the proper dissipative extensions of a dual pair let a and widetilde a be dissipative operators on a hilbert space mathcal h and let a widetilde a form a dual pair i e a subset widetilde a resp widetilde a subset a we present a method of determining the proper dissipative extensions widehat a of this dual pair i e a subset widehat a subset widetilde a provided that mathcal d a cap mathcal d widetilde a is dense in mathcal h applications to symmetric operators symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed finally we investigate the stability of the numerical range of the different dissipative extensions

on inverse spectral theory for self adjoint extensions mixed types of spectra abstract lethbe a symmetric operator in a separable hilbert space h suppose thathhas some gapj we shall investigate the question about what spectral properties the self adjoint extensions ofhcan have inside the gapjand provide methods on how to construct self adjoint extensions ofhwith prescribed spectral properties insidej under some weak assumptions about the operatorhwhich are satisfied e g provided the deficiency indices ofhare infinite and the operator h 1is compact for one regular point ofh we shall show that for every auxiliary self adjoint operatorm in the hilbert space h and every open subsetj0of the gapjofhthere exists a self adjoint extensionhofhsuch that insidejthe self adjoint extensionhofhhas the same absolutely continuous and the same point spectrum as the given operatorm and the singular continuous spectrum ofhinjequals the closure ofj0inj moreover we shall present a method of how to construct such a self adjoint extensionh via our methods it is possible to construct new kinds of self adjoint realizations of the laplacian on a bounded domain in r d d 1 with spectral properties very different from the spectral properties of the self adjoint realizations known before mathematics subject classification 1991 47a10 47a60 47b25 47e05 47f05

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

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

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 the number of negative eigenvalues of the laplacian on a metric graph the number of negative eigenvalues of self adjoint laplacians on metric graphs is calculated in terms of the boundary conditions and the underlying geometric structure this extends and complements earlier results by kostrykin and schrader 2006 contemp math 415 201 25

stochastic resonance in neuron models periodically stimulated sensory neurons typically exhibit a kind of statistical phase locking to the stimulus they tend to fire at a preferred phase of the stimulus cycle but not at every cycle hence the histogram of interspike intervals isih i e of times between successive firings is multimodal for these neurons with peaks centered at integer multiples of the driving period a particular kind of residence time histogram for a large class of noisy bistable systems has recently been shown to exhibit the major features of the neural data in the present paper we show that an excitable cell model the fitzhugh nagumo equations also reproduces these features when driven by additive periodic and stochastic forces this model exhibits its own brand of stochastic resonance as the peaks of the isih successively go through a maximum when the noise intensity is increased further the presence of a noise induced limit cycle introduces a third time scale in the problem this limit cycle is found to modify qualitatively the phase locking picture e g by suppressing certain peaks in the isih finally the role of noise and possibly of stochastic resonance sr in the neural encoding of sensory information is discussed

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

non self adjoint graphs on finite metric graphs we consider laplace operators subject to various classes of non self adjoint boundary conditions imposed at graph vertices we investigate spectral properties existence of a riesz basis of projectors and similarity transforms to self adjoint laplacians among other things we describe a simple way to relate the similarity transforms between laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions

on the necessary use of abstract set theory in this paper we present some independence results from the zermelo frankel axioms of set theory with the axiom of choice zfc which differ from earlier such independence results in three major respects firstly these new propositions that are shown to be independent of zfc i e neither provable nor refutable from zfc form mathematically natural assertions about bore1 functions of several variables from the hilbert cube i into the unit interval or back into the hilbert cube as such they are of a level of abstraction significantly below that of the earlier independence results secondly these propositions are not only independent of zfc but also of zfc together with the axiom of constructibility v l the only earlier examples of intelligible statements independent of zfc v l either express properties of formal systems such as zfc e g the consistency of zfc or assert the existence of very large cardinalities e g inaccessible cardinals the great bulk of independence results from zfclthe ones that involve standard mathematical concepts and constructions are about sets of limited cardinality most commonly that of at most the continuum and are obtained using the forcing method introduced by paul j cohen see 2 it is now known in virtually every such case that these independence results are eliminated if v l is added to zfc finally some of our propositions can be proved in the theory of classes as formalized by the morse kelley class theory with the axiom of choice for sets mkc but not in zfc mkc still formalizes only commonly accepted principles of mathematical reasoning thus these propositions provide examples of interesting theorems whose proofs necessarily involve the outer limits of what is commonly accepted as valid principles of mathematical reasoning the starting point for the development of these new propositions was our consideration in 1976 of certain aspects of cantor s basic theorem that the set of all real numbers is not countable thus given any sequence of real numbers there is a real number which is not a term of the sequence by using nested sequence of closed intervals with rational endpoints it is easy to

trace formulas for additive and non additive perturbations abstract trace formulas for pairs of self adjoint maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained in particular the existence of a complex valued spectral shift function for a pair h h of maximal accumulative operators has been proved we investigate also the existence of a real valued spectral shift function moreover we treat in detail the case of additive trace class perturbations assuming that h and h h v are maximal accumulative and v is trace class we prove the existence of a summable complex valued spectral shift function we also obtain trace formulas for pairs h h assuming only that h and h are resolvent comparable in this case the determinant of the characteristic function of h is involved in trace formulas the results rely on the technique of boundary triplets for densely and non densely defined closed symmetric operators the approach allows to express the spectral shift function of a pair of extensions in terms of the abstract weyl function and the boundary operator we improve and generalize certain classical results of m g kre n for pairs of self adjoint and dissipative operators results of a v rybkin for such pairs as well as of v m adamjan b s pavlov and m g kre n for pairs h h with a maximal dissipative operator h

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

j self adjoint operators with mathcal c symmetries an extension theory approach a well known tool in conventional von neumann quantum mechanics is the self adjoint extension technique for symmetric operators it is used e g for the construction of dirac hermitian hamiltonians with point interaction potentials here we reshape this technique to allow for the construction of pseudo hermitian j self adjoint hamiltonians with complex point interactions we demonstrate that the resulting hamiltonians are bijectively related to the so called hypermaximal neutral subspaces of the defect krein space of the symmetric operator this symmetric operator is allowed to have arbitrary but equal deficiency indices n n general properties of the operators for these hamiltonians are derived a detailed study of operator parametrizations and krein type resolvent formulae is provided for j self adjoint extensions of symmetric operators with deficiency indices 2 2 the technique is exemplified on 1d pseudo hermitian schr dinger and dirac hamiltonians with complex point interaction potentials

canonical systems of differential equations with self adjoint interface conditions on graphs for n canonical systems of differential equations the corresponding n copies of their domain 0 are thought of as a graph with vertex 0 an interface condition at 0 is given by a so called nevanlinna pair explicit formulae are deduced for the spectral representation of the corresponding underlying self adjoint relation and the generalized fourier transformation furthermore results on compressions of the fourier transformation to closed linear subspaces and the multiplicity of the eigenvalues if the spectrum is discrete are presented

shorting parallel addition and form sums of nonnegative selfadjoint linear relations abstract we extend the operations of shorting and parallel addition from the cone of bounded nonnegative selfadjoint operators in a hilbert space to the set of all nonnegative selfadjoint linear relations new properties of these operations and connections with the cayley transforms are established it is shown that for a pair of nonnegative selfadjoint linear relations there exist in general two arithmetic harmonic means applications of the arithmetic harmonic and arithmetic harmonic means to the theory of nonnegative selfadjoint extensions of nonnegative symmetric linear relations are given

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

a remark on discrete wave scattering we consider the spectral problem for the adjacency matrix of a discrete star graph composed of a compact cluster and a few semi infinite periodic leads attached based on the spectral properties of the adjacency matrix we develop the lax phillips scattering theory for the corresponding discrete wave equation

generalized q functions and dirichlet to neumann maps for elliptic differential operators the classical concept of q functions associated to symmetric and selfadjoint operators due to m g krein and h langer is extended in such a way that the dirichletto neumann map in the theory of elliptic differential equations can be interpreted as a generalized q function for couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit krein type formulas for the difference of the resolvents and trace formulas in an h 2 framework are obtained

influence of time delayed feedback in the firing pattern of thermally sensitive neurons we explore the dynamics of a hodgkin huxley type model for thermally sensitive neurons that exhibit intrinsic oscillatory activity the model is modified to include a feedback loop that is represented by two parameters the synaptic strength and the transmission delay time we analyze the dynamics of the neuron depending on the temperature the synaptic strength and the delay time we find parameter regions where the effect of the recurrent connexion is excitatory inducing spikes or trains of spikes and regions where it is inhibitory reducing or eliminating completely the spiking behavior we characterize the complex interplay of the intrinsic dynamics of the neuron with the recurrent feedback input and a noisy input

