Let (G e ) e>0 be a family of ‘e-thin’ Riemannian manifolds modeled on a finite metric graph G, for example, the e-neighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on G e , as e→0, for various boundary conditions. We obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions, uniformly for k≤Ce −1 , in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family (G e ). Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and the use of a priori estimates to show that all eigenfunctions are obtained in this way.
As already mentioned, the Neumann problem was treated in @cite_19 , @cite_8 , @cite_0 , KucZen:ASNLTD @cite_3 , @cite_17 . For Dirichlet boundary conditions, Post @cite_14 derived the first two terms of in the case of 'small' vertex neighborhoods, see Theorem . In the recent preprint @cite_5 Molchanov and Vainberg study the Dirichlet problem and show that, in the context of Theorem , the @math converge to eigenvalues of the quantum graph described in Theorem ; this was conjectured in @cite_4 , where also some results on the scattering theory on non-compact graphs are obtained. However, their statements are unclear as to whether the multiplicities coincide; also, they do not consider the effect of @math eigenvalues on @math or uniform asymptotics for large @math . In @cite_13 a related model is considered. The method in the previously cited papers is to compare quadratic forms or to show resolvent convergence of some sort, and in all cases only the leading asymptotic behavior is obtained.
@cite_14 @cite_4 @cite_8 @cite_3 @cite_0 @cite_19 @cite_5 @cite_13 @cite_17
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Many multiscale methods are based on the idea of extracting macroscopic behavior of solutions by solving an array of microscale models over small domains. A key ingredient in such multiscale methods is the boundary condition and the size of the computational domain over which the microscale problems are solved. This problem is systematically investigated in the present paper in the context of modeling strongly heterogeneous media. Three different boundary conditions are considered: the periodic boundary condition, Dirichlet boundary condition, and the Neumann boundary condition. Each is applied to several benchmark problems: the random checker-board problem, periodic problem with isotropic macroscale behavior, periodic problem with anisotropic macroscale behavior and periodic laminated media. In each case, convergence studies are conducted as the domain size for the microscale problem is changed. Convergence rates as well as the size of fluctuations in the computed effective coefficients are compared for the different formulations. In addition, we will discuss a mixed Dirichlet-Neumann boundary condition that is often used in porous medium modeling. We explain why that leads to unsatisfactory results and how it can be corrected. Also discussed are the different averaging methods used in extracting the effective coefficients. We consider a family of open sets M? which shrinks with respect to an appropriate parameter ? to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of M? converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature. We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+?j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I?I?Rd,f?TdI?I?Rd,f?Td, p,q?Rnp,q?Rn, t?T1t?T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0<e«10<e«1, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O(1)O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of and . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I?I?RdI?I?Rd. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from and . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in and By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same state space. This extends the ideas introduced in Diaconis and Saloff-Coste (1993), where random walks on finite groups were considered. The bounds involve geometric properties such as the diameter and covering number of an associated graph along the lines of Diaconis and Stroock (1991). The main application gives a sharp upper bound on the second eigenvalue of the symmetric exclusion process. Thus, let S0 be a connected undirected graph with n vertices. For simplicity, we assume in this introduction that SW is regular. To start, r unlabelled particles are placed in an initial configuration, 1 < r < n. At each step, a particle is chosen at random; then one of the neighboring sites of this particle is chosen at random. If the neighboring site is unoccupied, the chosen particle is moved there; if the neighboring site is occupied, the system stays as it was. This is a reversible Markov chain on the r-sets of 1, 2, . . ., n with uniform stationary distribution. Liggett (1985) gives background and motivation (he focuses on infinite systems). Fill (1991) gives bounds on the second eigenvalue of the labeled exclusion process on the finite circle ZZn 1 We study this chain by comparison with a second Markov chain on r-sets that proceeds by picking a particle at random, picking an unoccupied site at random (not necessarily a neighboring site) and moving the particle to the unoccupied site. This is a well studied chain (the Bernoulli-Laplace model for diffusion). Its eigenvalues are known. We show that the comparison techniques apply to give upper bounds on the eigenvalues of the exclusion We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ1L(∂Σ)⩽2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ1(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09]. Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. A recently developed abstract framework for asymptotically compatible schemes is applied to prove convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems. Intermediate results also provide new convergence results for the nonlocal diffusion problem. Special attention is paid to limiting behaviors as the interaction radius increases and the spatial grid size decreases, regardless of how these parameters may or may not be dependent. In particular, we show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are developed with minimal regularity assumptions on the solution and are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only required to be square integrable. Furthermore, our results also solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases. This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrodinger or Klein–Gordon equations in three dimensions. We demonstrate by a rigorous method that the linearized scalar operators which arise in this setting, traditionally denoted by L±, satisfy the gap property, at least over the radial functions. This means that the interval (0, 1] does not contain any eigenvalues of L± and that the threshold 1 is neither an eigenvalue nor a resonance. The gap property is required in order to prove scattering to the ground states for solutions starting on the centre-stable manifold associated with these states. This paper therefore provides the final instalment in the proof of this scattering property for the cubic Klein–Gordon equation in the radial case, see the recent theory of Nakanishi and the third author. The method developed here is quite general, and applicable to other spectral problems which arise in the theory of nonlinear equations. We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards.