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

self adjoint elliptic operators with boundary conditions on not closed hypersurfaces abstract the theory of self adjoint extensions of symmetric operators is used to construct self adjoint realizations of a second order elliptic differential operator on r n with linear boundary conditions on a relatively open part of a compact hypersurface our approach allows to obtain kre n like resolvent formulae where the reference operator coincides with the free operator with domain h 2 r n this provides an useful tool for the scattering problem from a hypersurface concrete examples of this construction are developed in connection with the standard boundary conditions dirichlet neumann robin and type assigned either on a n 1 dimensional compact boundary or on a relatively open part schatten von neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven these give existence and completeness of the wave operators of the associated scattering systems

a trace formula for canonical differential expressions we prove a trace formula for pairs of self adjoint operators associated to canonical differential expressions an important role is played by the associated weyl function

area minimizers and boundary rigidity of almost hyperbolic metrics this paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones we show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings we also provide a more invariant view on the approach used in the above mentioned paper

krein s resolvent formula for self adjoint extensions of symmetric second order elliptic differential operators given a symmetric semi bounded second order elliptic differential operator a on a bounded domain with c1 1 boundary we provide a kre n type formula for the resolvent difference between its friedrichs extension and an arbitrary self adjoint one

rings modules and algebras in infinite loop space theory we give a new construction of the algebraic k theory of small permutative categories that preserves multiplicative structure and therefore allows us to give a unified treatment of rings modules and algebras in both the input and output this requires us to define multiplicative structure on the category of small permutative categories the framework we use is the concept of multicategory elsewhere also called colored operad a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction our method ends up in the hovey shipley smith category of symmetric spectra with an intermediate stop at a category of functors out of a particular wreath product

on asymptotic volume of finsler tori minimal surfaces in normed spaces and symplectic filling volume the main unconditional result of this paper theorem 3 states that every two dimensional affine disc in a normed space that is a disc contained in a two dimensional affine subspace is an area minimizing surface among all immersed discs with the same boundary with respect to the symplectic holmes thompson surface area to emphasize that this is not at all obvious it may be worth mentioning that a similar statement with rational chains in place of immersed discs is incorrect theorem 2 and that it is not known for surfaces that may not be topological discs the result still may not sound too exciting to the reader who never looked at the problem before even though it goes back to busemann s works in the 50 s see bes th and references there and the proof heavily relies on asymptotic geometry of tori we believe that it is more important that we embed this problem into a whole area of mostly open problems as well as give some partial results and suggest certain directions of how to attack them we begin with a trivial statement

an isoperimetric problem for point interactions we consider a hamiltonian with n point interactions in all with the same coupling constant placed at vertices of an equilateral polygon it is shown that the ground state energy is locally maximized by a regular polygon the question whether the maximum is global is reduced to an interesting geometric problem

approximation by point potentials in a magnetic field we discuss magnetic schrodinger operators perturbed by measures from the generalized kato class using an explicit krein like formula for their resolvent we prove that these operators can be approximated in the strong resolvent sense by magnetic schrodinger operators with point potentials since the spectral problem of the latter operators is solvable one in fact gets an alternative way to calculate discrete spectra we illustrate it by numerical calculations in the case when the potential is supported by a circle

the mixed boundary value problem krein resolvent formulas and spectral asymptotic estimates abstract for a second order symmetric strongly elliptic operator a on a smooth bounded open set in r n the mixed problem is defined by a neumann type condition on a part of the boundary and a dirichlet condition on the other part we show a kre n resolvent formula where the difference between its resolvent and the dirichlet resolvent is expressed in terms of operators acting on sobolev spaces over this is used to obtain a new weyl type spectral asymptotics formula for the resolvent difference where upper estimates were known before namely s j j 2 n 1 c 0 2 n 1 where c 0 is proportional to the area of in the case where a is principally equal to the laplacian

an isoperimetric problem for leaky loops and related mean chord inequalities we consider a class of hamiltonians in l2 r2 with attractive interaction supported by piecewise c2 smooth loops of a fixed length l formally given by x with 0 it is shown that the ground state of this operator is locally maximized by a circular we also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of

