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

matrix representations of sturm liouville problems with transmission conditions we identify a class of sturm liouville equations with transmission conditions such that any sturm liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem conversely given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self adjoint boundary condition and transmission condition we construct a class of sturm liouville problems with this specified boundary condition and transmission condition each of which is equivalent to the given matrix eigenvalue problem

the fixed point property in every weak homotopy type we prove that for any connected compact cw complex k there exists a space x weak homotopy equivalent to k which has the fixed point property that is every continuous map x to x has a fixed point the result is known to be false if we require x to be a polyhedron the space x we construct is a non hausdorff space with finitely many points

brief new characterization and solution of input to state stabilization a state dependent scaling approach this paper proposes a novel approach to input to state stabilization and integral input to state stabilization problems for nonlinear systems based on the concept of state dependent scaling design both state feedback and output feedback controllers are constructed in a unified way the method provides global solutions whenever the system is in the strict feedback or output feedback form the paper allows the system to have structured static and dynamic uncertainties an advanced recursive procedure of observer design is the key to input to state stabilization by output feedback

continuous dependence of the n th eigenvalue of self adjoint discrete sturm liouville problems on the problem abstract this paper is concerned with continuous dependence of the n th eigenvalue of self adjoint discrete sturm liouville problems on the problem the n th eigenvalue is considered as a function on the space of the problems called the n th eigenvalue function a necessary and sufficient condition for all the n th eigenvalue functions to be continuous and several properties of the n th eigenvalue function on a subset of the space of the problems are given they play an important role in the study of continuous dependence of the n th eigenvalue function on the problem continuous dependence of the n th eigenvalue function on the equation and on the boundary condition is studied separately consequently the continuity and discontinuity of the n th eigenvalue function are completely characterized on the whole space of the problems especially asymptotic behavior of the n th eigenvalue function near each discontinuity point is given

high order approximations of the eigenvalues of regular sturm liouville problems abstract in this article we provide high order approximations to the eigenvalues of regular sturm liouville problems with general separable boundary conditions together with their error bounds the approach is based on sampling theory

matrix representations of fourth order boundary value problems with periodic boundary conditions we show that a class of regular self adjoint fourth order boundary value problems bvps is equivalent to a certain class of matrix problems conversely for any given matrix problem in this class there exist fourth order self adjoint bvps which are equivalent to the given matrix problem equivalent here means that they have exactly the same eigenvalues such an equivalence is considered with the special coupled boundary conditions i e periodic boundary conditions

on some eigenvalue problems in fuel cell dynamics an eigenvalue problem arising from the study of gas dynamics in a loaded tubular solid oxide fuel cell is considered an asymptotic theory from which the equations are derived is reviewed and the results of analysis on the small and large parameter asymptotics are presented these results suggest an interesting and hitherto unknown property of a class of sturm liouville problems in which the first eigenvalue approaches zero but subsequent ones approach infinity as a parameter approaches zero this was first discovered numerically and later confirmed asymptotically and rigorously

dependence of eigenvalues on the diffusion operators with random jumps from the boundary abstract this paper deals with a non self adjoint eigenvalue problem a x y x b x y x y x y 0 0 1 y x d 0 x y 1 0 1 y x d 1 x which is associated with the generator of one dimensional diffusions with random jumps from the boundary we focus on the dependence of spectral gap eigenvalues and eigenfunctions on the coefficients a b and the probability distributions 0 1 to prove this we show that all the eigenvalues are confined to a parabolic neighborhood of the real axis moreover we also prove that zero is an algebraically simple eigenvalue of the problem

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

fourth order boundary value problems with finite spectrum abstract for every positive integer m we construct a class of regular self adjoint and non self adjoint fourth order boundary value problems with at most 3 m 1 eigenvalues counting multiplicity

dependence of eigenvalues on the boundary conditions of sturm liouville problems with one singular endpoint abstract this paper is concerned with continuous and differentiable dependence of isolated eigenvalues on the boundary conditions of self adjoint sturm liouville problems with one singular endpoint locally continuous dependence of eigenvalues on the boundary conditions is proved especially in the limit point case the continuous dependence of isolated eigenvalues is shown by the relationships between weyl titchmarsh m function and the spectrum of the singular problems then continuous eigenvalue branches through each isolated eigenvalue over the space of boundary conditions are formed it is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound in l w 2 norm its proof is different from that in the regular sturm liouville problems the derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given and thus its monotonicity with respect to some parameters is derived

optimal partitioning of an interval and applications to sturm liouville eigenvalues abstract we study the optimal partitioning of a possibly unbounded interval of the real line into n subintervals in order to minimize the maximum of certain set functions under rather general assumptions such as continuity monotonicity and a radon nikodym property we prove existence and uniqueness of a solution to this minimax partition problem showing that the values of the set functions on the intervals of any optimal partition must coincide we also investigate the asymptotic distribution of the optimal partitions as n tends to infinity several examples of set functions fit in this framework including measures weighted distances and eigenvalues we recover in particular some classical results of sturm liouville theory the asymptotic distribution of the zeros of the eigenfunctions the asymptotics of the eigenvalues and the celebrated weyl law on the asymptotics of the counting function

existence and nonexistence of solutions of second order nonlinear boundary value problems abstract we study the nonlinear boundary value problem consisting of the equation y q t y w t f y on a b and a general separated homogeneous linear boundary condition by comparing this problem with a corresponding linear sturm liouville problem we obtain conditions for the existence and nonexistence of solutions of this problem more specifically let n n 0 1 2 be the n th eigenvalues of the corresponding linear sturm liouville problem then under certain assumptions the boundary value problem has a solution with exactly n zeros in a b if n is in the interior of the range of f y y y 0 and does not have any solution with exactly n zeros in a b if n is outside of the range of f y y y 0 these conditions become necessary and sufficient when f y y is monotone the existences of multiple and even an infinite number of solutions are derived as consequences we also discuss the changes of the number and the types of nontrivial solutions as the interval a b shrinks as q increases in a given direction and as the boundary condition changes

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

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 finite spectrum of sturm liouville problems with transmission conditions for any positive integer n and a set of positive integers m i i 1 2 ldots n 1 we construct a class of regular sturm liouville problems with n transmission conditions which have exactly sum n 1 i 1 m i n 1 eigenvalues and further we show that these sum n 1 i 1 m i n 1 eigenvalues can be distributed arbitrarily throughout the complex plane in the non self adjoint case and anywhere along the real line in the self adjoint case

inverse sturm liouville problems with finite spectrum abstract we study inverse sturm liouville problems of atkinson type whose spectrum consists entirely of a finite set of eigenvalues we show that given two finite sets of interlacing real numbers there exists a class of sturm liouville equations of atkinson type such that the two sets of numbers are the eigenvalues of their associated sturm liouville problems with two different separated boundary conditions parallel results are also obtained for real coupled boundary conditions our approach is to use the equivalence between sturm liouville problems of atkinson type and matrix eigenvalue problems and to apply our development of the well known theory for inverse matrix eigenvalue problems

on sturm liouville operators with discontinuity conditions inside an interval abstract in this study transformation operator some properties of kernel properties of spectral characteristics and also uniquiness theorems for the inverse problems are studied for a class of sturm liouville differential operators which have discontinuity conditions inside a finite interval

matrix representations of sturm liouville problems with coupled eigenparameter dependent boundary conditions abstract we study the matrix representations of sturm liouville problems with coupled eigenparameter dependent boundary conditions with a finite spectrum we prove for any positive integer n the considered problems have at most n 3 eigenvalues and show that this kind of sturm liouville problems with coupled eigenparameter dependent boundary conditions is equivalent to a class of matrix eigenvalue problems in the sense that they have exactly the same eigenvalues

the study of jacobi and cyclic jacobi matrix eigenvalue problems using sturm liouville theory abstract we study the eigenvalues of matrix problems involving jacobi and cyclic jacobi matrices as functions of certain entries of particular interest are the limits of the eigenvalues as these entries approach infinity our approach is to use the recently discovered equivalence between these problems and a class of sturm liouville problems and then to apply the sturm liouville theory

