Isospectral potentials and conformally equivalent isospectral metrics on spheres,balls and Lie groups

We construct pairs of conformally equivalent isospectral Riemannian metrics ?1g and ?2g on spheres Sⁿ and balls Bⁿ⁺¹ for certain dimensions n, the smallest of which is n=7, and on certain compact simple Lie groups. In the case of Lie groups, the metric g is left-invariant. In the case of spheres and balls, the metric g not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds (M, g) we also show that the functions ?₁ and ?₂ are isospectral potentials for the Schrödinger operator ?₂\gD + \gf. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.     

Second Order Semiclassics with Self-Generated Magnetic Fields

We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field B. We also add the field energy bòB²{\beta \int B^{2}} and we minimize over all magnetic fields. The parameter β effectively determines the strength of the field. We consider the weak field regime with β h ² ≥ const > 0, where h is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor h^1+e{h^{1+\varepsilon}} , i.e. the subleading term vanishes. However for potentials with a Coulomb singularity, the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper (Erdős et al. in Scott correction for large molecules with a self-generated magnetic field, Preprint, 2011) to prove the second order Scott correction to the ground state energy of large atoms and molecules.     

Trace of heat kernel,spectral zeta function and isospectral problem for sub-laplacians

In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ℂ ⁿ⁺¹. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part I. Four‐Potential Case

In the paper, we first investigate symmetries of isospectral and non‐isospectral four‐potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of u_n,t= L^mH⁽⁰⁾ , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero‐curvature representations of the isospectral and non‐isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non‐isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac‐Moody‐Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four‐potential and two‐potential Ablowitz–Ladik hierarchies. The even order members in the four‐potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two‐potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L² .     

Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

For geometrically finite hyperbolic manifolds Γ\ℍ ⁿ⁺¹, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of ℍ ⁿ⁺¹ in terms of the Hausdorff dimension of the limit set of Γ.     

Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for Certain Sums of Squares of Vector Fields

We consider an operator P which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form σ is not constant on Char P. Moreover the Hamilton foliation of the non-symplectic stratum of the Poisson-Treves stratification for P consists of closed curves in a ring-shaped open set around the origin. We prove that then P is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for P is G ^k+1 and that this is sharp.      

Global smooth solutions to a class of semilinear wave equaiions with strong nonlinearities

We consider the Cauchy problem for semilinear wave equationsu _tt−Δu=g(u) in 3+1 dimensions with smooth but possibly large data. Ifg isC ^2,α and bounded from above everywhere and from below for negative arguments the existence of a global classical solution is shown. If moreoverg is nonpositive and vanishes at least of order 2+∈ at the origin and if the data decay sufficiently rapidly at infinity the scattering operator exists.     

Random Projections of Smooth Manifolds

We propose a new approach for nonadaptive dimensionality reduction of manifold-modeled data, demonstrating that a small number of random linear projections can preserve key information about a manifold-modeled signal. We center our analysis on the effect of a random linear projection operator Φ:ℝ ^N →ℝ ^M , M<N, on a smooth well-conditioned K-dimensional submanifold ℳ⊂ℝ ^N . As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on ℳ are well preserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in ℝ ^N . Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifold-modeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data. This research was supported by ONR grants N00014-06-1-0769 and N00014-06-1-0829; AFOSR grant FA9550-04-0148; DARPA grants N66001-06-1-2011 and N00014-06-1-0610; NSF grants CCF-0431150, CNS-0435425, CNS-0520280, and DMS-0603606; and the Texas Instruments Leadership University Program. Web: dsp.rice.edu/cs.     

Spectral Properties of Four-Dimensional Compact Flat Manifolds

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on p-forms, we determine all p-isospectral and L-isospectral pairs and we show that in this class of manifolds, isospectrality on functions and isospectrality on p-forms for all values of p are equivalent to each other. The list shows for any p, 1 ≤ p ≤ 3, many p-isospectral pairs that are not isospectral on functions and have different lengths of closed geodesics. We also determine all length isospectral pairs (i.e. with the same length multiplicities), showing that there are two weak length isospectral pairs that are not length isospectral, and many pairs, p-isospectral for all p and not length isospectral. Mathematics Subject Classifications (2000): 58J53, 58C22, 20H15.     

Energy dependence of the scattering operator

We prove that the scattering operator S(E) depends continuously on the energy E for a certain class of Schrodinger operators, by an abstract method using trace conditions and the dilation group. We also obtain pointwise bounds on S(E) −1₁as E → ∞, and even as E → 0 in the case of repulsive potentials.     

Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians

Sur les fonctions entièresq-arithmétiques

Letq ɛ Z, |q|>1. In this paper, we study entire functions of a complex variable such thatf(q ^n+m)≡f(qⁿ) (modq ^m-1), ∀n ɛ N andm>0. We prove that iff is of sufficiently small growth, then it is a polynomial.      

On the maximal operator of (<Emphasis Type="Italic">C</Emphasis>, α)-means of Walsh–Kaczmarz–Fourier series

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σ^α,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H _p to the space L _p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σ^α,κ,* is bounded from the martingale Hardy space H _1/(1+α) to the space weak- L _1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H _p such that the maximal operator σ^α,κ,* f does not belong to the space L _p .     

Quantum three-body problems

A scheme for dealing with the quantum three-body problem is presented to separate the rotational degrees of freedom completely from the internal ones. In this method, the three-body Schrodinger equation is reduced to a system of coupled partial differential equations, depending only upon three internal variables. For arbitrary total orbital angular momentum / and the parity (− 1) ^l+λ (λ = 0 or 1), the number of the equations in this system isl = 1 −λ. By expanding the wavefunction with respect to a complete set of orthonormal basis functions, the system of equations is further reduced to a system of linear algebraic equations.     

$${\mathbb{Z}}_2^k$$ -Manifolds are isospectral on forms

We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, Δ _f , acting on sections of the full exterior bundle over an arbitrary compact flat Riemannian n-manifold M with holonomy group , 1 ≤ k ≤ n − 1. This formula implies that any two such manifolds having isospectral lattices of translations are isospectral with respect to Δ _f . As a consequence, we construct a large family of pairwise Δ _f -isospectral and nonhomeomorphic n-manifolds of cardinality greater than . Supported by Conicet, Secyt-UNC.     

Higher-order conservation laws for the nonlinear Poisson equation via characteristic cohomology

We study higher-order conservation laws of the nonlinearizable elliptic Poisson equation as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. We show that the \mathbbS¹{\mathbb{S}^1} -symmetry of the PDE leads to a normal form for the undifferentiated conservation laws. Zhiber and Shabat (in Sov Phys Dokl Akad 24(8):607–609, 1979) determine which potentials of nonlinearizable Poisson equations admit nontrivial Lie–B?cklund transformations. In the case that such transformations exist, they introduce a pseudo-differential operator that can be used to generate infinitely many such transformations. We obtain similar results using the theory of characteristic cohomology: we show that for higher-order conservation laws to exist, it is necessary that the potential satisfies a linear second-order ODE. In this case, at most two new conservation laws in normal form appear at each even prolongation. By using a recursion motivated by Killing fields, we show that, for the simplest class of potentials, this upper bound is attained. The recursion circumvents the use of pseudo-differential operators. We relate higher-order conservation laws to generalized symmetries of the exterior differential system by identifying their generating functions. This Noether correspondence provides the connection between conservation laws and the canonical Jacobi fields of Pinkall and Sterling.     

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T₁(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x₁ ..., x_p} (2p=q+1), and Γ, a subgroup of finite index in G. T₁(Γ, 1) is defined on the subspace L²(Γ/G, 1) ⊖ E_ex, where E_ex is the subspace of eigenfunctions of T₁(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 92–109. Translated by A. M. Nikitin.     

On the invertibility of the operator <Emphasis Type="Italic">d/dt</Emphasis> + <Emphasis Type="Italic">A</Emphasis> in certain functional spaces

We prove that the operator d/dt + A constructed on the basis of a sectorial operator A with spectrum in the right half-plane of ℂ is continuously invertible in the Sobolev spaces W _p ¹ (ℝ, D _α), α ≥ 0. Here, D _α is the domain of definition of the operator A ^α and the norm in D _α is the norm of the graph of A ^α. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1020–1025, August, 2007.     

Spectral properties of higher order anharmonic oscillators

We discuss spectral properties of the selfadjoint operator - \fracd²dt² + ( \fract^{k + 1}k + 1 - a )² \begin{gathered} - \frac{{{d^2}}}{{d{t^2}}} + {\left( {\frac{{{t^{k + 1}}}}{{k + 1}} - \alpha } \right)^2} \hfill \\ \hfill \\ \end{gathered} in L ²(ℝ) for odd integers k. We prove that the minimum over α of the ground state energy of this operator is attained at a unique point which tends to zero as k tends to infinity. We also show that the minimum is nondegenerate. These questions arise naturally in the spectral analysis of Schr?dinger operators with magnetic field. Bibliography: 13 titles. Illustrations: 2 figures.