properties of the range of q and the description of a natural complement of this range

the effect of finite rank perturbations on jordan chains of linear operators abstract a general result on the structure and dimension of the root subspaces of a linear operator under finite rank perturbations is proved the increase of dimension from the kernel of the n th power to the kernel of the n 1 th power of the perturbed operator differs from the increase of dimension of the kernels of the corresponding powers of the unperturbed operator by at most the rank of the perturbation this bound is sharp

signed frames and hadamard products of gram matrices abstract this paper concerns redundant representations in a hilbert space h of the form f j c j f j j f h these are more general than those obtained from a tight frame and we develop a general theory based on what are called signed frames we are particularly interested in the cases where the scaling factors cj are unique and the geometric interpretation of negative cj this is related to results about the invertibility of certain hadamard products of gram matrices which are of independent interest e g we show for almost every v 1 v n c d rank v i v j r v i v j s min r d 1 d 1 s d 1 d 1 n r s 0 applications include the construction of tight frames of bivariate jacobi polynomials on a triangle which preserve symmetries and numerical results and conjectures about the class of tight signed frames in a finite dimensional space

a krein space approach to symmetric ordinary differential operators with an indefinite weight function on considere les proprietes spectrales du probleme differentiel l f 1 n p 0 f n n 1 n 1 p 1 f n 1 n 1 p n f rf sur un intervalle fini ou infini a b avec des coefficients reels localement sommables 1 p 0 p 1 p n r sous l hypothese que p 0 0 et que la fonction poids r change de signe sur a b

changing the spectrum of an operator by perturbation abstract we discuss the following problem how can the spectrum a of a linear operator a be changed upon perturbation of a with a one dimensional operator k in particular we study this question for normal self adjoint and unitary operators

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

singular left definite sturm liouville problems abstract we study singular left definite sturm liouville problems with an indefinite weight function the existence of eigenvalues is established based on the existence of eigenvalues of corresponding right definite problems furthermore for each singular left definite problem with limit circle non oscillatory endpoints we construct a regular left definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions inequalities among eigenvalues recently established for regular left definite problems are extended to the singular case

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

sharp eigenvalue estimates for rank one perturbations of nonnegative operators in krein spaces abstract let a and b be selfadjoint operators in a krein space and assume that the resolvent difference of a and b is of rank one in the case that a is nonnegative and i is an open interval such that a i consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of b in i the general result is illustrated with eigenvalue estimates for singular indefinite sturm liouville problems

bounds on the non real spectrum of differential operators with indefinite weights ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non negative operators in krein spaces under the assumption that 0 and infty are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non real spectrum is contained in a compact set and with definite type real spectrum outside this set the main results are quantitative estimates for this set which are applied to sturm liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains

symmetric matrices with respect to sesquilinear forms abstract simple forms are obtained for matrices that are symmetric with respect to degenerate sesquilinear forms on finite dimensional complex linear spaces of column vectors symmetric matrices and the sesquilinear forms are then representable as block diagonals having simple forms as the diagonal blocks the notion of indecomposability for symmetric matrices is studied an example shows that in contrast with the nondegenerate sesquilinear forms an indecomposable symmetric matrix with respect to a degenerate sesquilinear form may have arbitrarily many jordan blocks all indecomposable symmetric matrices are characterized in two situations when the sesquilinear form has only one degree of degeneracy and when the form is semidefinite

definitizable operators and quasihyperbolic operator polynomials abstract the main concern of this paper is bounded operators a on a hilbert space with inner product which are selfadjoint in an indefinite scalar product say x y gx y and have entirely real spectrum in addition all points of spectrum are required to have determinate type a notion refining earlier ideas of krein langer et al which implies a strong stability property under perturbations the central result states that the spectrum is of this type if and only if the operator in question is uniformly definitizable i e gp a for some polynomial p as a first application characterizations of compact uniformly definitizable operators on pontrjagin spaces are obtained then the basic ideas are extended to selfadjoint monic operator polynomials via their linearizations in particular a new class of quasihyperbolic polynomials qhp with real and determinate spectrum is introduced it is shown that qhp have nontrivial monic factors the special cases of strongly hyperbolic and quadratic polynomials are also discussed in particular a factorization theorem is proved for a class of gyroscopically stabilized quadratic polynomials which originate in recent investigations of problems in mechanics

riesz mcnaughton functions and riesz mv algebras of nonlinear functions abstract we focus on riesz mv algebras which are mv algebras equipped with a multiplication by numbers in the real interval 0 1 we consider for every integer n the riesz mv algebra of all continuous functions from the n th power of 0 1 to 0 1 and the riesz mv subalgebras thereof in particular we study the riesz mv subalgebras isomorphic to free riesz mv algebras with finitely many generators possibly different from the usual linear models given by what we call riesz mcnaughton functions and which generalize mcnaughton functions used in the case of mv algebras in doing this we characterize zerosets of riesz mcnaughton functions by means of polyhedra and we extend to riesz mv algebras a duality for mv algebras exposed in a paper by marra and spada

perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations abstract for selfadjoint matrices in an indefinite inner product possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation genericity is understood in the sense of algebraic geometry special attention is paid to the perturbation behavior of the sign characteristic typically under such a perturbation for every given eigenvalue the largest jordan block of the eigenvalue is destroyed and in case the eigenvalue is real all other jordan blocks keep their sign characteristic the new eigenvalues i e those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix are typically simple and in some cases information is provided about their sign characteristic if the new eigenvalue is real the main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product a version of the brunovsky canonical form and on general results concerning rank one perturbations obtained

on a class of j self adjoint operators with empty resolvent set in the present paper we investigate the set j of all j self adjoint extensions of an operator s which is symmetric in a hilbert space h with deficiency indices 2 2 and which commutes with a non trivial fundamental symmetry j of a krein space h sj js our aim is to describe different types of j self adjoint extensions of s which in general are non self adjoint operators in the hilbert space h one of our main results is the equivalence between the presence of j self adjoint extensions of s with empty resolvent set and the commutation of s with a clifford algebra cl2 j r where r is an additional fundamental symmetry with jr rj this enables one to parameterize in terms of cl2 j r the set of all j self adjoint extensions of s with stable c symmetry here an extension has stable c symmetry if it commutes with a fundamental symmetry and in turn this fundamental symmetry commutes with s such a situation occurs naturally in many applications here we discuss the case of indefinite sturm liouville operators and the case of a one dimensional dirac operator with point interaction

strongly cohen macaulay schemes and residual intersections this paper studies the local properties of closed subschemes y in cohen macaulay schemes x such that locally the defining ideal of y in x has the property that its koszul homology is cohen macaulay whenever this occurs y is said to be strongly cohen macaulay in x this paper proves several facts about such embeddings chiefly with reference to the residual intersections of y in x the main result states that any residual intersection of y in x is again cohen macaulay introduction our purpose in this paper is to investigate a property of a closed subscheme y in a noetherian cohen macaulay scheme x which we call strong cohen macaulayness this is a local property which we first describe for a local ring let x spec r where r is a cohen macaulay local ring and let y spec r i

rank one perturbations approximations and selfadjoint extensions rank one perturbationsa a of a semibounded selfadjoint operatoraare studied with the help of distribution theory it is shown that such perturbations can be defined for finite values of even if the element does not belong to h 1 a approximations of the rank one perturbations are constructed in the strong operator topology it is proven that rank one h 2perturbations can be defined uniquely for the homogeneous operators the results are applied to a schrodinger operator with a delta interaction in dimension 3

selfadjoint operators in s spaces abstract we study s spaces and operators therein an s space is a hilbert space s with an additional inner product given by u where u is a unitary operator in s we investigate spectral properties of selfadjoint operators in s spaces we show that their spectrum is symmetric with respect to the real axis as a main result we prove that for each selfadjoint operator a in an s space we find an inner product which turns s into a krein space and a into a selfadjoint operator therein as a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in krein spaces which provides a different insight into this well know and in general unsolved problem

an analytic characterization of the eigenvalues of self adjoint extensions let a be a self adjoint extension in k of a fixed symmetric operator a in k k an analytic characterization of the eigenvalues of a is given in terms of the q function and the parameter function in the krein naimark formula here k and k are krein spaces and it is assumed that a locally has the same spectral properties as a self adjoint operator in a pontryagin space the general results are applied to a class of boundary value problems with dependent boundary conditions

spectral enclosures for a class of block operator matrices abstract we prove new spectral enclosures for the non real spectrum of a class of 2 2 block operator matrices with self adjoint operators a and d on the diagonal and operators b and b as off diagonal entries one of our main results resembles gershgorin s circle theorem the enclosures are applied to j frame operators

gershgorin s theorem for matrices of operators let a aij be an n n matrix of operators acting on the bannach space x x1 x2 xn endowed with the x norm gershgorin circle theorem extends to this setting gi aii aii and aii 1 1 j li jn aij then a i 1ngi moreover assume that j is a proper nonempty subset of 1 2 n if i jgj and i jgj are disjoint then there exist invariant subspaces y1 and y2 for a such that a y1 i jgiand a y2 i jgi where y1 i jxiand y2 i jxi the notion of minimal gershgorin sets that follows is one possible generalization of what varga studied in the scalar case for partitioned matrices he used a more refined generalization for any positive n dimensional vector x x1 xn the operator 1 x1 i 1 xn i a x1i xni ax is similar to a let gix a aii z z aii and aii z 1 1 j lj in xj xi aij the minimal gershgorin set g a is defined to be g a x 0 i angix a as in the scalar case it has the property that a b a b g a where a b bii bii aiifori 1 nand bij aij ifi j it is proved that if each xi is a hilbert space and each aii is normal i 1 n then g a a where g a denotes the boundary of g a it is worth remarking that the closure of a is necessary