extremal values of eigenvalues of sturm liouville operators with potentials in l 1 balls abstract this paper is a continuation of zhang m zhang continuity in weak topology higher order linear systems of ode sci china ser a 51 2008 1036 1058 m zhang extremal values of smallest eigenvalues of hill s operators with potentials in l 1 balls j differential equations 246 2009 4188 4220 given a potential q l p 0 1 r p 1 we use m q to denote the m th dirichlet eigenvalue of the sturm liouville operator with potential q t where m n the minimal value l m p r and the maximal value m m p r of m q with potentials q in the l p ball of radius r are well defined in this paper we will exploit the continuity of m q in q with weak topologies and the variational method to give characterizations of l m p r and m m p r when p 1 by using the limiting approach as p 1 we find that the most important extremal values l m 1 r and m m 1 r can be evaluated explicitly using some elementary functions of r the corresponding extremal problems for neumann eigenvalues and some periodic eigenvalues will be reduced to l m p r and m m p r

computation of the eigenvalues of sturm liouville problems with parameter dependent boundary conditions using the regularized sampling method the purpose in this paper is to compute the eigenvalues of sturm liouville problems with quite general separated boundary conditions nonlinear in the eigenvalue parameter using the regularized sampling method an improvement on the method based on shannon sampling theory which does not involve any multiple integration and provides higher order estimates of the eigenvalues at a very low cost a few examples shall be presented to illustrate the power of the method and a comparison made with the the exact eigenvalues obtained as squares of the zeros of the exact characteristic functions

simple homotopy types and finite spaces whitehead s theory of simple homotopy types is inspired by tietze s theorem in combinatorial group theory which states that any finite presentation of a group could be deformed into any other by a finite sequence of elementary moves which are now called tietze transformations whitehead translated these algebraic moves into the well known geometric moves of elementary collapses and expansions of finite simplicial complexes

an exactly solvable periodic schrodinger operator the authors explicitly determine the band spectrum of a periodically strongly singular schrodinger operator in l2 r associated with the differential expression d2 dx2 s2 1 4 cos2 x s 2 1 4 sin2 x x epsilon r x epsilon pi z 2 0 s s 1 the corresponding density of states is calculated analytically

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 with discontinuities at two points in this paper we extend some spectral properties of regular sturm liouville problems to those which consist of a sturm liouville equation with piecewise continuous potentials together with eigenparameter dependent boundary conditions and four supplementary transmission conditions by modifying some techniques of c t fulton two point boundary value problems with eigenvalue parameter contained in the boundary conditions proc roy soc edinburgh sect a 77 1977 293 308 e tunc o sh muhtarov fundamental solutions and eigenvalues of one boundary value problem with transmission conditions appl math comput 157 2004 347 355 o sh mukhtarov e tunc eigenvalue problems for sturm liouville equations with transmission conditions israel j math 144 2004 367 380 and o sh mukhtarov m kadakal f s muhtarov eigenvalues and normalized eigenfunctions of discontinuous sturm liouville problem with transmission conditions rep math phys 54 2004 41 56 we give an operator theoretic formulation for the considered problem and obtain asymptotic formulae for the eigenvalues and eigenfunctions

equality of multiplicities of a sturm liouville eigenvalue abstract we give a new and unified proof of the fact that for any eigenvalue of a self adjoint sturm liouville problem with limit circle end points its analytic and geometric multiplicities are equal this proof is geometric in nature and can be generalized to the case of high order differential equations

a note on eigenvalues of a class of singular continuous and discrete linear hamiltonian systems abstract in this paper we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear hamiltonian systems are equal where both endpoints are in the limit circle cases the proof is fundamental and is given for both continuous and discrete hamiltonian systems the method used in this paper also works for both endpoints are regular or one endpoint is regular and the other is in the limit circle case

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

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

stieltjes sturm liouville equations eigenvalue dependence on problem parameters abstract we examine properties of eigenvalues and solutions to a 2 n dimensional stieltjes sturm liouville eigenvalue problem existence and uniqueness of a solution has been established previously an earlier paper considered the corresponding initial value problem and established conditions which guarantee that solutions depend continuously on the coefficients l e battle solution dependence on problem parameters for initial value problems associated with the stieltjes sturm liouville equations electron j differential equations 2005 2 2005 1 18 here we find conditions which guarantee that the eigenvalues and solutions depend continuously on the coefficients endpoints and boundary data for a simplified two dimensional problem we find conditions which guarantee the eigenvalues to be differentiable functions of the problem data

eigenvalues of limit point sturm liouville problems abstract the dependence of the eigenvalues of self adjoint sturm liouville problems on the boundary conditions when each endpoint is regular or in the limit circle case is now due to some surprisingly recent results well understood here we study this dependence for singular problems with one endpoint in the limit point case

