on the minimizing triple collision orbits in the planar newtonian three body problem in this paper we study the minimizing triple collision orbits in the planar newtonian three body problem with arbitrary masses we show that for a given non collinear initial configuration the minimizing triple collision orbit is collision free until a simultaneous collision and its limiting configuration is the lagrangian configuration with the same orientation as the initial configuration for the collinear initial configuration under a certain technical assumption there exist two minimizing orbits the limiting configurations of these orbits are the two opposite lagrangian configurations

monte carlo results on several new and existing tests for the error component model abstract the breusch pagan test is a popular test for testing zero variances in a random error component model realizing the one sided nature of this problem honda 1985 derived the one sided lm test for the one way error component model and suggested a handy one sided candidate for the two way error component model in this paper we derive a new one sided test for testing the two way error component model based upon the results of gourieroux holly and monfort 1982 next we derive a lm test for testing zero individual time specific effects assuming that the other time period individual effects are present this is in contrast to the usual lm test which assumes that under the null hypothesis both effects are absent exact power comparisons whenever possible as well as monte carlo experiments are employed to compare the performance of these tests with the corresponding likelihood ratio and anova f tests

on multiple diagnostic procedures for the linear model abstract multiple diagnostic procedures entail a partition of the sample space into a set of critical regions when the procedure is designed in a piecemeal fashion these regions bear no particular relation to one another and the procedure as a whole can be logically inconsistent we use a simple example to illustrate these difficulties and also to suggest criteria that should be satisfied by any diagnostic procedure that is to meet the needs of the practitioner

infinitely many syzygies we show that any bounded zero angular momentum solution of the newtonian three body problem suffers infinitely many syzygies collinearities provided that it does not suffer a triple collision our motivation comes from the dream of building a symbolic dynamics for the three body problem the proof relies on the conformal geometry of the shape sphere

an index theory for asymptotic motions under singular potentials abstract we develop an index theory for parabolic and collision solutions to the classical n body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities both problems suffer from a lack of compactness and can be brought in a similar form of a lagrangian system on the half time line by a regularising change of coordinates which preserve the lagrangian structure we then introduce a maslov type index which is suitable to capture the asymptotic nature of these trajectories as half clinic orbits by taking into account the underlying hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

self adjoint extensions for linear hamiltonian systems with two singular endpoints abstract this paper is concerned with self adjoint extensions for a linear hamiltonian system with two singular endpoints the domain of the closure of the corresponding minimal hamiltonian operator h 0 is described by properties of its elements at the endpoints of the discussed interval decompositions of the domains of the corresponding left and right maximal hamiltonian operators are provided and expressions of the defect indices of h 0 in terms of those of the left and right minimal operators are given based on them characterizations of all the self adjoint extensions for a hamiltonian system are obtained in terms of square integrable solutions as a consequence the characterizations of all the self adjoint extensions are given for systems in several special cases

potential theory on lipschitz domains in riemannian manifolds sobolev besov space results and the poisson problem abstract we continue a program to develop layer potential techniques for pde on lipschitz domains in riemannian manifolds building on l p and hardy space estimates established in previous papers here we establish sobolev and besov space estimates on solutions to the dirichlet and neumann problems for the laplace operator plus a potential on a lipschitz domain in a riemannian manifold with a metric tensor smooth of class c 1 for some 0 we treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant coefficient laplace operator on a lipschitz domain in euclidean space with the dirichlet boundary condition by d jerison and c kenig

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

self adjoint extensions for second order symmetric linear difference equations abstract in this paper self adjoint extensions for second order symmetric linear difference equations with real coefficients are studied by applying the glazman krein naimark theory for hermitian subspaces both self adjoint subspace extensions and self adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions where the two endpoints may be regular or singular

tests for the error component model in the presence of local misspecification it is well known that most of the standard specification tests are not valid when the alternative hypothesis is misspecified this is particularly true in the error component model when one tests for either random effects or serial correlation without taking account of the presence of the other effect in this paper we study the size and power of the standard rao s score tests analytically and by simulation when the data are contaminated by local misspecification these tests are adversely affected under misspecification we suggest simple procedures to test for random effects or serial correlation in the presence of local serial correlation or random effects and these tests require ordinary least squares residuals only our monte carlo results demonstrate that the suggested tests have good finite sample properties for local misspecification and in some cases even for far distant misspecification our tests are also capable of detecting the right direction of the departure from the null hypothesis we also provide some empirical illustrations to highlight the usefulness of our tests

dirichlet problems on varying domains abstract the aim of the paper is to characterise sequences of domains for which solutions to an elliptic equation with dirichlet boundary conditions converge to a solution of the corresponding problem on a limit domain necessary and sufficient conditions are discussed for strong and uniform convergence for the corresponding resolvent operators examples are given to illustrate that most results are optimal