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

a krein like formula for singular perturbations of self adjoint operators and applications abstract given a self adjoint operator a d a h h and a continuous linear operator d a x with range h 0 x a banach space we explicitly construct a family a of self adjoint operators such that any a coincides with the original a on the kernel of such a family is obtained by giving a krein like formula where the role of the deficiency spaces is played by the dual pair x x the parameter belongs to the space of symmetric operators from x to x when x c one recovers the h 2 construction of kiselev and simon and so to some extent our results can be regarded as an extension of it to the infinite rank case considering the situation in which h l 2 r n and is the trace restriction operator along some null subset we give various applications to singular perturbations of non necessarily elliptic pseudo differential operators thus unifying and extending previously known results

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 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

dirichlet to neumann maps on bounded lipschitz domains the dirichlet to neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying dirichlet problem is not uniquely solvable the main objective of this paper is to present a systematic study of the dirichlet to neumann map and its inverse the neumann to dirichlet map in the framework of linear relations in hilbert spaces our treatment is inspired by abstract methods from extension theory of symmetric operators utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded lipschitz domains

response of autaptic hodgkin huxley neuron with noise to subthreshold sinusoidal signals abstract in this work we investigated the response of a stochastic hodgkin huxley hh neuron with an autapse to subthreshold sinusoidal signals it is found that the autapse not only adjusts the stochastic responses but also improves the detection of subthreshold signals in the case of weak noise the autapse facilitates the response of neuron to the subthreshold sinusoidal signals with a small parameter region in t d e l a y space the increased noise intensity enlarges this parameter region and increases the corresponding response frequency in such range as the autaptic intensity increases however this parameter region shrunks we also observed that there is an optimal range of the delay time of autapse within which the stochastic hh neuron fires action potentials with high frequency the corresponding response spike train for the optimal delay time is nearly a regular sequence with the interspike intervals approximated to the delay time the current results reveal a novel resonance phenomenon facilitated by autapse named autaptic delay induced coherence resonance

the functional model for maximal dissipative operators translation form an approach in the spirit of operator knots in this article we develop a functional model for a general maximal dissipative operator we construct the selfadjoint dilation of such operators unlike previous functional models our model is given explicitly in terms of parameters of the original operator making it more useful in concrete applications for our construction we introduce an abstract framework for working with a maximal dissipative operator and its anti dissipative adjoint and make use of the straus characteristic function in our setting explicit formulae are given for the selfadjoint dilation its resolvent a core and the completely non selfadjoint subspace minimality of the dilation is shown the abstract theory is illustrated by the example of a schrodinger operator on a half line with dissipative potential and boundary condition and connections to existing theory are discussed

spectral theory for perturbed krein laplacians in nonsmooth domains abstract we study spectral properties for h k the krein von neumann extension of the perturbed laplacian v defined on c 0 where v is measurable bounded and nonnegative in a bounded open set r n belonging to a class of nonsmooth domains which contains all convex domains along with all domains of class c 1 r r 1 2 in particular in the aforementioned context we establish the weyl asymptotic formula j n k j 2 n v n n 2 o n 1 2 2 as where v n n 2 n 2 1 denotes the volume of the unit ball in r n and k j j n are the non zero eigenvalues of h k listed in increasing order according to their multiplicities we prove this formula by showing that the perturbed krein laplacian i e the krein von neumann extension of v defined on c 0 is spectrally equivalent to the buckling of a clamped plate problem and using an abstract result of kozlov from the mid 1980s our work builds on that of grubb in the early 1980s who has considered similar issues for elliptic operators in smooth domains and shows that the question posed by alonso and simon in 1980 pertaining to the validity of the above weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting we also study certain exterior type domains r n k n 3 with k r n compact and vanishing bessel capacity b 2 2 k 0 to prove equality of friedrichs and krein laplacians in l 2 d n x that is c 0 has a unique nonnegative self adjoint extension in l 2 d n x