spectral theory of higher order discrete vector sturm liouville problems abstract this paper is concerned with spectral problems of higher order vector difference equations with self adjoint boundary conditions where the coefficient of the leading term may be singular a suitable admissible function space is constructed so that the corresponding difference operator is self adjoint in it and the fundamental spectral results are obtained rayleigh s principles and minimax theorems in two special linear spaces are given as an application comparison theorems for eigenvalues of two sturm liouville problems are presented especially the dual orthogonality and multiplicity of eigenvalues are discussed

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

inverse indefinite sturm liouville problems with three spectra abstract we prove that the potential q x of an indefinite sturm liouville problem on the closed interval a b with the indefinite weight function w x can be determined uniquely by three spectra which are generated by the indefinite problem defined on a b and two right definite problems defined on a 0 and 0 b where point 0 lies in a b and is the turning point of the weight function w x

floquet theory for left definite sturm liouville problems abstract we determine the essential spectrum of left definite sturm liouville problems with periodic coefficients and a weight function which changes sign

matrix representations of sturm liouville problems with eigenparameter dependent boundary conditions abstract we show that a class of regular self adjoint sturm liouville problems with eigenparameter dependent boundary conditions are equivalent to a certain class of matrix problems equivalent here means that they have exactly the same eigenvalues

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

an isoperimetric type inequality for the principal eigenvalue of schrodinger operators depending on the curvature of a loop abstract let be the curvature of a smooth closed curve of length 2 the 2 periodic eigenvalue problems u g 2 u u 0 where g is a real parameter are considered and upper and lower bounds for their principal eigenvalues are obtained the bounds are of isoperimetric type and characterize uniquely the curve in the case where equality occurs

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

dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary in this paper we consider the dependence of eigenvalues of a class of fourth order sturm liouville problems on the boundary we show that the eigenvalues depend not only continuously but smoothly on boundary points and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation in addition we prove that as the length of the interval shrinks to zero all higher fourth order dirichlet eigenvalues march off to plus infinity this is also true for the first i e lowest eigenvalue

regular third order boundary value problems abstract in this paper we consider some boundary value problems generated by third order formally symmetric self adjoint regular differential expression and separated real coupled and complex coupled boundary conditions it is shown that these problems generate self adjoint operators moreover the dependence of eigenvalues of these problems on the data are studied and some derivatives of the eigenvalues with respect to some elements of data are introduced

singularity of the n th eigenvalue of high dimensional sturm liouville problems abstract it is natural to consider continuous dependence of the n th eigenvalue on d dimensional d 2 sturm liouville problems after the results on 1 dimensional case by kong wu and zettl 14 in this paper we find all the boundary conditions such that the n th eigenvalue is not continuous and give complete characterization of asymptotic behavior of the n th eigenvalue this renders a precise description of the jump phenomena of the n th eigenvalue near such a boundary condition furthermore we divide the space of boundary conditions into 2 d 1 layers and show that the n th eigenvalue is continuously dependent on sturm liouville equations and on boundary conditions when restricted into each layer in addition we prove that the analytic and geometric multiplicities of an eigenvalue are equal finally we obtain derivative formula and positive direction of eigenvalues with respect to boundary conditions

existence of multiple positive solutions for sturm liouville like four point boundary value problem with p laplacian abstract in this paper we consider the sturm liouville like four point boundary value problem with p laplacian p u t f t u t 0 t 0 1 u 0 u 0 u 1 u 0 where p s s p 2 s p 1 by means of a fixed point theorem for operators on a cone the existence of multiple at least three positive solutions to the above boundary value problem is obtained

characterizations of integral input to state stability for systems with multiple invariant sets we generalize the theory of input to state stability iss and of its characterizations by means of lyapunov dissipation inequalities to the study of systems admitting invariant sets which are not necessarily stable in the sense of lyapunov but admit a suitable hierarchical decomposition it is the latter which allows to greatly extend the class of systems to which iss theory can be applied allowing in a unified treatement to deal with oscillators in euclidean coordinates almost globally asymptotically stable systems on manifolds systems with multiple equilibria in bbr n just to name a few