lens rigidity for manifolds with hyperbolic trapped set for a riemannian manifold m g with strictly convex boundary m the lens data consists in the set of lengths of geodesics with endpoints on m together with their endpoints x x m m and tangent exit vectors v v t x m t x m we show deformation lens rigidity for manifolds with hyperbolic trapped set and no conjugate points a class which contains all manifolds with negative curvature and strictly convex boundary including those with non trivial topology and trapped geodesics for the same class of manifolds in dimension 2 we prove that the set of endpoints and exit vectors of geodesics ie the scattering data determines the riemann surface up to conformal diffeomorphism

lengths and volumes in riemannian manifolds we consider the question of when an inequality between lengths of corresponding geodesics implies a corresponding inequality between volumes we prove this in a number of cases for compact manifolds with and without boundary in particular we show that for two riemannian metrics of negative curvature on a compact surface without boundary an inequality between the marked length spectra implies the same inequality between the areas with equality implying isometry

rigidity of broken geodesic flow and inverse problems consider broken geodesics alpha 0 l on a compact riemannian manifold m g with boundary of dimension n geq 3 the broken geodesics are unions of two geodesics with the property that they have a common end point assume that for every broken geodesic alpha 0 l starting at and ending to the boundary partial m we know the starting point and direction alpha 0 alpha 0 the end point and direction alpha l alpha l and the length l we show that this data determines uniquely up to an isometry the manifold m g this result has applications in inverse problems on very heterogeneous media for situations where there are many scattering points in the medium and arises in several applications including geophysics and medical imaging as an example we consider the inverse problem for the radiative transfer equation or the linear transport equation with a nonconstant wave speed assuming that the scattering kernel is everywhere positive we show that the boundary measurements determine the wave speed inside the domain up to an isometry

triangulations orientals and skew monoidal categories abstract a concrete model of the free skew monoidal category fsk on a single generating object is obtained the situation is clubbable in the sense of g m kelly so this allows a description of the free skew monoidal category on any category as the objects of fsk are meaningfully bracketed words in the skew unit i and the generating object x it is necessary to examine bracketings and to find the appropriate kinds of morphisms between them this leads us to relationships between triangulations of polygons the tamari lattice left and right bracketing functions and the orientals a consequence of our description of fsk is a coherence theorem asserting the existence of a strictly structure preserving faithful functor fsk where is the skew monoidal category of finite non empty ordinals and first element and order preserving functions this in turn provides a complete solution to the word problem for skew monoidal categories

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

bound states in point interaction star graphs we discuss the discrete spectrum of the hamiltonian describing a two dimensional quantum particle interacting with an infinite family of point interactions we suppose that the latter are arranged into a star shaped graph with n arms and a fixed spacing between the interaction sites we prove that the essential spectrum of this system is the same as that of the infinite straight polymer but in addition there are isolated eigenvalues unless n 2 and the graph is a straight line we also show that the system has many strongly bound states if at least one of the angles between the star arms is small enough examples of eigenfunctions and eigenvalues are computed numerically

