nodal surface conjectures for the convex quantum billiard stemming from known properties of one dimensional 1 d and 2 d quantum billiards it is conjectured that the nodal surface of the first excited state of the convex 3 d quantum billiard intersects the billiard surface in a single simple closed curve examples of the validity of this conjecture are given for a number of elementary 3 d billiard configurations from these examples a second conjecture is introduced that addresses convex quantum billiards which are figures of rotation and contain one and only one plane of mirror symmetry normal to the axis of rotation two characteristic displacement parameters are defined which are labeled an axis length l and diameter a it is conjectured that a parameter 1 exists whose exact value depends on the properties of the billiard such that for l a prolatelike the nodal surface of the first excited state of a quantum billiard is a plane surface of mirror symmetry which divides the length of the billiard in half for l a oblatelike the nodal surface of the first excited state is a plane surface of mirror symmetry which contains the rotation axis and divides the diameter of the billiard in half arguments are given in support of a third conjecture which addresses the regular polyhedra quantum billiards termed spherical like it is hypothesized that the nodal surface of the first excited state for any of these billiards is any plane of reflection symmetry of the given polyhedron

determining conductivity by boundary measurements ii interior results abstract in a recent paper the authors showed that an unknown real analytic conductivity gamma may be determined from static boundary measurements in this document they extend this analysis by demonstrating that a similar result holds for piecewise real analytic conductivities in addition for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements originator supplied keywords include inversion convergence algorithms estimates coefficients

on continuous dependence for an inverse initial boundary value problem for the wave equation abstract we show that the bounded coefficient q x of the wave equation u tt u qu x t 0 t u x 0 u t x 0 0 x u v x x x t 0 t can be continuously determined by its neumann to dirichlet map in certain function spaces

stability of the inversion of 3d fixed frequency scattering data abstract numerical methods for solving 3d inverse scattering problems with fixed energy data are described exact inversion formulas are given stability of the inversion methods based on these formulas is investigated an explicit formula is given for construction of the dirichlet to neumann map from the scattering amplitude this formula is used as a basis for another numerical approach to the inverse scattering problem several open problems are formulated

universal schwinger cocycles of current algebras in d 1 dimensions geometry and physics we discuss the universal version of the schwinger terms of current algebra we call it the universal schwinger cocycle forp 3 herep denotes the class of the schatten ideali p which is related to the d 1 space time dimensions byp d 1 2 in detail and give a conjecture of the general form of the cocycle for anyp we also discuss the infinite charge renormalizations the highest weight vector and state vectors forp 3 last we give brief comments on the problems caused by the difficulties to construct the measure of infinite dimensional grassmann manifolds

analytically solvable dynamical systems which are not integrable abstract the central two fold aim of this paper is to expose a class of nonintegrable systems whose solutions can be analytically expressed in terms of computationally tractable algorithms and to explicitly construct a demonstrably meaningful general solution for one member of this class toward this end information theoretic results are reviewed which show that dynamical systems of null metric entropy have general solutions of null algorithmic complexity hence all systems exhibiting null metric entropy including ergodic only and mixing only systems have orbits which can be specified by algorithmically meaningful analytic expressions whose information content is very much less than that of the orbits they describe in consequence we may quite legitimately define the class of systems having null metric entropy ot be algorithmically integrable called a integrable herein after establishing the existence of a integrability this paper turns to the problem of exposing a specific nonintegrable but nonetheless a integrable system whose algorithmically meaningful solution can be explicitly derived the billiard moving within a plane polygon having the shape of a 60 120 rhombus is shown to be a simple example indeed the whole class of billiards moving in polygons whose angles are all rational multiples of have null metric entropy and are therefore a integrable systems moreover all rational billiards appear to have explicitly derivable solutions as is discussed herein and in a recent preprint by bjorn birnir because rational billiards are dense in the set of all plane billiards including chaotic ones the present paper suggests that a integrable billiards may provide a long sought analytic route to chaos finally it is pointed out that null entropied a integrable systems are perhaps the broadest category of systems which can reasonably be termed integrable for systems having positive metric entropy are truly nonintegrable due to the presence of orbits which are their own simplest description and their own fastest computer

