geometric aspects of sturm liouville problems iii level surfaces of the nth eigenvalue for a given regular sturm liouville equation and any natural number n we explicitly describe the level surfaces of the nth eigenvalue of the equation regarded as a function on the space of self adjoint boundary conditions the description of each level surface uses only a particular value of the transfer matrix of the equation the shapes of these surfaces are also determined

dissipative eigenvalue problems for a singular dirac system abstract dissipative singular dirac operators are studied in the space l a 2 a b c 2 that the extensions of a minimal symmetric operator in weyl s limit point case we construct a selfadjoint dilation and its incoming and outgoing spectral representations which makes it possible to determine the scattering matrix of dilation we construct a functional model of the dissipative operator and specify its characteristic function in terms of the titchmarsh weyl function of selfadjoint operator we prove theorems on completeness of the system of eigenvectors and associated vectors of the dissipative dirac operators

dependence of eigenvalues of sturm liouville problems with eigenparameter dependent boundary conditions abstract this paper is concerned with regular sturm liouville problems with eigenparameter dependent boundary conditions we obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem in particular eigenparameter dependent boundary condition matrix moreover the differential expression of the eigenvalues with respect to the data is given the dependence of the nth eigenvalue as a function of the parameters of the problem is also investigated by pr u fer transformation and the inequalities of eigenvalues

spectral analysis of dissipative dirac operators with general boundary conditions abstract a space of boundary values is constructed for minimal symmetric dirac operator in l a 2 c 2 with defect index 2 2 in weyl s limit circle cases at a description of all maximal dissipative accretive selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at we investigate maximal dissipative operators with generally speaking nonseparated nondecomposed boundary conditions in particular if we consider separated boundary conditions at the nonselfadjoint dissipative boundary conditions are prescribed simultaneously we construct a selfadjoint dilation and its incoming and outgoing spectral representations which makes it possible to determine the scattering matrix of the dilation we also construct a functional model of the dissipative operator and define its characteristic function we prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative dirac operators

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

viability solutions to structured hamilton jacobi equations under constraints structured hamilton jacobi partial differential equations are hamilton jacobi equations where the time variable is replaced by a vector valued variable structuring the system it could be the time age pair hamilton jacobi mckendrick equations or candidates for initial or terminal conditions hamilton jacobi cournot equations among a manifold of examples here we define the concept of viability solution which always exists and can be computed by viability algorithms this is a constructive approach allowing us to derive from the tools of viability theory dealing with sets instead of functions some known as well as new properties of its solutions regarding functions as their epigraphs we bypass the regularity issues to arrive directly at the concept of barron jensen frankowska viscosity solutions above all we take into account viability constraints and extend classical boundary conditions to other internal conditions beyond that we use the fenchel legendre transform to associate a lagrangian with the hamiltonian and uncover an underlying variational problem it is proved that the viability solution to the structured hamilton jacobi equation is the valuation function of this variational principle the optimal evolutions of which are regulated by a regulation map which is constructed from the viability solution and can be computed by viability algorithms in other words the viability solution solves all at once a class of first order partial differential equations of intertemporal optimization problems and the regulation of their optimal evolutions

on the deficiency indices and self adjointness of symmetric hamiltonian systems abstract the main purpose of this paper is to investigate the formal deficiency indices n of a symmetric first order system jf bf h f on an interval i where i r or i r here j b h are n n matrix valued functions and the hamiltonian h 0 may be singular even everywhere we obtain two results for such a system to have minimal numbers n 0 if i r resp n n if i r and a criterion for their maximality n 2n for i r as well as the quasi regularity this covers the kac krein and de branges trans amer math soc 99 1961 118 theorems on 2 2 canonical systems and some results from kogan and rofe beketov proc roy soc edinburgh sect a 74 1974 75 5 some conditions for a canonical system to have intermediate formal deficiency indices are presented too we also obtain a generalization of the well known titchmarsh sears theorem for second order sturm liouville type equations this contains results due to lidskii and krein as special cases we present two approaches to the above problems one dealing with formal deficiency indices and one dealing with ordinary deficiency indices our main non formal approach is based on the investigation of a symmetric linear relation smin which is naturally associated to a first order system this approach works in the framework of extension theory and therefore we investigate in detail the domain d s min of s min in particular we prove the so called regularity theorem for d s min as a byproduct of the regularity result we obtain very short proofs of generalizations of the main results of the paper by kogan and rofe beketov 1974 75