local boundary rigidity of a compact riemannian manifold with curvature bounded above this paper considers the boundary rigidity problem for a compact convex riemannian manifold m g with boundary o9m whose curvature satisfies a general upper bound condition this includes all nonpositively curved manifolds and all sufficiently small convex domains on any given riemannian manifold it is shown that in the space of metrics g on m there is a c3 neighborhood of g such that g is the unique metric with the given boundary distance function i e the function that assigns to any pair of boundary points their distance as measured in m more precisely given any metric g in this neighborhood with the same boundary distance function there is diffeomorphism o which is the identity on o9m such that g g there is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance function 1 statement of the result the general boundary rigidity problem reads to which extent is a riemannian metric on a compact manifold with boundary determined from the distances between boundary points more precisely it can be formulated as follows let m g be a compact riemannian manifold with boundary om let g be another riemannian metric on m we say that g and g have the same boundary distance function if d1 x y d9 x y for arbitrary boundary points x y e om where d1 resp d9 represents distance in m with respect to g resp g it is easy to give examples of pairs of metrics with the same boundary distance function indeed if m m is an arbitrary diffeomorphism of m onto itself which is the identity on the boundary then the metrics g and g g have the same boundary distance function here g og is the pull back of g under i e for arbitrary vectors 1 e txm we have r1 71 x i where p txm t x m is the differential of at x and resp is the inner product with respect to the metric g resp g we say that a compact riemannian manifold is boundary rigid if this is the only type of nonuniqueness more precisely m g is boundary rigid if for every received by the editors august 10 1998 2000 mathematics subject classification primary 53c65 53c22 53c20 secondary 58c25 58c40 the first and third authors were supported by crdf grant rm2 143 the first author was also supported by nsf grants mds 9505175 and mds 9626232 the third author was also supported by intas rfbr grant 95 0763 2000 american matlhematical society 3937 this content downloaded from 207 46 13 167 on thu 21 jul 2016 04 20 00 utc all use subject to http about jstor org terms 3938 c b croke n s dairbekov and v a sharafnjtdinov riemannian metric g on m1with the same boundary distance function as g there is a diffeomorphism o m m which is the identity on the boundary and for which g 0 of in ward and outward vectors here v is the unit outward norn al to om1 given x c qmvi we denote by k x the maximum of the sectional curvatures of all two planes oc t1 m such that c a for m g a cdrm we define the following invariant

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

rigidity of the scattering length spectrum in this paper we consider properties of obstacles satisfying some non degeneracy conditions that can be recovered from the scattering length spectrum sls clearly the latter tells us whether the obstacle k is trapping or non trapping if the set of trapped points is relatively small then the sls also determines the volume of the obstacle the number of its connected components and whether its boundary is convex everywhere or it has non trivial concavities under the additional assumption that the curvature of the obstacle does not vanish of infinite order it is proved that from the sls one can recover certain information about the number of reflection points of any simply reflecting ray in the exterior of the obstacle finally for some special classes of obstacles e g star shaped ones it is shown that the sls completely determines the obstacle

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

stability for the lens rigidity problem let g be a riemannian metric for mathbb r d d geqq 3 which differs from the euclidean metric only in a smooth and strictly convex bounded domain m the lens rigidity problem is concerned with recovering the metric g inside m from the corresponding lens relation on the boundary partial m in this paper the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non trapping metric satisfying the strong fold regular condition a metric g is called strong fold regular if for each point x in m there exists a set of geodesics passing through x whose conormal bundle covers t x m moreover these geodesics contain either no conjugate points or only fold conjugate points with a non degeneracy condition examples of strong fold regular metrics are constructed our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points the approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow which is a weighted geodesic x ray transform problem for symmetric 2 tensor fields a key ingredient is to show that the kernel of the x ray transform on symmetric solenoidal 2 tensor fields is of finite dimension it remains open whether the kernel space is trivial or not

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

the boundary rigidity problem in the presence of a magnetic field abstract for a compact riemannian manifold with boundary endowed with a magnetic potential we consider the problem of restoring the metric g and the magnetic potential from the values of the mane action potential between boundary points and the associated linearized problem we study simple magnetic systems in this case knowledge of the mane action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction this problem can only be solved up to an isometry and a gauge transformation of for the linearized problem we show injectivity up to the natural obstruction under explicit bounds on the curvature and on we also show injectivity and stability for g and in a generic class g including real analytic ones for the nonlinear problem we show rigidity for real analytic simple g rigidity for metrics in a given conformal class and locally near any g g we also show that simple magnetic systems on two dimensional manifolds are always rigid

hiatus perturbation for a singular schrodinger operator with an interaction supported by a curve in r3 we consider schrodinger operators in l2 r3 with a singular interaction supported by a finite curve we present a proper definition of the operators and study their properties in particular we show that the discrete spectrum can be empty if is short enough if it is not the case we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length 2 we derive an asymptotic expansion with the leading term which a multiple of ln