rank one perturbations at infinite coupling in pontryagin spaces in this paper we relate the operators in the operator representations of a generalized nevanlinna function n z and of the function n z 1 under the assumption that z is the only generalized pole of nonpositive type the results are applied to the q function for s and h and the q function for s and h where h is a self adjoint operator in a pontryagin space with a cyclic element w h is the self adjoint relation obtained from h and w via a rank one perturbation at infinite coupling and s is the symmetric operator given by s h h

weakly divisible mv algebras and product abstract we introduce weakly divisible mv algebras and we show that every weakly divisible complete mv algebra is isomorphic to the system of all continuous fuzzy functions defined on some compact hausdorff space which generalizes a result of di nola and sesse in non classical logics and their application to fuzzy subsets kluwer academic dordrecht to appear this enables us to define an associative and commutative product on weakly divisible complete mv algebras with a multiplicative unity 1 as an application we show that any system of observables on this mv algebra admits a joint observable with respect to this product

spectral analysis of singular ordinary differential operators with indefinite weights abstract in this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions we prove a general perturbation result on the local spectral properties of selfadjoint operators in krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares this result is applied to singular indefinite sturm liouville operators and higher order singular ordinary differential operators with indefinite weight functions

bounds on real and imaginary parts of non real eigenvalues of a non definite sturm liouville problem abstract in this paper we obtain bounds on the real and imaginary parts of non real eigenvalues of a non definite sturm liouville problem with dirichlet boundary conditions that improve on corresponding results obtained in behrndt et al 7

on domains of cal p cal t symmetric operators related to y x 1 nx2ny x in recent years a generalization of hermiticity has been investigated using a complex deformation h p2 x2 ix of the harmonic oscillator hamiltonian where is a real parameter these complex hamiltonians possessing symmetry the product of parity and time reversal can have a real spectrum we will consider the most simple case even in this paper we describe all self adjoint hermitian and at the same time symmetric operators associated with h p2 x2 ix surprisingly it turns out that there is a large class of self adjoint operators associated with h p2 x2 ix which are not symmetric

scattering and inverse scattering for a left definite sturm liouville problem this work develops a scattering and an inverse scattering theory for the sturm liouville equation u qu wu where w may change sign but q 0 thus the left hand side of the equation gives rise to a positive quadratic form and one is led to a left definite spectral problem the crucial ingredient of the approach is a generalized transform built on the jost solutions of the problem and hence termed the jost transform and the associated paley wiener theorem linking growth properties of transforms with support properties of functions one motivation for this investigation comes from the camassa holm equation for which the solution of the cauchy problem can be achieved by the inverse scattering transform for formula

spectral theory of elliptic differential operators with indefinite weights the spectral properties of a class of non self adjoint second order elliptic operators with indefinite weight functions on unbounded domains are investigated it is shown under an abstract regularity assumption that the non real spectrum of the associated elliptic operators in l 2 is bounded in the special case where r n decomposes into subdomains and with smooth compact boundaries and the weight function is positive on and negative on it turns out that the non real spectrum consists only of normal eigenvalues that can be characterized with a dirichlet to neumann map

perturbation theory of symmetric hamiltonians in the framework of perturbation theory the reality of the perturbed eigenvalues of a class of symmetric hamiltonians is proved using stability techniques we apply this method to symmetric unperturbed hamiltonians perturbed by symmetric additional interactions

on the topology of graph picture spaces abstract we study the space x d g of pictures of a graph g in complex projective d space the main result is that the homology groups with integer coefficients of x d g are completely determined by the tutte polynomial of g one application is a criterion in terms of the tutte polynomial for independence in the d parallel matroids studied in combinatorial rigidity theory for certain special graphs called orchards the picture space is smooth and has the structure of an iterated projective bundle we give a borel presentation of the cohomology ring of the picture space of an orchard and use this presentation to develop an analogue of the classical schubert calculus

spectral properties of singular sturm liouville operators with indefinite weight sgn x we consider a singular sturm liouville expression with the indefinite weight sgnx to this expression there is naturally a self adjoint operator in some krein space associated we characterize the local definitizability of this operator in a neighbourhood of 1 moreover in this situation the point 1 is a regular critical point we construct an operator a sgnx id 2 dx 2 q with non real spectrum accumulating to a real point the obtained results are applied to several classes of sturm liouville operators

rank one perturbations in a pontryagin space with one negative square let n 1 denote the class of generalized nevanfinna functions with one negative square and let n 1 n 0 be the subclass of functions q z is an element of n with the additional properties lim y infinity q i y y 0 and lim sup y infinity y im q i y i these classes form an analytic framework for studying generalized rank one perturbations a a in a pontryagin space setting many functions appearing in quantum mechanical models of point interactions either belong to the subclass n 1 n 0 or can be associated with the corresponding generalized friedrichs extension in this paper a spectral theoretical analysis of the perturbations a and the associated friedrichs extension is carried out many results such as the explicit characterizations for the critical eigenvalues of the perturbations a are based on a recent factorization result for generalized nevanlinna functions c 2002 elsevier science usa