the homotopy types of the posets of p subgroups of a finite group abstract we study the homotopy properties of the posets of p subgroups s p g and a p g of a finite group g viewed as finite topological spaces we answer a question raised by r e stong in 1984 about the relationship between the contractibility of the finite space a p g and that of s p g negatively and describe the contractibility of a p g in terms of algebraic properties of the group g

sturm liouville problems with impulse effects abstract this paper deals with the computation of the eigenvalues of sturm liouville problems with several discontinuity conditions inside a finite interval and parameter dependent boundary conditions using the regularized sampling method a few examples are worked out by computing several eigenvalues and their corresponding eigenfunctions

computing the indices of sturm liouville eigenvalues for coupled boundary conditions the eigenind slp codes the main purpose of the eigenind slp codes is to compute the indices of known eigenvalues of self adjoint sturm liouville problems with coupled boundary conditions bcs the spectrum of the problems can be unbounded from both below and above using some recent theoretical results the computation is converted to that of the indices of the same eigenvalues for appropriate separated bcs and is then carried out in terms of the prufer angle the algorithm so generated and its implementation are discussed and numerous examples are presented to illustrate the theoretical results and various aspects of the implementation

geometric aspects of sturm liouville problems ii space of boundary conditions for left definiteness for a given regular sturm liouville equation with an indefinite weight function we explicitly describe the space of left definite selfadjoint boundary conditions the description only uses one value of a fundamental solution of the matrix form of the equation as a consequence we show that this space has the shape of a solid consisting of two cones sharing a common base

accurate solutions of fourth order sturm liouville problems recently we introduced a new method which we call the extended sampling method to compute the eigenvalues of second order sturm liouville problems with eigenvalue dependent potential we shall see in this paper how we use this method to compute the eigenvalues of fourth order sturm liouville problems and present its practical use on a few examples

matrix representations of fourth order boundary value problems with finite spectrum abstract we show that a class of regular self adjoint fourth order boundary value problems is equivalent to a certain class of matrix problems equivalent here means that they have exactly the same eigenvalues such an equivalence was previously known only in the second order case

velocity observers for non linear mechanical systems subject to non smooth impacts this paper is concerned with the estimation of the velocity variables when the position variables are the measured outputs for non linear mechanical systems subject to non smooth impacts both elastic and inelastic i e with coefficient of restitution e 1 and e 0 1 respectively a reduced order observer is proposed which guarantees that the corresponding error system despite the possible presence of an infinite sequence of non smooth impacts is locally exponentially stable an estimate of the basin of attraction is also given

error estimate of eigenvalues of perturbed second order discrete sturm liouville problems abstract this paper is concerned with eigenvalues of perturbed second order vector discrete sturm liouville problems by some variational properties of eigenvalues of discrete sturm liouville problems error estimates of eigenvalues of perturbed problems sufficiently close to a given sturm liouville problem are given under a certain non singularity condition perturbations of the coefficient functions of the difference equation the weight function and the coefficients of the boundary condition are all considered this together with higher dimension involved results in a certain complexity of the problem and difficulty of study as a direct consequence continuous dependence of eigenvalues on problems is obtained under the non singularity condition in addition an example is presented to illustrate the necessity of the non singularity condition

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

multiplicity of sturm liouville eigenvalues the geometric multiplicity of each eigenvalue of a self adjoint sturm liouville problem is equal to its algebraic multiplicity this is true for regular problems and for singular problems with limit circle endpoints including the case when the leading coefficient changes sign

poset fiber theorems suppose that f p q is a poset map whose fibers f 1 qless than or equal to q are sufficiently well connected our main result is a formula expressing the homotopy type of p in terms of q and the fibers several fiber theorems from the literature due to babson baclawski and quillen are obtained as consequences or special cases homology cohen macaulay and equivariant versions are given and some applications are discussed

sturm liouville problems with finite spectrum abstract for every positive integer n we construct a class of regular self adjoint and non self adjoint sturm liouville problems with exactly n eigenvalues these n eigenvalues can be located anywhere in the complex plane in the non self adjoint case and anywhere along the real line in the self adjoint case the latter complements the well known general result for right definite sturm liouville problems with an infinite number of eigenvalues which must go to infinity asymptotically like n2 does with an appropriate and natural interpretation of a zero the eigenfunctions have the usual zero properties

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