eigenvalue problems on exterior domains and dirichlet to neumann maps we consider a schroedinger equation on an exterior domain in the case where the potential which may be complex valued has a limit at infinity associated with the problem is a dirichlet to neumann map on the inner boundary we examine approximations to the problem obtained by a truncating the domain b replacing the potential by its value at infinity outside some set c a combination of a and b these approximations give rise to approximate dirichlet to neumann maps on the inner boundary we analyze the convergence of these maps and deduce results on the approximation of the spectrum of the original problem by the spectra of the approximating problems we show that in theory spurious eigenvalues cannot be generated by these procedures numerical experiments show however that the inherent ill conditioning of a problem may cause spurious eigenvalues to appear due to the discretization of the truncated problems even when the potential decays exponentially fast

the geodesic ray transform on riemannian surfaces with conjugate points we study the geodesic x ray transform x on compact riemannian surfaces with conjugate points regardless of the type of the conjugate points we show that we cannot recover the singularities and therefore this transform is always unstable ill posed we describe the microlocal kernel of x and relate it to the conjugate locus we present numerical examples illustrating the cancellation of singularities we also show that the attenuated x ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic but it is still ill posed if there are three or more conjugate points those results follow from our analysis of the weighted x ray transform

integral geometry of tensor fields on a class of non simple riemannian manifolds we study the geodesic x ray transform ig of tensor fields on a compact riemannian manifold m with non necessarily convex boundary and with possible conjugate points we assume that ig is known for geodesics belonging to an open set g with endpoints on the boundary we prove generic s injectivity and a stability estimate under some topological assumptions and under the condition that for any x t m there is a geodesic in g through x normal to without conjugate points

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

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

semiglobal boundary rigidity for riemannian metrics let m g be a riemannian manifold with boundary in this paper we consider the problem of determining the metric g from its associated boundary distance function dg x y dist x y x y m that is the geodesic distance between boundary points this problem arose in rigidity questions in riemannian geometry 12 6 8 for the case in which m is a bounded domain of euclidean space and the metric is conformal to the euclidean one this problem is called the inverse kinematic problem which arose in geophysics and has a long history starting at least in the early part of the 20th century with herglotz 10 he considered the case where m is a ball x r r x r equipped with a spherically symmetric metric ds2 dx2 c2 r where c r is a positive function depending only on the radius r x herglotz found a formula to determine c r from the boundary distance function physically this corresponds to the case of a spherically symmetric earth model with an index of refraction depending only on the radius the boundary distance function corresponds to the travel times of e g acoustic waves going through the earth and measured at the surface the general problem for the case that the sound speed depends on all variables has been extensively studied see for instance 17 and the references given there also this problem has a close connection for other inverse problems related to determining the sound speed from boundary measurements see 20

leaky quantum graphs approximations by point interaction hamiltonians we prove an approximation result showing how operators of the type x in where is a graph can be modelled in the strong resolvent sense by point interaction hamiltonians with an appropriate arrangement of the potentials the result is illustrated on finding the spectral properties in cases when is a ring or a star furthermore we use this method to indicate that scattering on an infinite curve which is locally close to a loop shape or has multiple bends may exhibit resonances due to quantum tunnelling or repeated reflections

boundary rigidity and filling volume minimality of metrics close to a flat one we say that a riemannian manifold m g with a non empty boundary m is a minimal orientable filling if for every compact orientable m g with m m the inequality d g x y d g x y for all x y m implies vol m g vol m g we show that if a metric g on a region m c r n with a connected boundary is sufficiently c 2 close to a euclidean one then it is a minimal filling by studying the equality case vol m g vol m g we show that if d g x y d g x y for all x y m then m g is isometric to m g this gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of michel s conjecture

boundary and lens rigidity of lorentzian surfaces let g be a lorentzian metric on the plane 2 that agrees with the standard metric g0 dx2 dy2 outside a compact set and so that there are no conjugate points along any time like geodesic of 2 g then 2 g and 2 g0 are isometric further if m g and m p are two dimensional compact time oriented lorentzian manifolds with space like boundaries and so that all time like geodesies of m g maximize the distances between their points and m g and m g are boundary isometric then there is a conformal diffeomorphism between m g and m g and they have the same areas similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds

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

skew monoidal categories and bialgebroids abstract skew monoidal categories arise when the associator and the left and right units of a monoidal category are in a specific way not invertible we prove that the closed skew monoidal structures on the category of right r modules are precisely the right bialgebroids over the ring r these skew monoidal structures induce quotient skew monoidal structures on the category of r r bimodules and this leads to the following generalization opmonoidal monads on a monoidal category correspond to skew monoidal structures with the same unit object which are compatible with the ordinary monoidal structure by means of a natural distributive law pursuing a theorem of day and street we also discuss monoidal lax comonads to describe the comodule categories of bialgebroids beyond the flat case

dirac structures and their composition on hilbert spaces dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures we study these structures and their properties from an operator theoretic point of view in particular we find necessary and sufficient conditions for the composition of two dirac structures to be a dirac structure and we show that they can be seen as lagrangian hyper maximal neutral subspaces of krein spaces moreover special emphasis is laid on dirac structures associated with operator colligations it turns out that this class of dirac structures is linked to boundary triplets and that this class is closed under composition

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

an inverse problem in elastodynamics uniqueness of the wave speeds in the interior abstract we consider the unique determination of internal properties of a nonhomogeneous isotropic elastic object from measurements made at the surface the 3 dimensional object is modelled by solutions of the linear hyperbolic system of equations for elastodynamics whose leading coefficients correspond to the internal properties of the object its density and elasticity we model surface measurements by the dirichlet to neumann map on a finite time interval in a previous paper the author has shown that the density and elastic properties of the surface of the object are uniquely determined by the dirichlet to neumann map here we apply that result to conclude that certain properties of the interior of the object the wave speeds are also determined we then observe that the elastodynamic polarization is determined outside the object by the dirichlet to neumann map we conclude in the case that the wave speeds are constant that the polarization data do not determine the density in the interior this problem and techniques used in its study are closely related to those in for example seismology and medical imaging the techniques used here though from geometric optics integral geometry and microlocal analysis lead to the solution of this fully three dimensional problem

inverse scattering problem in anisotropic media a new family of scattering invariants is constructed that differs from the scattering invariants obtained by the geometric optics approximations a local uniqueness modulo a diffeomorphism of the metric with a given hodograph function is proven under some restrictions on the curvature of the metric

skew monoidal categories and skew multicategories abstract we describe a perfect correspondence between skew monoidal categories and certain generalised multicategories called skew multicategories that arise in nature

schrodinger operators with singular interactions a model of tunnelling resonances we discuss a generalized schr odinger operator in l 2 r d d 2 3 with an attractive singular interaction supported by a d 1 dimensional hyperplane and a finite family of points it can be regarded as a model of a leaky quantum wire and a family of quantum dots if d 2 or surface waves in the presence of a finite number of impurities if d 3 we analyse the discrete spectrum and furthermore we show that the resonance problem in this setting can be explicitly solved by birman schwinger method it is cast into a form similar to the friedrichs model

heat flow for dirichlet to neumann operator with critical growth abstract in this article we study the heat flow equation for dirichlet to neumann operator with critical growth by assuming that the initial value is lower energy we obtain the existence blowup and regularity on the other hand a concentration phenomenon of the solution when the time goes to infinity is proved

on tori without conjugate points in this paper we consider riemannian metrics without conjugate points on an n torus recent work of j heber established that the gradient vector fields of busemann functions on the universal cover of such a manifold induce a natural foliation akin to the weak stable foliation for a riemannian manifold with negative sectional curvature on the unit tangent bundle the main result in the paper is that the metric is flat if this foliation is lipschitz we also prove that this foliation is lipschitz if and only if the metric has bounded asymptotes this confirms a conjecture of e hopf in this case