schrodinger equation for convex plane polygons a tiling method for the derivation of eigenvalues and eigenfunctions motivated by a recently advanced conjecture on the ergodic properties of quantum systems the problem of solving the schrodinger equation for a free particle in a plane polygonal enclosure is revisited it will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves without making use of advanced group theoretical techniques it turns out inter alia that these polygons considered as classical billiards are all and only those which are completely integrable in the sense of arnold s theorem motivated by a recently advanced conjecture on the ergodic properties of quantum systems the problem of solving the schrodinger equation for a free particle in a plane polygonal enclosure is revisited it will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves without making use of advanced group theoretical techniques it turns out inter alia that these polygons considered as classical billiards are all and only those which are completely integrable in the sense of arnold s theorem

the radial curvature of an end that makes eigenvalues vanish in the essential spectrum i the concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the laplacian in particular this paper discusses the question of whether there is an eigenvalue of the laplacian embedded in the essential spectrum or not the borderline behavior of the radial curvatures for this problem will be determined we will assume that the radial curvature k rad of an end converges to a constant 1 at infinity with the decay order k rad 1 o r 1 and prove the absence of eigenvalues embedded in the essential spectrum furthermore in order to show that this decay order k rad 1 o r 1 is sharp we will construct a manifold with the radial curvature decay k rad 1 o r 1 and with an eigenvalue frac n 1 2 4 1 embedded in the essential spectrum frac n 1 2 4 infty of the laplacian

the inverse scattering problem for time harmonic acoustic waves in an inhomogeneous medium with complex refraction index abstract the inverse scattering problem of determining the complex refraction index of a compact inhomogeneous medium from a knowledge of the far field patterns of the scattered fields corresponding to many incident time harmonic plane acoustic waves is solved by using the colton monk method again by using the same method the analogue of karp s theorem for the scattering of acoustic waves through an inhomogeneous medium with compact support is proven

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

multidimensional inverse problems and completeness of the products of solutions to pde a method is given for proving uniqueness theorems for some inverse problems the method is based on a result on completeness of the products of solutions to pde as an example the following uniqueness theorems are proved 1 the scattering amplitude a k known for all s2 and a fixed k 0 determines the compactly supported q x l2 d uniquely 2 the surface data u x y k known for all x y p x x3 0 and a fixed k 0 determine the compactly supported x l2 d d r 3 x x3 0 is arbitrarily small determine aj x j 1 2 uniquely here 2u k2u k2a1x a2 x u x y in r3 a1 l2 d a2 h2 d the same conclusion holds if the surface data are known at two distinct frequencies 4 the surface data u x y k known for all x y p and all k 0 determine v x and h k uniquely here 2 k2 k2 x u x y h k x l2 d h k is fourier transform of a wavelet of compact support 5 the conductivity x w2 2 d x c 0 is uniquely determined by the measurements of u and un on d here n is the outward normal to d d r3 is a bounded domain with a smooth boundary d x u 0 in d 6 necessary and sufficient conditions are given for a function a k s2 k 0 is fixed to be the scattering amplitude corresponding to a local potential from a certain class

numerical solution of some inverse scattering problems of geophysics some results are reported of the numerical solution of the inverse problem which consists of finding an inhomogeneity v x in the 3d region r 3 x x3 0 under the surface of the earth from the knowledge of the acoustic field u x y k at low frequencies k for all positions of the source y and the receiver x on the surface p x x3 0 of the earth the inhomogeneity v x in the refraction coefficient is related to the data of u x y k on p by an integral equation of the first kind this inverse problem is extremely ill conditioned the numerical computations in this paper show how to successfully obtain v x which was a priori assumed to be a known constant in an a priori unknown region b and zero outside b an ad hoc b searching procedure together with a regularization method have been used to solve the resulting ill condioned linear system for the discretized problem a subsequent direct comparison finds b more accurately this type of computation can be used to detect piecewise constant inhomogeneity in a medium for example to detect a hole or a crack in a rock or in a construction element

circular sector quantum billiard and allied configurations the circular sector quantum billiard problem is studied numerical evaluation of the zeros of first order bessel functions finds that there is an abrupt change in the nodal line structure of the first excited state of the system equivalently second eigenstate of the laplacian at the critical sector angle c 0 354 for sector angle 0 in the domain 0 0 c the nodal curve of the first excited state is a circular arc segment for c 0 the nodal curve of the first excited state is the bisector of the sector otherwise nondegenerate first excited states become twofold degenerate at the critical angle c the ground and first excited state energies eg e1 increase monotonically as 0 decreases from its maximum value a graph of e1 vs 0 reveals an inflection point at 0 c which is attributed to the change in bessel function contribution to the development of e1 a proof is given for the existence of a common zero for two bessel functions whose respective orders differ by a noninteger applicat

sum rules and spectral measures of schrodinger operators with l 2 potentials necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half line schrodinger operator with square integrable potential

function mixing hypothesis and quantum chaos abstract billiards are studied whose boundaries comprise two or more segments which allow local but not global separation of the helmholtz equation related local solutions are labeled sep functions such billiards comprise the and sets a definition of quantum chaos for these sets of billiards is presented based on the ratio of the fluctuation length of the wave function nodal pattern to the c diameter of the billiard the function mixing hypothesis states that a sufficient condition for a billiard to be chaotic is that the billiard be an element of one of these sets it further ascribes such chaotic behavior to be due to mixing of dissimilar sep functions examples of the application of this hypothesis are described a set theoretic formalism is introduced to describe perturbation theory for infinite potentials and applied to the concave sided square billiard of sufficiently small concavity it is concluded that the adiabatic theorem of quantum mechanics does not apply to this configuration in the limit of large quantum numbers

necessary and sufficient condition for l to have property c abstract let l m j 0aj j be a formal differential operator with constant coefficients in r n n 2 j is a multi index we say that l has property c iff for any bounded domain d r n the set of products uw u w nd l u l u 0 in d u hm d is complete in l2 d here hm d is the sobolev space a necessary and sufficient condition is given for l to have property c

finite gap potentials and wkb asymptotics for one dimensional schrodinger operators consider the schrodinger operator h d2 dx2 v x with power decaying potential v x o x we prove that a previously obtained dimensional bound on exceptional sets of the wkb method is sharp in its whole range of validity the construction relies on pointwise bounds on finite gap potentials these bounds are obtained by an analysis of the jacobi inversion problem on hyperelliptic riemann surfaces

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

numerical recovery of the layered medium from the surface data abstract in this paper an inverse problem is formulated and its numerical solution is given the problem is of interest for example in geophysics numerical results show that the algorithm is efficient for noisy data practical recommendations are given the problem consists of finding n z the refraction coefficient of the medium given the surface data u x k x p x x3 0 and all k 0 k0 where k0 0 is an arbitrary small wave number the governing equation is 2 k2n z u x x x1 x2 z analytical solution to this inverse problem is given a numerical implementation is suggested and the results of the numerical experiments are reported

formal solutions of inverse scattering problems iv error estimates the formal solutions of certain three dimensional inverse scattering problems presented in papers i iii in this series j math phys 10 1819 1969 17 1175 1976 21 2648 1980 are employed here to obtain quantitative estimates on the error resulting from the use of the born approximations in both direct and inverse potential scattering problems these estimates are uniformly valid at all energies and for all sufficiently weak potentials

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

on periodic billiard trajectories in obtuse triangles in 1775 j f de tuschis a fagnano observed that in every acute triangle the orthoptic triangle represents a periodic billiard trajectory but to the present day it is not known whether or not in every obtuse triangle a periodic billiard trajectory exists the limiting case of right triangles was settled in 1993 by f holt who proved that all right triangles possess periodic trajectories the same result had appeared independently in the russian literature in 1991 namely in the work of g a gal perin a m stepin and y b vorobets the latter authors discovered in 1992 a class of obtuse triangles which contain particular periodic billiard paths in this article we review the above mentioned results and some of the techniques used in the proofs and at the same time show for an extended class of obtuse triangles that they contain periodic billiard trajectories

on the coexistence of absolutely continuous and singular continuous components of the spectral measure for some sturm liouville operators with square summable potential abstract we prove that there exist some sturm liouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive half line the hausdorff dimension of the support of this singular measure can be arbitrary number from 0 to 1

scattering theory for riemannian laplacians we introduce a notion of scattering theory for the laplace beltrami operator on non compact connected and complete riemannian manifolds a principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity another condition is certain bounds of derivatives up to order one of the trace of this quantity these conditions are shown to be optimal for existence and completeness of a wave operator our theory does not involve prescribed asymptotic behavior of the metric at infinity like asymptotic euclidean or hyperbolic metrics studied previously in the literature a consequence of the theory is spectral theory for the laplace beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum

wkb and spectral analysis of one dimensional schrodinger operators with slowly varying potentials consider a schrodinger operator on l 2 of the line or of a half line with appropriate boundary conditions if the potential tends to zero and is a finite sum of terms each of which has a derivative of some order in l 1 l p for some exponent p 2 then an essential support of the the absolutely continuous spectrum equals almost every generalized eigenfunction is bounded and satisfies certain wkb type asymptotics at infinity if moreover these derivatives belong to l p with respect to a weight x with 0 then the hausdorff dimension of the singular component of the spectral measure is strictly less than one

behavior of the solutions to exterior boundary value problems at low frequencies abstract a simple method is given to study the low frequency behavior of the solutions to exterior boundary value problems for elliptic equations of second order

imbedded singular continuous spectrum for schrodinger operators 1 1 hv v x and some self adjoint boundary condition at zero the operator 1 1 describes a charged particle such as an electron in the electric field v x when v x is decaying quickly one expects the spectral and dynamical properties of hy to remain close to those of the free operator ho recall that with the operator 1 1 one can associate in a canonical way a spectral measure p see for example 6 2 which contains much information about the quantum system the classical result in this direction going back to the beginning of the century is that if v g l1 the spectral measure on the positive semi axis remains purely absolutely continuous in one dimension an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle a natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of hy may happen as far as v x has any decay at all the essential spectrum coincides with 0 oo but the quality of the spectrum and dynamics may change in 1928 wigner and von neumann 27 showed that there exist potentials v x satisfying v x jsuch that hy has positive imbedded eigenvalue e 1 this is a purely quantum resonance phenomenon since v can be chosen arbitrarily small and the bound state originates from long range correlations in the potential rather than the usual confining effect naboko 17 and simon 25 provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than coulomb namely for any positive function h x which grows at infinity there exist potentials v such that v x yjgand hy has a dense set of positive eigenvalues due to some recent results 3 21 it is known that the absolutely continuous spectrum on the positive semi axis is preserved for v o c l x l 2 e and in fact for v g l2 8 more precisely for such potentials the absolutely continuous part of the spectral measure iac gives positive weight to any subset of 0 oo of positive lebesgue measure therefore the naboko

radial rescaling approach for the eigenvalue problem of a particle in an arbitrarily shaped box in the present work we introduce a methodology for solving a quantum billiard with dirichlet boundary conditions the procedure starts from the exactly known solutions for the particle in a circular disk which are subsequently radially rescaled in such a way that they obey the new boundary conditions in this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy

the polygon quantum billiard problem the present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates the polygon ground state theorem is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the elemental polygons defined in the text the polygon first excited state theorem is established which states that for every n sided regular polygon n equivalent first excited states exist each of which contains a nodal curve that is a line of mirror symmetry of the related polygon a vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the n sided regular polygon is spanned by any two of these n nodal diagonal eigenstates i e the first excited state is twofold degenerate at various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations thus for example correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive nonoverlapping classes of polygons stemming from the preceding conclusions a discussion is included on the possibility of employing transverse magnetic tm modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic

on the ground state of regular polygonal billiards the helmholtz equation for regular polygons is investigated by means of the conformal mapping from the circle which provides an expansion parameter for the approximate evaluation of the lowest eigenvalue and the corresponding eigenvector