inverse spectral theory for symmetric operators with several gaps scalar type weyl functions let s be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non zero deficiency indices let j be an open subset of r if there exists a self adjoint extension s0 of s such that j is contained in the resolvent set of s0 and the associated weyl function of the pair s s0 is monotone with respect to j then for any self adjoint operator r there exists a self adjoint extension s such that the spectral parts s j and rj are unitarily equivalent it is shown that for any extension s of s the absolutely continuous spectrum of s0 is contained in that one of s moreover for a wide class of extensions the absolutely continuous parts of s and s are even unitarily equivalent

on the many dirichlet laplacians on a non convex polygon and their approximations by point interactions abstract by birman and skvortsov it is known that if is a planar curvilinear polygon with n non convex corners then the laplace operator with domain h 2 h 0 1 is a closed symmetric operator with deficiency indices n n here we provide a kre n type resolvent formula for any self adjoint extensions of such an operator i e for the set of self adjoint non friedrichs dirichlet laplacians on and show that any element in this set is the norm resolvent limit of a suitable sequence of friedrichs dirichlet laplacians with n point interactions

spontaneous spiking in an autaptic hodgkin huxley setup the effect of intrinsic channel noise is investigated for the dynamic response of a neuronal cell with a delayed feedback loop the loop is based on the so called autapse phenomenon in which dendrites establish connections not only to neighboring cells but also to its own axon the biophysical modeling is achieved in terms of a stochastic hodgkin huxley model containing such a built in delayed feedback the fluctuations stem from intrinsic channel noise being caused by the stochastic nature of the gating dynamics of ion channels the influence of the delayed stimulus is systematically analyzed with respect to the coupling parameter and the delay time in terms of the interspike interval histograms and the average interspike interval the delayed feedback manifests itself in the occurrence of bursting and a rich multimodal interspike interval distribution exhibiting a delay induced reduction in the spontaneous spiking activity at characteristic frequencies moreover a specific frequency locking mechanism is detected for the mean interspike interval

a remark on schatten von neumann properties of resolvent differences of generalized robin laplacians on bounded domains in this note we investigate the asymptotic behavior of the s numbers of the resolvent difference of two generalized self adjoint maximal dissipative or maximal accumulative robin laplacians on a bounded domain with smooth boundary for this we apply the recently introduced abstract notion of quasi boundary triples and weyl functions from extension theory of symmetric operators together with krein type resolvent formulae and well known eigenvalue asymptotics of the laplace beltrami operator on it is shown that the resolvent difference of two generalized robin laplacians belongs to the schatten von neumann class of any order p for which

on kreĭn s formula abstract kre n s formula provides a parametrization of the generalized resolvents and straus extensions of a closed symmetric operator with equal possibly infinite defect numbers in a hilbert space in terms of nevanlinna families in a parameter space the aim of this note is to give a simple complete analytical proof of kre n s formula

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

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

on eisenbud s and wigner s r matrix a general approach abstract the main objective of this paper is to give a rigorous treatment of wigner s and eisenbud s r matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices in the framework of boundary triplets and associated weyl functions an abstract generalization of the r matrix method is developed and the results are applied to schrodinger operators on the real axis

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

pt symmetry cartan decompositions lie triple systems and krein space related clifford algebras gauged quantum mechanics ptqm and corresponding krein space setups are studied for models with constant non abelian gauge potentials and extended parity inversions compact and noncompact lie group components are analyzed via cartan decompositions a lie triple structure is found and an interpretation as symmetrically generalized jaynes cummings model is possible with close relation to recently studied cavity qed setups with transmon states in multilevel artificial atoms for models with abelian gauge potentials a hidden clifford algebra structure is found and used to obtain the fundamental symmetry of krein space related j self adjoint extensions for ptqm setups with ultra localized potentials

boundary value problems for elliptic partial differential operators on bounded domains for a symmetric operator or relation a with infinite deficiency indices in a hilbert space we develop an abstract framework for the description of symmetric and self adjoint extensions a of a as restrictions of an operator or relation t which is a core of the adjoint a this concept is applied to second order elliptic partial differential operators on smooth bounded domains and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

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

spectral analysis of selfadjoint elliptic differential operators dirichlet to neumann maps and abstract weyl functions abstract the spectrum of a selfadjoint second order elliptic differential operator in l 2 r n is described in terms of the limiting behavior of dirichlet to neumann maps which arise in a multi dimensional glazman decomposition and correspond to an interior and an exterior boundary value problem this leads to pde analogs of renowned facts in spectral theory of odes the main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding weyl functions and are applied to the pde setting afterwards

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

autaptic pacemaker mediated propagation of weak rhythmic activity across small world neuronal networks abstract we study the effects of an autapse which is mathematically described as a self feedback loop on the propagation of weak localized pacemaker activity across a newman watts small world network consisting of stochastic hodgkin huxley neurons we consider that only the pacemaker neuron which is stimulated by a subthreshold periodic signal has an electrical autapse that is characterized by a coupling strength and a delay time we focus on the impact of the coupling strength the network structure the properties of the weak periodic stimulus and the properties of the autapse on the transmission of localized pacemaker activity obtained results indicate the existence of optimal channel noise intensity for the propagation of the localized rhythm under optimal conditions the autapse can significantly improve the propagation of pacemaker activity but only for a specific range of the autaptic coupling strength moreover the autaptic delay time has to be equal to the intrinsic oscillation period of the hodgkin huxley neuron or its integer multiples we analyze the inter spike interval histogram and show that the autapse enhances or suppresses the propagation of the localized rhythm by increasing or decreasing the phase locking between the spiking of the pacemaker neuron and the weak periodic signal in particular when the autaptic delay time is equal to the intrinsic period of oscillations an optimal phase locking takes place resulting in a dominant time scale of the spiking activity we also investigate the effects of the network structure and the coupling strength on the propagation of pacemaker activity we find that there exist an optimal coupling strength and an optimal network structure that together warrant an optimal propagation of the localized rhythm

a bound for the eigenvalue counting function for krein von neumann and friedrichs extensions for an arbitrary open nonempty bounded set rn n n and sufficiently smooth coefficients a b q we consider the closed strictly positive higher order differential operator a 2m a b q in l2 defined on w02m 2 associated with the differential expression 2m a b q j k 1n i j bj aj k i k bk q m m n and its krein von neumann extension ak 2m a b q in l2 denoting by n ak 2m a b q 0 the eigenvalue counting function corresponding to the strictly positive eigenvalues of ak 2m a b q we derive the bound n ak 2m a b q cvn 2 n 1 2m2m n n 2m n 2m 0 where c c a b q 0 with c in 0 0 is connected to the eigenfunction expansion of the self adjoint operator a 2m a b q in l2 rn defined on w2m 2 rn corresponding to 2m a b q here vn n 2 n 2 2 denotes the euclidean volume of the unit ball in rn our method of proof relies on variational considerations exploiting the fundamental link between the krein von neumann extension and an underlying abstract buckling problem and on the distorted fourier transform defined in terms of the eigenfunction transform of a 2 a b q in l2 rn we also consider the analogous bound for the eigenvalue counting function for the friedrichs extension af 2m a b q in l2 of a 2m a b q no assumptions on the boundary of are made

products of generalized nevanlinna functions with symmetric rational functions abstract new classes of generalized nevanlinna functions which under multiplication with an arbitrary fixed symmetric rational function remain generalized nevanlinna functions are introduced characterizations for these classes of functions are established by connecting the canonical factorizations of the product function and the original generalized nevanlinna function in a constructive manner also a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function the operator theoretic treatment of these realizations is based on the notions of rigged spaces boundary triplets and associated weyl functions

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

projective hilbert space structures at exceptional points a non hermitian complex symmetric 2 2 matrix toy model is used to study projective hilbert space structures in the vicinity of exceptional points eps the bi orthogonal eigenvectors of a diagonalizable matrix are puiseuxexpanded in terms of the root vectors at the ep it is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the ep limit can be resolved by projectively extending the original hilbert space the complementary normalization conditions correspond then to two different affine charts of this enlarged projective hilbert space geometric phase and phase jump behaviour are analysed and the usefulness of the phase rigidity as measure for the distance to ep configurations is demonstrated finally ep related aspects of pt symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated

completion extension factorization and lifting of operators the well known results of m g kre n concerning the description of selfadjoint contractive extensions of a hermitian contraction t 1 and the characterization of all nonnegative selfadjoint extensions widetilde a of a nonnegative operator a via the inequalities a k le widetilde a le a f where a k and a f are the kre n von neumann extension and the friedrichs extension of a are generalized to the situation where widetilde a is allowed to have a fixed number of negative eigenvalues these generalizations are shown to be possible under a certain minimality condition on the negative index of the operators i t 1 t 1 and a respectively these conditions are automatically satisfied if t 1 is contractive or a is nonnegative respectively the approach developed in this paper starts by establishing first a generalization of an old result due to yu l shmul yan on completions of nonnegative block operators the extension of this fundamental result allows us to prove analogs of the above mentioned results of m g kre n and in addition to solve some related lifting problems for j contractive operators in hilbert pontryagin and kre n spaces in a simple manner also some new factorization results are derived for instance a generalization of the well known douglas factorization of hilbert space operators in the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non contractive selfadjoint operators under cayley transforms properly

regressive partition relations for infinite cardinals the regressive partition relation which turns out to be important in incompleteness phenomena is completely characterized in the transfinite case this work is related to schmerl si whose characterizations we complete regressive functions arise naturally in the study of infinite cardinals from fodor s well known lemma to contexts involving large cardinals for example the n subtle cardinals of baumgartner b2 in kanamori and mcaloon km a regressive function version of the theorem of erd6s and rado er on canonical partitions was miniaturized and shown to be independent of peano arithmetic this result in turn reverberated to the infinite context to raise new questions here we completely characterize the corresponding partition symbol for infinite cardinals in contrast to the erdos rado theorem for the ordinary partition symbol we show that these partition relations actually provide a characterization of cardinals in the finite mahlo hierarchy thus just as with the finite miniaturization an elementary combinatorial property leads to a necessary transcendence our work confirms some speculations in mcaloon m where an infinitary analogue of the paris harrington partition relation is considered after we had already established some characterizations we became aware of the close relationship of this work to the results of schmerl s the third author then saw how to sharpen the characterization of schmerl s property as well as ours and this paper is written so as to approach these optimal results most directly the sharpening uses ideas of todorcevic t who noted that our 3 4 for n 0 can be derived directly from his work in 1 we begin the study of our partition symbol and establish the straightforward positive results about mahlo cardinals in 2 we develop some technical formulations and lemmata finally we apply this work in 3 to establish the optimal results on the necessity of mahlo cardinals we discuss the connections with schmerl s at the end we would like to thank jim schmerl for several expository suggestions particularly for providing the statement of 3 3 1 preliminaries we first formulate the regressive partition symbol let x be a set of ordinals and n a natural number if f is a function with domain xi n we write f ao an 1 for f ao an 1 with the understanding that ao k n 1 such a function is called regressive if f a0o an i ao received by the editors december 26 1985 1980 mathematics subject classification 1985 revision primary 03e05 secondary 03e55

dirichlet to neumann maps abstract weyl titchmarsh m functions and a generalized index of unbounded meromorphic operator valued functions abstract we introduce a generalized index for certain meromorphic unbounded operator valued functions the class of functions is chosen such that energy parameter dependent dirichlet to neumann maps associated to uniformly elliptic partial differential operators particularly non self adjoint schrodinger operators on bounded lipschitz domains and abstract operator valued weyl titchmarsh m functions and donoghue type m functions corresponding to closed extensions of symmetric operators belong to it the principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of robin realizations of non self adjoint schrodinger operators and more abstract pairs of closed operators in hilbert spaces with the generalized index of the corresponding energy dependent dirichlet to neumann maps and abstract weyl titchmarsh m functions respectively

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

p adic schrodinger type operator with point interactions a p adic schrodinger type operator d vy is studied d 0 is the operator of fractional differentiation and vy i j 1nbij xj xi bij c is a singular potential containing the dirac delta functions x concentrated on a set of points y x1 xn of the field of p adic numbers qp it is shown that such a problem is well posed for 1 2 and the singular perturbation vy is form bounded for 1 in the latter case the spectral analysis of self adjoint operator realizations of d vy in l2 qp is carried out

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

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

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