boundary rigidity and stability for generic simple metrics let m g be a riemannian manifold with boundary denote by pg the distance function in the metric g we consider the inverse problem of whether pg x y known for all x y on 9m determines the metric uniquely this problem arose in geophysics in an attempt to determine the inner structure of the earth by measuring the travel times of seismic waves it goes back to herglotz h and wiechert and zoeppritz wz although the emphasis has been in the case that the medium is isotropic the anisotropic case has been of interest in geophysics since it has been found that the inner core of the earth exhibits anisotropic behavior cr in differential geometry this inverse problem has been studied because of rigidity questions and is known as the boundary rigidity problem it is clear that one cannot determine the metric uniquely any isometry which is the identity at the boundary will give rise to the same measurements furthermore the boundary distance function only takes into account the shortest paths and it is easy to find counterexamples to unique determination so one needs to pose some restrictions on the metric michel mi conjectured that a simple metric g is uniquely determined up to an action of a diffeomorphism fixing the boundary by the boundary distance function pg x y known for all x and y on dm we recall definition 1 1 we say that the riemannian metric g is simple in m if dm is strictly convex w r t g and for any x g m the exponential map exp exp 1 m m is a diffeomorphism

on deformation boundary rigidity and spectral rigidity of riemannian surfaces with no focal points although the main results of the paper are obtained in the two dimensional case the problems under consideration can be posed in an arbitrary dimension let m g be a compact riemannian manifold with boundary the boundary rigidity problem consists of determining the riemannian metric g on m up to isometries which are the identity on the boundary by measuring the geodesic distance dg between boundary points it is easy to see that the answer to this question is no in general we can find metrics g such that there exists x0 m satisfying dist x0 m supx y mdg x y then we can change the metric g in a neighborhood of x0 without changing the boundary distance function a compact riemannian manifold m g with boundary is called boundary rigid if the metric g is uniquely determined by dg up to isometries which are the identity on the boundary there are a few classes of metrics which are known to be boundary rigid see 13 for references in this paper we consider the linearized problem let g be a one parameter family of riemannian metrics on m such that dg dg0 e e for a geodesic a b m of the metric g with a b m it is easy to see that

local lens rigidity with incomplete data for a class of non simple riemannian manifolds let be the scattering relation on a compact riemannian manifold m with nonnecessarily convex boundary that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction let be the length of that geodesic ray we study the question of whether the metric g is uniquely determined up to an isometry by knowledge of and restricted on some subset d we allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from d covers t m and that those geodesics have no conjugate points under an additional topological assumption we prove that and restricted to d uniquely recover an isometric copy of g locally near generic metrics and in particular near real analytic ones

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

estimates for the lowest eigenvalue of a star graph we derive new estimates for the lowest eigenvalue of the schrodinger operator associated with a star graph in r2 we achieve this by a variational method and a procedure for identifying test functions which are sympathetic to the geometry of the star graph

propagation of polarization in elastodynamics with residual stress and travel times we show that knowing the displacement to traction map associated to the equations of isotropic elastodynamics with residual stress we can determine the lens maps of compressional and shear waves we derive several consequences of this for the inverse problem of determining the residual stress and the lame parameters from the displacement to traction map

inverse scattering on matrices with boundary conditions we describe inverse scattering for the matrix schrodinger operator with general self adjoint boundary conditions at the origin using the marchenko equation our approach allows the recovery of the potential as well as the boundary conditions it is easily specialized to inverse scattering on star shaped graphs with boundary conditions at the node

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

m operators a generalisation of weyl titchmarsh theory the theory of the weyl titchmarsh m function for second order ordinary differential operators is generalized and applied to partial differential operators of the form q x acting in three space dimensions weyl operators m z are defined as maps from l2 s 1 to l2 s 1 s 1 unit sphere in r3 for exterior and interior boundary value problems and for the partial differential operator acting in l2 r 3 with the standard weyl titchmarsh m function recovered in the special case that q is spherically symmetric the analysis is carried out rather explicitly allowing for the determination of precise norm bounds for m operators and for the proof of higher dimensional analogues of a number of the fundamental results of standard weyl titchmarsh theory

boundary rigidity with partial data we study the boundary rigidity problem with partial data consisting of determining locally the riemannian metric of a riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary we show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information in particular this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary the local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed