stability of reducing subspaces abstract we characterize the stability of reducing subspaces of a rectangular matrix pencil of complex matrices b a except for the special case in which the pencil has no eigenvalues and only has one row and one column minimal indices and both are different from zero

a rigorous theory of finite size scaling at first order phase transitions a large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered the dependence of the finite volume magnetizationm per h l in cubes of sizel d under periodic boundary conditions on the external fieldh is analyzed for the case where two phases coexist at the infinite volume transition pointh t we prove that independent of the details of the model the finite volume magnetization per lattice site behaves likem per h t m tanh ml d h ht withm per h denoting the infinite volume magnetization and m 1 2 m per h t 0 m per h t 0 introducing the finite size transition pointh m l as the point where the finite volume susceptibility attains the maximum we show that in the case of asymmetric field driven transitions its shift ish t h m l o l 2d in contrast to claims in the literature starting from the obvious observation that the number of stable phases has a local maximum at the transition point we propose a new way of determining the pointh t from finite size data with a shift that is exponentially small inl finally the finite size effects are discussed also in the case where more than two phases coexist

finite size scaling for potts models recently borgs and koteck developed a rigorous theory of finite size effects near first order phase transitions here we apply this theory to the ferromagneticq state potts model which forq large andd 2 undergoes a first order phase transition as the inverse temperature is varied we prove a formula for the internal energy in a periodic cube of side lengthl which describes the rounding of the infinite volume jump e in terms of a hyperbolic tangent and show that the position of the maximum of the specific heat is shifted by m l inq e l d o l 2d with respect to the infinite volume transition point t we also propose an alternative definition of the finite volume transition temperature t l which might be useful for numerical calculations because it differs only by exponentially small corrections from t

finite size effects for the potts model with weak boundary conditions using the pirogov sinai theory we study finite size effects for the ferromagnetic q state potts model in a cube with boundary conditions that interpolate between free and constant boundary conditions if the surface coupling is about half of the bulk coupling and q is sufficiently large we show that only small perturbations of the ordered and disordered ground states are dominant contributions to the partition function in a finite but large volume this allows a rigorous control of the finite size effects for these weak boundary conditions in particular we give explicit formulae for the rounding of the infinite volume jumps of the internal energy and magnetization as well as the position of the maximum of the finite volume specific heat while the width of the rounding window is of order l d the same as for periodic boundary conditions the shift is much larger of order l 1 for strong boundary conditions the surface coupling is either close to zero or close to the bulk coupling the finite size effects at the transition point are shown to be dominated by either the disordered or the ordered phase respectively in particular it means that sufficiently small boundary fields lead to the disordered and not to the ordered gibbs state this gives an explicit proof of a van enter s result that the phase transition in the potts model is not robust

birkhoff normal forms for fourier integral operators ii in this work we construct logarithms and birkhoff normal forms for elliptic fourier integral operators in the semi classical limit under more general assumptions than in a previous work by the first author the methods are similar but slightly different

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

quantum ergodicity of boundary values of eigenfunctions suppose that n is a bounded piecewise smooth domain we prove that the boundary values cauchy data of eigenfunctions of the laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle b is ergodic our proof is based on the classical observation that the boundary values of an interior eigenfunction 2 is an eigenfunction of an operator fh on the boundary of with h 1 in the case of the neumann boundary condition fh is the boundary integral operator induced by the double layer potential we show that fh is a semiclassical fourier integral operator quantizing the billiard map plus a small remainder the quantum dynamics defined by fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of zelditch zworski 1996 novelties include the facts that fh is not unitary and consequently the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by s ozawa 1993 and was almost simultaneously proved by gerard leichtnam 1993 in the case of convex c1 1 domains with continuous tangent planes and with dirichlet boundary conditions our methods seem to be quite different motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form

first order phase transitions in unbounded spin systems i construction of the phase diagram the phase diagram and the corresponding infinite volume gibbs states are constructed for a large class of continuous unbounded spin models our construction relies on a partition of unity mapping our system onto an interacting contour system a generalisation of zahradnik s approach to piragov sinai theory to interacting contour systems and a suitable mean field expansion around the minimas of the hamiltonian

finite size scaling for the 2d ising model with minus boundary conditions we study the magnetization m l h for the ising model on a large but finite lattice square under the minus boundary conditions using known large deviation results evaluating the balance between the competing effects of the minus boundary conditions and the external magnetic field h we describe the details of its dependence on h as exemplified by the finite size rounding of the infinite volume magnetization discontinuity and its shift with respect to the infinite volume transition point

critical region for droplet formation in the two dimensional ising model we study the formation dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence specifically we consider the 2d ising model in volumes of size l2 inverse temperature c and overall magnetization conditioned to take the value m l2 2m vl where c 1 is the critical temperature m m is the spontaneous magnetization and vl is a sequence of positive numbers we find that the critical scaling for droplet formation dissolution is when vl3 2l 2 tends to a definite limit specifically we identify a dimensionless parameter proportional to this limit a non trivial critical value c and a function such that the following holds for c there is a single wulff shaped droplet containing a fraction c 2 3 of the magnetization deficit and there are no other droplets beyond the scale of log l moreover and are related via a universal equation that apparently is independent of the details of the system

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

crossover finite size scaling at first order transitions in a recent paper we developed a method which allows one to control rigorously the finite size behavior in long cylinders near first order phase transitions at low temperature here we apply this method to asymmetric transitions with two competing phases and to theq state potts model as a typical model of a temperature driven transition whereq low temperature phases compete with one high temperature phase we obtain the finite size scaling of the firstn eigenvalues wheren is the number of competing phases of the transfer matrix in a periodic box of volumel l t and as a corollary the finite size scaling of the shape of the order parameter in a hypercubic box t l the infinite cylinder t and the crossover regime from hypercubic to cylindrical scaling for the two phase case n 2 we find that the crossover length l is given by o lw exp lv where is the inverse temperature is the surface tension and w 1 2 if v 1 2 whilew 0 if v 1 2 for the standard ising model we also consider free boundary conditions showing that l exp lv o lv 1 for any dimension v 1 2 for v 1 2 we finally discuss a class of boundary conditions which interpolate between free corresponding to the interpolating parameter g 0 and periodic boundary conditions corresponding to g 1 finding that l o lw exp lv withw 0 forg 0 andw 1 2 for 0 g 1

sums of characteristic values of compact polynomial operator pencils we consider polynomial pencils whose coefficients are compact operators bounds for the sums of absolute values real and imaginary parts of characteristic values are derived applications to differential and difference equations are discussed

finite size scaling and surface tension from effective one dimensional systems we develop a method for precise asymptotic analysis of partition functions near first order phase transitions working in a 1 dimensional cylinder of volumel l t we show that leading exponentials int can be determined from a simple matrix calculation providedt v logl through a careful surface analysis we relate the off diagonal matrix elements of this matrix to the surface tension andl while the diagonal matrix elements of this matrix are related to the metastable free energies of the model for the off diagonal matrix elements which are related to the crossover length from hypercubic l t to cylindrical t scaling this includes a determination of the pre exponential power ofl as a function of dimension the results are applied to supersymmetric field theory and in a forthcoming paper to the finite size scaling of the magnetization and inner energy at field and temperature driven first order transitions in the crossover region from hypercubic to cylindrical scaling

inverse spectral problem for analytic domains i balian bloch trace formula this is the first in a series of papers z3 z4 on inverse spectral resonance problems for analytic plane domains in this paper we present a rigorous version of the balian bloch trace formula bb1 bb2 it is an asymptotic formula for the trace tr1 r k i log k of the regularized resolvent of the dirichlet or neumann laplacian of as k with 0 when the support of open image in new window contains the length l of precisely one periodic reflecting ray then the asymptotic expansion of tr1 r k i log k is essentially the same as the wave trace expansion at the raison d etre for this approach is that it leads to relatively simple explicit formulae for wave invariants for example we give the first formulae for wave invariants of bouncing ball orbits of plane domains the details will appear in z3 although we only present details in dimension 2 the methods and results extend with few modifications to all dimensions

bounds for characteristic values of entire matrix pencils entire matrix valued functions of a complex argument entire matrix pencils are considered upper bounds for sums of characteristic values and a lower bound for the smallest characteristic value are derived in the terms of the coefficients of the taylor series these results are new even for polynomial pencils

bounds for eigenvalues of matrix polynomials upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial or matrix the bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices they generalize various existing bounds for scalar polynomials and single matrices a variety of tools are used in the derivations including block companion matrices gershgorin s theorem the numerical radius and associated scalar polynomials numerical experiments show that the bounds can be surprisingly sharp on practical problems

escape function conditions for the observation control and stabilization of the wave equation for the linear wave equation with time invariant coefficients on a connected compact riemannian manifold omega g with c 3 boundary the geodesics condition of bardos lebeau and rauch siam j control optim 30 1992 pp 1024 1065 is characterized in terms of escape functions which are some lyapunov functions on the phase space s bar omega the unit sphere cotangent bundle differentiable escape functions yield a sufficient condition which is slightly less sharp but does not refer to geodesics the escape function condition yields a straightforward geometric proof that the geodesics condition holds in the situations where first order differential multiplier methods apply using microlocal control results it allows us to generalize some control results that were obtained by multiplier methods to variable coefficients and lower order terms it also allows us to prove in some class of simple situations e g in mathbb r 2 with constant coefficients that no first order differential multiplier method can reach the optimal control time or control regions

inverse spectral problems for schrodinger operators in this article we find some explicit formulas for the semi classical wave invariants at the bottom of the well of a schrodinger operator as an application of these new formulas for the wave invariants we improve the inverse spectral results proved by guillemin and uribe in gu they proved that under some symmetry assumptions on the potential v x the taylor expansion of v x near a non degenerate global minimum can be recovered from the knowledge of the low lying eigenvalues of the associated schrodinger operator in mathbb r n we prove similar inverse spectral results using fewer symmetry assumptions we also show that in dimension 1 no symmetry assumption is needed to recover the taylor coefficients of v x

inverse spectral problem for analytic domains ii ℤ2 symmetric domains this paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit the algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary and on a feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral the algorithm is particularly effective for euclidean plane domains possessing a 2 symmetry which reverses the orientation of a bouncing ball orbit it is also very effective for domains with dihedral symmetries for simply connected analytic euclidean plane domains in either symmetry class we prove that the domain is determined within the class by either its dirichlet or neumann spectrum this improves and generalizes the best prior inverse result that simply connected analytic plane domains with two symmetries are spectrally determined within that class

identities for sums of characteristic values of entire matrix pencils abstract the paper deals with an entire matrix valued function of a complex argument an entire matrix pencil f of order f identities for the following sums of the characteristic values z k f k 1 2 of f are established k 1 1 z k m f m f and k 1 im 1 z k f 2 f 2 the suggested results are new even for polynomial pencils

first order phase transitions in unbounded spin systems ii completeness of the phase diagram we continue our analysis of unbounded spin systems with nearest neighbor interactionw and a single spin potentialv which hasn deep and widely separated minima in this second part we show that all translation invariant phases obeying a certain regularity condition are convex combinations of the stable phases determined in the first part of this paper for periodic boundary conditions each stable phase contributes with the same weight in the infinite volume limit

a proof of the gibbs thomson formula in the droplet formation regime we study equilibrium droplets in two phase systems at parameter values corresponding to phase coexistence specifically we give a self contained microscopic derivation of the gibbs thomson formula for the deviation of the pressure and the density away from their equilibrium values which according to the interpretation of the classical thermodynamics appears due to the presence of a curved interface the general albeit heuristic reasoning is corroborated by a rigorous proof in the case of the two dimensional ising lattice gas

a unified approach to phase diagrams in field theory and statistical mechanics we construct the phase diagram of any system which admits a low temperature polymer or cluster expansion such an expansion turns the system into a hard core interacting contour model with small but not necessarily positive activities the method uses some of zahradnik s ideas z1 but applies equally well to systems with complex interactions we give two applications first to low temperaturep 2 models with complex couplings and second to a computation of asymptotics of partition functions in periodic volumes if the index of a supersymmetric field theory is known the second application would help determine the number of phases in infinite volume

surface induced finite size effects for first order phase transitions we consider classical lattice models describing first order phase transitions and study the finite size scaling of the magnetization and susceptibility in order to model the effects of an actual surface in systems such as small magnetic clusters we consider models with free boundary conditions for a field driven transition with two coexisting phases at the infinite volume transition pointh h t we prove that the low temperature finite volume magnetizationmfree l h per site in a cubic volume of sizel d behaves like m free l h frac m m 2 frac m m 2 tanh left frac m m 2 l d h h chi l right o left frac 1 l right whereh x l is the position of the maximum of the finite volume susceptibility andm are the infinite volume magnetizations ath h t 0 andh h t 0 respectively we show thath x l is shifted by an amount proportional to 1 l with respect to the infinite volume transition pointh t provided the surface free energies of the two phases at the transition point are different this should be compared with the shift for periodic boundary conditions which for an asymmetric transition with two coexisting phases is proportional only to 1 l2d one can consider also other definitions of finite volume transition points for example the positionh u l of the maximum of the so called binder cumulantufree l h whileh u l is again shifted by an amount proportional to 1 l with respect to the infinite volume transition pointh t its shift with respect toh l is of the much smaller order 1 l2d we give explicit formulas for the proportionality factors and show that in the leading 1 l2d term the relative shift is the same as that for periodic boundary conditions

on the ising model with random boundary condition the infinite volume limit behavior of the 2d ising model under possibly strong random boundary conditions is studied the model exhibits chaotic size dependence at low temperatures and we prove that the and phases are the only almost sure limit gibbs measures assuming that the limit is taken along a sparse enough sequence of squares in particular we provide an argument to show that in a sufficiently large volume a typical spin configuration under a typical boundary condition contains no interfaces in order to exclude mixtures as possible limit points a detailed multi scale contour analysis is performed

nonsymmetric first order transitions finite size scaling and tests for infinite range models finite size rounding of first order transitions is studied for the general case of nonsymmetric phases and nonperiodic boundary conditions the main features include the surface induced shift of the rounded transition on the scale 1 l while the order parameter discontinuity is rounded on the scale 1 ld this rounding is described by the universal scaling forms with scaling functions identical to those for the periodic symmetric case the proposed formalism applies to scalar order parameter single domain systems it is tested by exact calculations for a class of infinite range models