asymptotic relative efficiency of the classical test statistics under misspecification abstract asymptotic properties of the three classical test statistics the likelihood ratio lagrange multiplier efficient score and wald statistics are studied in the case of two types of misspecification the first misspecification is that the alternative used to construct the test is different from the true nonnull model in the second case the alternative hypothesis is correct but some of the nuisance parameters satisfy restrictions which are not taken into account in the test pitman s local alternatives and the concept of asymptotic relative efficiency are used to derive the asymptotic distributions of the test statistics and to examine the effects of the misspecifications the general results of the paper are illustrated by an example on the dynamic specification of a regression model

an unusual self adjoint linear partial differential operator hhin an american mathematical society memoir published in 2003 the authors everitt and markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions and the generation of self adjoint differential operators in sobolev hilbert spaces in the case when the differential expression has smooth coefficients on the closure of a bounded open region in euclidean space and when the region has a smooth boundary this theory leads to the construction of certain self adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region this present paper concerns the spectral properties of one of these unusual self adjoint operators sometimes called the harmonic operator the boundary value problems considered in the memoir see above and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region also the region is bounded and has a smooth boundary under these and some additional technical conditions it is shown in the memoir and emphasized in this present paper that all the self adjoint operators considered are explicitly determined on their domains by the partial differential expression this property makes a remarkable comparison with the case of symmetric ordinary differential expressions in the regular ordinary case the spectrum of all the self adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation and each eigenvalue is of finite multiplicity thus the essential spectrum of all these operators is empty this spectral property extends to the present partial differential case for the classical dirichlet and neumann operators but not to the harmonic operator it is shown in this paper that the harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane thus the essential spectrum of this operator is not empty both the weak and strong formulations of the harmonic boundary value problem are considered these two formulations are shown to be equivalent in the final section of the paper examples are considered which show that the harmonic operator defined by the methods of symplectic algebra has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region

formulation and estimation of dynamic models using panel data abstract this paper presents a statistical analysis of time series regression models for longitudinal data with and without lagged dependent variables under a variety of assumptions about the initial conditions of the processes being analyzed the analysis demonstrates how the asymptotic properties of estimators of longitudinal models are critically dependent on the manner in which samples become large by expanding the number of observations per person holding the number of people fixed or by expanding the number of persons holding the number of observations per person fixed the paper demonstrates which parameters can and cannot be identified from data produced by different sampling plans

action minimizing orbits in the parallelogram four body problem with equal masses in this paper we prove the existence of a new periodic solution for the planar newtonian four body problem with equal masses on this orbit two mass points travel on one star shaped closed curve while the other two travel on another and have the opposite orientation the configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time

spectral analysis of a two body problem with zero range perturbation abstract we consider a class of singular zero range perturbations of the hamiltonian of a quantum system composed by a test particle and a harmonic oscillator in dimension one two and three and we study its spectrum in fact we give a detailed characterization of point spectrum and its asymptotic behavior with respect to the parameters entering the hamiltonian we also partially describe the positive spectrum and scattering properties of the hamiltonian

weyl asymptotic formula for the laplacian on domains with rough boundaries we study asymptotic distribution of eigenvalues of the laplacian on a bounded domain in n our main results include an explicit remainder estimate in the weyl formula for the dirichlet laplacian on an arbitrary bounded domain sufficient conditions for the validity of the weyl formula for the neumann laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula in particular we show that the weyl formula holds true for the neumann laplacian on a lip domain whenever d 1 d prove that the remainder in this formula is o d 1 and give an example where the remainder estimate o d 1 is order sharp we use a new version of the variational technique which does not require the extension theorem

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

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

testing for heteroskedasticity and serial correlation in a random effects panel data model this paper considers a panel data regression model with heteroskedastic as well as serially correlated disturbances and derives a joint lm test for homoskedasticity and no first order serial correlation the restricted model is the standard random individual error component model it also derives a conditional lm test for homoskedasticity given serial correlation as well as a conditional lm test for no first order serial correlation given heteroskedasticity all in the context of a random effects panel data model monte carlo results show that these tests along with their likelihood ratio alternatives have good size and power under various forms of heteroskedasticity including exponential and quadratic functional forms

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

binary decompositions for planar n body problems and symmetric periodic solutions a binary decomposition for a system of n masses is a way of treating the system as big n atop 2 big binaries with the total action exactly the same as that of the original system by considering binary decompositions we are able to provide effective lower bound estimates for the action of collision paths in several spaces of symmetric loops as applications we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the n body problem

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

symmetric trajectories for the 2n body problem with equal masses we consider the problem of 2n bodies of equal masses in mathbb r 3 for the newtonian like weak force potential r and we prove the existence of a family of collision free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations in addition the rotation number with respect to the vertical axis ranges in a suitable interval these solutions have the hip hop symmetry a generalization of that introduced in 19 for the case of many bodies and taking account of a topological constraint the argument exploits the variational structure of the problem and is based on the minimization of lagrangian action on a given class of paths

existence and minimizing properties of retrograde orbits to the three body problem with various choices of masses poincare made the first attempt in 1896 on applying variational calculus to the three body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions little progress had been made on resolving this difficulty until a recent breakthrough by chenciner and montgomery afterward variational methods were successfully applied to the jv body problem to construct new classes of solutions in order to avoid collisions the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal a question for the variational approach on planar problems naturally arises are minimizing methods useful only when some masses are identical this article addresses this question for the three body problem for various choices of masses it is proved that there exist infinitely many solutions with a certain topological type called retrograde orbits that minimize the action functional on certain path spaces cases covered in our work include triple stars in retrograde motions double stars with one outer planet and some double stars with one planet orbiting around one primary mass our results largely complement the classical results by the poincare continuation method and conley s geometric approach

model specification tests a simultaneous approach abstract in econometrics specification tests have been constructed to verify the validity of one specification at a time it is argued that most of these tests are not in general robust in the presence of other misspecifications so their application may result in misleading conclusions using the lagrange multiplier principle we develop efficient test procedures that are capable of testing a number of specifications simultaneously these tests will confirm the validity or invalidity of a general model requiring the estimates of the restricted model only through an extensive monte carlo experiment we study the performance of these tests and some commonly used one directional tests we also suggest a multiple comparison procedure to identify different sources of errors this we hope will lead to a better specification of econometric models

a bayesian robust chi squared test for testing simple hypotheses abstract in this paper we introduce a new bayesian chi squared test based on an adjusted quadratic loss function for testing a simple null hypothesis we show that the asymptotic null distribution of our suggested test is a central chi squared distribution under some assumptions required for the bayesian large sample theory we refer to our test as the bayesian robust chi squared test since it is robust to parametric misspecification in the alternative model that is the limiting null distribution of our test is a central chi squared distribution irrespective of parametric misspecification in the alternative model in addition to being robust to parametric misspecification our test also shares properties of the test suggested by li et al 2015 based on a quadratic loss function we provide four examples to illustrate the implementation of our suggested bayesian test statistic

a continuum of unusual self adjoint linear partial differential operators in an earlier publication a linear operator thar was defined as an unusual self adjoint extension generated by each linear elliptic partial differential expression satisfying suitable conditions on a bounded region of some euclidean space in this present work the authors define an extensive class of thar like self adjoint operators on the hilbert function space l2 but here for brevity we restrict the development to the classical laplacian differential expression with now the planar unit disk it is demonstrated that there exists a non denumerable set of such thar like operators each a self adjoint extension generated by the laplacian each of which has a domain in l2 that does not lie within the usual sobolev hilbert function space w2 these thar like operators cannot be specified by conventional differential boundary conditions on the boundary of and may have non empty essential spectra

the sturm liouville group in several recent publications everitt and markus have developed a theory of ordinary and partial differential boundary value problems through their novel methods of complex symplectic algebra for instance in the case of the regular sturm liouville problem the corresponding boundary complex symplectic space has four dimensions and can be used to classify all the self adjoint boundary conditions by means of two dimensional lagrangian subspaces in this investigation the groups auto s of all symplectic automorphisms are analyzed for all finite dimensional complex symplectic spaces s say of dimension n 2 and these are demonstrated to be noncommutative connected lie groups of real dimension n2 this global topological analysis provides information on continuous families of canonical coordinates in s and their deformations

globally minimizing parabolic motions in the newtonian n body problem we consider the n body problem in mathbb r d with the newtonian potential 1 r we prove that for every initial configuration xi and for every minimizing normalized central configuration x0 there exists a collision free parabolic solution starting from xi and asymptotic to x0 this solution is a minimizer in every time interval the proof exploits the variational structure of the problem and it consists in finding a convergent subsequence in a family of minimizing trajectories the hardest part is to show that this solution is parabolic and asymptotic to x0

wavelet based testing for serial correlation of unknown form in panel models wavelet analysis is a new mathematical method developed as a unified field of science over the last decade or so as a spatially adaptive analytic tool wavelets are useful for capturing serial correlation where the spectrum has peaks or kinks as can arise from persistent dependence seasonality and other kinds of periodicity this paper proposes a new class of generally applicable wavelet based tests for serial correlation of unknown form in the estimated residuals of a panel regression model where error components can be one way or two way individual and time effects can be fixed or random and regressors may contain lagged dependent variables or deterministic stochastic trending variables our tests are applicable to unbalanced heterogenous panel data they have a convenient null limit n 0 1 distribution no formulation of an alternative model is required and our tests are consistent against serial correlation of unknown form even in the presence of substantial in homogeneity in serial correlation across individuals this is in contrast to existing serial correlation tests for panel models which ignore inhomogeneity in serial correlation across individuals by assuming a common alternative and thus have no power against the alternatives where the average of serial correlations among individuals is close to zero we propose and justify a data driven method to choose the smoothing parameter the finest scale in wavelet spectral estimation making the tests completely operational in practice the data driven finest scale automatically converges to zero under the null hypothesis of no serial correlation and diverges to infinity as the sample size increases under the alternative ensuring the consistency of our tests simulation shows that our tests perform well in small and finite samples relative to some existing tests

moment based tests for individual and time effects in panel data models abstract this paper proposes two hausman type tests respectively for individual and time effects in a two way error component regression model by comparing estimators of the variance of the idiosyncratic error at different robust levels they are both robust to the presence of the other effect and the test for the individual effect has a larger asymptotic power than the corresponding anova f test when the effects are correlated with covariates tests jointly for both effects are also discussed monte carlo evidence shows their good size properties and better power properties than competing tests and the application to the crime rate study gives further support

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

point interactions in two dimensions basic properties approximations and applications to solid state physics on etudie rigoureusement l operateur de schrodinger formellement donne par h y dans l 2 r 2 et on montre comment l approcher en termes d hamiltoniens a courte portee transformes d echelles on considere des questions semblables pour l operateur h j 1 n j y j n n on etudie les proprietes spectrales du modele a un electron d un cristal a 2 dimensions avec des interactions ponctuelles

a remarkable spectral feature of the schrodinger hamiltonian of the harmonic oscillator perturbed by an attractive δ interaction centred at the origin double degeneracy and level crossing we rigorously define the self adjoint hamiltonian of the harmonic oscillator perturbed by an attractive interaction of strength centred at 0 the bottom of the confining parabolic potential by explicitly providing its resolvent our approach is based on a coupling constant renormalization related to a technique originated in quantum field theory and implemented in the rigorous mathematical construction of the self adjoint operator representing the negative laplacian perturbed by the interaction in two and three dimensions the way the interaction enters in our hamiltonian corresponds to the one originally discussed for the free hamiltonian instead of the harmonic oscillator one by p s ba it should not be confused with the potential perturbation of the harmonic oscillator discussed e g in a recent paper by gadella glasser and nieto also introduced by p s ba as a perturbation of the one dimensional free laplacian and recently investigated in that context by golovaty hryniv and zolotaryuk we investigate in detail the spectrum of our perturbed harmonic oscillator the spectral structure differs from that of the one dimensional harmonic oscillator perturbed by an attractive interaction centred at the origin the even eigenvalues are not modified at all by the interaction moreover all the odd eigenvalues regarded as functions of exhibit the rather remarkable phenomenon called level crossing after first producing the double degeneracy of all the even eigenvalues for the value b being the beta function

testing ar 1 against ma 1 disturbances in an error component model this paper derives three lm statistics for an error component model with first order serially correlated errors the first lm statistic jointly tests for zero first order serial correlation and random individual effects the second lm statistic tests for zero first order serial correlation assuming fixed individual effects and the third lm statistic tests for zero first order serial correlation assuming random individual effects in all three cases the corresponding lm statistic is the same whether the alternative is ar 1 or ma 1 this paper also derives two extensions of the burke godfrey and termayne 1990 test from the time series to the panel data literature the first tests the null of ar 1 disturbances against ma 1 disturbances and the second tests the null of ma 1 disturbances against ar 1 disturbances in an error component model these tests are computationally simple requiring only ols or within residuals the small sample performance of these tests are studied using monte carlo experiments

implicit alternatives and the local power of test statistics the local power of test statistics is analyzed by extending the notion of pitman sequences to sequences of data generating processes dgp s that approach the null hypothesis without necessarily satisfying the alternative a space of probability densities is defined and endowed with the structure of an infinite dimensional hilbert manifold which permits a geometrical interpretation of hypothesis testing the three classical test statistics lr wald and lm are shown to tend asymptotically to the same random variable under all sequences of local dgp s the power of these statistics is seen to depend on the null the alternative and the sequence of dgp s in a simple and geometrically intuitive way moreover for any test statistic that is asymptotically chi squared under the null there exists an implicit alternative hypothesis against which that statistic will have highest power and which coincides with the explicit alternative for the classical test statistics

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

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