dependence of eigenvalues of sturm liouville problems with interface conditions this paper investigates sturm liouville problems with interface conditions we obtain that the eigen values depend not only continuously but also smoothly on the coefficient functions boundary conditions and interface conditions moreover the derivative of the eigenvalues with respect to each of these parameters is given

a hierarchical model of quantum anharmonic oscillators critical point convergence a hierarchical model of interacting quantum particles performing anharmonic oscillations is studied in the euclidean approach in which the local gibbs states are constructed as measures on infinite dimensional spaces the local states restricted to the subalgebra generated by fluctuations of displacements of particles are in the center of the study they are described by means of the corresponding temperature green matsubara functions the result of the paper is a theorem which describes the critical point convergence of such matsubara functions in the thermodynamic limit

the spectrum of differential operators and square integrable solutions abstract we give a comprehensive account of the relationship between the square integrable solutions for real values of the spectral parameter and the spectrum of self adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d and we solve an open problem stated by weidmann in his well known 1987 monograph according to a well known result if one endpoint is regular and for some real value of the spectral parameter the number of linearly independent square integrable solutions is less than d then is in the essential spectrum of every self adjoint realization of the equation weidmann extends this result to the two singular endpoint case provided an additional condition is satisfied here we prove this result without the additional condition

characterization of domains of self adjoint ordinary differential operators article history the gkn glazman krein naimark theorem characterizes all self adjoint realizations of linear symmetric formally self adjoint ordinary differential equations in terms of maximal domain functions these functions depend on the coefficients and this dependence is implicit and complicated in the regular case an explicit characterization in terms of two point boundary conditions can be given in the singular case when the deficiency index d is maximal the gkn characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter i n the much more difficult intermediate cases not all solutions contribute to the singular self adjoint conditions in 1986 sun found a representation of the self adjoint singular conditions in terms of certain solutions for nonreal values of in this paper we give a representation in terms of certain solutions for real this leads to a classification of solutions as limit point lp or limit circle lc in analogy with the celebrated weyl classification in the second order case the lc solutions contribute to the singular boundary conditions the lp solutions do not the advantage of using real is not only because it is in general easier to find explicit solutions but more importantly it yields information about the spectrum

inverse spectral problem for quantum graphs the inverse spectral problem for the laplace operator on a finite metric graph is investigated it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2 to prove the result a trace formula connecting the spectrum of the laplace operator with the set of periodic orbits for the metric graph is established

a dissipative singular sturm liouville problem with a spectral parameter in the boundary condition abstract in the hilbert space l w 2 a b we consider nonselfadjoint singular sturm liouville boundary value problem with two singular end points a and b in limit circle cases at a and b and with a spectral parameter in the boundary condition the approach is based on the use of the maximal dissipative operator and the spectral analysis of this operator is adequate for boundary value problem we construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations which make it possible to determine the scattering matrix of the dilation we also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding sturm liouville equation on the basis of the results obtained regarding the theory of the characteristic function we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and sturm liouville boundary value problem

two interval sturm liouville operators in modified hilbert spaces abstract by modifying the inner product in the direct sum of the hilbert spaces associated with each of two underlying intervals on which the sturm liouville equation is defined we generate self adjoint realizations for boundary conditions with any real coupling matrix whose determinant is positive this contrasts with the usual theory which requires the coupling matrix to have determinant one

dependence of the nth sturm liouville eigenvalue on the problem abstract we consider the nth eigenvalue as a function on the space of self adjoint regular sturm liouville problems with positive leading coefficient and weight functions the discontinuity of the nth eigenvalue is completely characterized

sturm liouville problems and discontinuous eigenvalues if a sturm liouville problem is given in an open interval of the real line then regular boundary value problems can be considered on compact sub intervals for these regular problems all with necessarily discrete spectra the eigenvalues depend on both the end points of the compact intervals and upon the choice of the real separated boundary conditions at these end points these eigenvalues are not in general continuous functionsof the end points and boundary conditions the paper shows the surprising form of these discontinuities the results have applications to the approximations of singular sturm liouville problems by regular problems and to the theoretical aspects of the sleign2 computer program

spectral analysis for discontinuous non self adjoint singular dirac operators with eigenparameter dependent boundary condition abstract in this paper a discontinuous non self adjoint dissipative dirac operator with eigenparameter dependent boundary condition and with two singular endpoints is studied the interface conditions are imposed on the discontinuous point firstly we pass the considered problem to a maximal dissipative operator l h by using operator theoretic formulation the self adjoint dilation t h of l h in the space h is constructed furthermore the incoming and outgoing representations of t h and functional model are also constructed hence in light of lax phillips theory we derive the scattering matrix using the equivalence between scattering matrix and characteristic function completeness theorem on the eigenvectors and associated vectors of this dissipative operator is proved

eigenvalues of sturm liouville problems with discontinuity conditions inside a finite interval abstract in this work we use the regularized sampling method to compute the eigenvalues of sturm liouville problems with discontinuity conditions inside a finite interval we work out an example by computing a few eigenvalues and their corresponding eigenfunctions

renormalization of dyson s hierarchical vector valued φ 4 model at low temperatures we investigate dyson s hierarchical vector valued 4 model at low temperatures the case 2 c sqrt 2 is considered the pure phase is constructed and the existence of its large scale limit is proved the limit is gaussian but an unusual normalization has to be chosen in the direction of the spontaneous magnetization one has to divide by the square root of the volume but in the orthogonal direction one has to divide by a different power of the volume for all low temperatures

dependence of eigenvalues of sturm liouville problems on the boundary abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on boundary points the derivative of the n th eigenvalue as a function of an endpoint satisfies a first order differential equation this for arbitrary separated or coupled self adjoint regular boundary conditions in addition as the length of the interval shrinks to zero all higher eigenvalues march off to plus infinity this is also true for the first i e lowest dirichlet eigenvalue but not for the lowest neumann eigenvalue the latter has a finite limit

a nonlinear dynamical system on the set of laguerre entire functions a nonlinear modification of the cauchy problem dtf t z dzf t z zd 2 zf t z t ir 0 z c 0 f 0 z g z l is considered the set l consists of laguerre entire functions which one obtains as a closure of the set of polynomials having real nonpositive zeros only in the topology of uniform convergence on compact subsets of c the modification means that the time half line ir is divided onto the intervals in n 1 n n in 0 and on each in the evolution is to be described by the above equation but at the endpoints the function f t z is changed f n z

eigenvalues of regular sturm liouville problems abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on the problem an expression for the derivative of an eigenvalue with respect to a given parameter an endpoint a boundary condition a coefficient or the weight function is found

weak pullback attractors of setvalued processes weak pullback attractors are defined for nonautonomous setvalued processes and their existence and upper semicontinuous convergence under perturbation is established unlike strong pullback attractors invariance and pullback attraction here are required only for at least one trajectory rather than all trajectories at each starting point the concept is useful in for example continuous time control systems and is related to that of viability

boundary eigenvalue problems for differential equations nη λpη with λ polynomial boundary conditions abstract the present paper deals with the spectral properties of boundary eigenvalue problems for differential equations of the form n p on a compact interval with boundary conditions which depend on the spectral parameter polynomially here n as well as p are regular differential operators of order n and p respectively with n p 0 the main results concern the completeness minimality and riesz basis properties of the corresponding eigenfunctions and associated functions they are obtained after a suitable linearization of the problem and by means of a detailed asymptotic analysis of the green s function the function spaces where the above properties hold are described by independent boundary conditions an application to a problem from elasticity theory is given

the viability kernel algorithm for computing value functions of infinite horizon optimal control problems we characterize in this paper the epigraph of the value function of a discounted infinite horizon optimal control problem as the viability kernel of an auxiliary differential inclusion then the viability kernel algorithm applied to this problem provides the value function of the discretized optimal control problem as the supremum of a nondecreasing sequence of functions iteratively defined we also use the fact that an upper painleve kuratowski limit of closed viability domains is a viability domain to prove the convergence of the discrete value functions

spectral analysis of eigenparameter dependent boundary value transmission problems abstract in this paper the singular second order differential operators are considered defined on the multi interval some boundary and transmission conditions are imposed on the maximal domain functions with the spectral parameter after constructing the differential operators associated with the boundary value transmission problems on the suitable hilbert spaces it is proved that these operators are the maximal dissipative operators finally constructing the model operators which are established with the help of the scattering functions it is proved that all root vectors of the maximal dissipative operators are complete in the hilbert spaces

regular approximation of linear hamiltonian operators with two singular endpoints abstract in this paper we investigate regular approximation of linear hamiltonian operators with two singular endpoints for any given singular self adjoint hamiltonian operator its spectrum can be approximated with the eigenvalues of a sequence of regular operators the inherited restriction operators are constructed by the limit circle solutions and the limit point solutions and proved to be self adjoint via the abstract operator theory we obtain the strongly resolvent convergence of the inherited regular operators and spectral inclusion of spectrum moreover for the case with both endpoints being limit circle spectral exactness of spectrum is obtained by hilbert schmidt norm convergence in the case of friedrichs extension of singular hamiltonian systems spectral exactness also holds due to variational principle of eigenvalues

a gkn theorem for symmetric operators abstract we prove a gkn theorem for symmetric operators the well known gkn theorem for self adjoint operators is a special case of this result

inverse spectral problems for discontinuous sturm liouville problems of atkinson type abstract we investigate inverse spectral problems for discontinuous sturm liouville problems of atkinson type whose spectrum consists of a finite set of eigenvalues for given two finite sets of interlacing real numbers there exists a class of sturm liouville equations such that the two sets of numbers are exactly the eigenvalues of their associated sturm liouville problems with two different separated boundary conditions the main approach is to give an equivalent relation between sturm liouville problems of atkinson type and matrix eigenvalue problems and the theory of inverse matrix eigenvalue problems

dependence of eigenvalues of fourth order differential equations with discontinuous boundary conditions on the problem abstract fourth order boundary value problems with discontinuous boundary conditions are studied we prove that the eigenvalues depend not only continuously but smoothly on the coefficients and on the boundary conditions and find formulas for the derivatives with respect to each of these parameters

some observations and remarks on differential operators generated by first order boundary value problems this paper deals with the study of the set of all self adjoint differential operators which are generated from first order linear ordinary boundary value problems these operators are defined on a weighted hilbert function space and are examined as an application of the result obtained by everitt and markus in their paper in 1997 an investigation is given so that first order self adjoint boundary value problems are transformed to a study of the nature of the spectrum of associated self adjoint operators however the analysis of this paper is restricted to consideration of conditions under which the spectral properties of these operators yield a discrete spectrum and consequently to the determination of conditions under which the construction of kramer analytic kernels from the above boundary value problems can be accomplished

hamiltonian systems of limit point or limit circle type with both endpoints singular abstract a linear hamiltonian system j y a b y is considered on an open interval a b where both a and b are singular the system is assumed to be of limit point or limit circle type at the endpoints a theory of boundary problems for such systems is developed explicit boundary conditions are given resolvent operators constructed and unique solutions established the results given extend to hamiltonian systems a theory of singular boundary value problems due to m h stone and k kodaira

a multiscale asymptotic analysis of time evolution equations on the complex plane using an appropriate norm on the space of entire functions we extend to the complex plane the renormalization group method as developed by bricmont et al the method is based upon a multiscale approach that allows for a detailed description of the long time asymptotics of solutions to initial value problems the time evolution equation considered here arises in the study of iterations of the block spin renormalization group transformation for the hierarchical n vector model we show that for initial conditions belonging to a certain frechet space of entire functions of exponential type the asymptotics is universal in the sense that it is dictated by the fixed point of a certain operator acting on the space of initial conditions

the glazman krein naimark theory for a class of discrete hamiltonian systems abstract in this paper the glazman krein naimark theory for a class of discrete hamiltonian systems is developed a minimal and a maximal operators gkn sets and a boundary space for the system are introduced algebraic characterizations of the domains of self adjoint extensions of the minimal operator are given a close relationship between the domains of self adjoint extensions and the gkn sets is established it is shown that there exist one to one correspondences among the set of all the self adjoint extensions the set of all the d dimensional lagrangian subspaces of the boundary space and the set of all the complete lagrangian subspaces of the boundary space

hierarchical ferromagnetic vector spin model possessing the lee yang property thermodynamic limit at the critical point and above the hierarchical ferromagneticn dimensional vector spin model as a sequence of probability measures onrn is considered the starting element of this sequence is chosen to belong to the lee yang class of measures that is defined in the paper and includes most known examples 4 measures gaussian measures and so on for this model we probe two thermodynamic limit theorems one of them is just the classical central limit theorem for weakly dependent random vectors it describes the convergence of classically normed sums of spins when temperature is sufficiently high the other theorem describes the convergence of more than normally normed sums that holds for some fixed temperature it corresponds to the strong dependence of spins which appears at the critical point of the model

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

gkn theory for linear hamiltonian systems abstract in this paper we give the definition of maximal and minimal operators for linear hamiltonian systems and investigate the relationship between the conjugate scalar product in a weighted hilbert space and the skew symmetric boundary form of the associated singular hamiltonian operator namely the one to one correspondence between the set of self adjoint extensions of the minimal operator and the set of lagrangian symplectic subspaces these results extend and improve the classical glazman krein naimark gkn theory for quasi differential operators

regular approximation of singular sturm liouville problems with transmission conditions in this paper we study regular approximation of singular sturm liouville problems with transmission conditions we construct all self adjoint extensions of the minimal operators and the induced restriction operators using the abstract operator theory in the hilbert space we obtain that the spectrum of singular sturm liouville problems with transmission conditions can be approximated by eigenvalues of a sequence of regular sturm liouville problems with transmission conditions

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

on the boundary conditions in self adjoint multi interval sturm liouville problems abstract in this paper we completely solve a problem whose two interval case has been studied recently by wang et al in a wang j sun a zettl two interval sturm liouville operators in modified hilbert spaces j math anal appl 328 2007 390 399 the problem is to give an explicit description of all boundary conditions defining the self adjoint differential operators associated with several sturm liouville equations together i e the so called multi interval sturm liouville problems the explicit description obtained in this paper only uses hermitian matrices as an application we then show that a generic self adjoint boundary condition can not come from any direct sum of self adjoint operators associated with the individual sturm liouville equations and it depends non trivially on the multiples used in the definition of the so called maximum operator

eigenparameter dependent sturm liouville problems in boundary conditions with transmission conditions abstract in this paper we investigate the nonselfadjoint dissipative boundary value transmission problems in weyl s limit circle case at first using the method of operator theoretic formulation we pass to a new operator after showing that this new operator is a maximal dissipative operator we construct a selfadjoint dilation of the maximal dissipative operator using the equivalence of the lax phillips scattering function and the sz nagy foias characteristic function we show that all eigenfunctions and associated functions are complete in the space l w 2

phase transitions and algebra of fluctuation operators in an exactly soluble model of a quantum anharmonic crystal a complete description of the fluctuation operator algebra is given for a quantum crystal showing displacement structural phase transitions in the one phase region the fluctuations are normal and its algebra is non abelian in the two phase region and on the critical line t c 0 the momentum fluctuation is normal the displacement is critical and the algebra is abelian att c 0 quantum phase transition this algebra is non abelian with abnormal displacement and supernormal squeezed momentum fluctuation operators both being dimension dependent