uniqueness theorems for multidimensional inverse problems with unbounded coefficients abstract in a g ramm inverse problems 3 1987 l77 l82 a method to prove uniqueness theorems is given the method uses an estimate of the solution to a partial differential equation with l d coefficients here the estimate is established for l2 d coefficients and the resulting uniqueness theorems are formulated

the hexagon quantum billiard a subset of eigenfunctions and eigenvalues for the hexagon quantum billiard are constructed by way of tessellation of the plane and incorporation of symmetries of the hexagon these eigenfunctions are given as a double fourier series obeying c6 symmetry a table of the lower lying eigen numbers for these states is included the explicit form for these eigenstates is given in terms of a sum of six exponentials each of which contains a pair of quantum numbers and a symmetry integer eigenstates so constructed are found to satisfy periodicity of the hexagon array contour read outs of a lower lying eigenstate reveal in each case hexagonal 6 fold symmetric arrays derived solutions satisfy either dirichlet or neumann boundary conditions and are irregular in neighborhoods about vertices this singular property is intrinsic to the hexagon quantum billiard dirichlet solutions are valid in the open neighborhood of the hexagon due to singular boundary conditions for integer phase factors neumann solutions are valid over the domain of the hexagon these doubly degenerate eigenstates are identified with the basis of a two dimensional irreducible representation of the c6v group a description is included on the application of these findings to the hexagonal nitride compounds

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

absence of embedded eigenvalues for riemannian laplacians in this paper we study absence of embedded eigenvalues for schrodinger operators on non compact connected riemannian manifolds a principal example is given by a manifold with an end possibly more than one in which geodesic coordinates are naturally defined in this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end another condition is an upper bound of the trace of this quantity while a third one is a bound of the derivatives of part of the trace some oscillatory behaviour of the trace is allowed in addition to geometric bounds we need conditions on the potential a regularity property of the domain of the schrodinger operator and the unique continuation property examples include ends endowed with asymptotic euclidean or hyperbolic metrics

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

potentials on the two torus for which the hamiltonian flow is ergodic we consider the motion of a test particle in a compound central potential field on a two dimensional torus we discuss three different classes of potentials attracting repelling and mixed that lead to hamiltonian systems which have positive lyapunov exponent almost everywhere and are ergodic included among the mixed potentials are smooth potentials without singularities

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

schrodinger equation for convex plane polygons ii a no go theorem for plane waves representation of solutions in the present article we discuss whether or not ansatzes more general than those considered hitherto for the representation in terms of plane waves of eigensolutions of the schrodinger equation for a free particle in a plane convex polygonal domain lead to the construction of a complete set for a class of domains larger than the well known one containing the polygons for which such a complete set can be obtained in terms of a superposition of a finite number of plane waves our conclusion is in the negative comments are also made on some features of a representation in terms of analytic functions of two complex variables

some geometrical models of chaotic dynamics the free motion of a particle on a surface of constant negative curvature a pseudosphere was one of the first models of chaotic motion it became the prototype for the theory of hyperbolic systems developed by bowen and sinai in these models geometry suggests a symbolic coding which already exhibits fully chaotic behaviour one can return to these models to seek possible manifestations of quantum chaos here the mathematical technique is harmonic analysis on hyperbolic space chaotic behaviour seems to appear both in the behaviour of individual eigenfunctions and in the sequence of spectral values

a linearized inverse boundary value problem for maxwell s equations abstract in this paper we consider the problem of determining the electromagnetic state of a body from measurements made on the surface of the body we study the full set of maxwell s equations that govern time harmonic electric and magnetic fields we seek to reconstruct the magnetic permeability the electric permitivity and the electric conductivity in the interior of the body from measurements made on the surface we exhibit appropriate boundary measurements in the form of a boundary mapping specifically the mapping from the tangential components of the electric field to the tangential components of the magnetic field this data can be used to reconstruct and approximately provided they deviate only slightly from known constants we also estimate the reconstruction errors

some schrodinger operators with power decaying potentials and pure point spectrum we construct deterministic potentials such that the schrodinger equation on has dense pure point spectrum in for almost all boundary conditions at as a by product we also obtain power decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces