Super-exponential Decay Research Articles

Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Finally, we prove that such phenomena remain, in the case of compactlysupported initial data.

Weighted Finite Fourier Transform Operator: Uniform Approximations of the Eigenfunctions, Eigenvalues Decay and Behaviour

In this paper, we first give two uniform asymptotic approximations of the eigenfunctions of the weighted finite Fourier transform operator, defined by ${\displaystyle \mathcal F_c^{(\alpha)} f(x)=\int_{-1}^1 e^{icxy} f(y)\,(1-y^2)^{\alpha}\, dy,\,}$ where $ c >0, \alpha > -1$ are two fixed real numbers. The first uniform approximation is given in terms of a Bessel function, whereas the second one is given in terms of a normalized Jacobi polynomial. These eigenfunctions are called generalized prolate spheroidal wave functions (GPSWFs). By using the uniform asymptotic approximations of the GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the operator $\mathcal F_c^{(\alpha)}$ in the case where $0<\alpha < 3/2.$ Finally, by computing the trace and an estimate of the norm of the operator ${\displaystyle \mathcal Q_c^{\alpha}=\frac{c}{2\pi} \mathcal F_c^{{\alpha}^*} \mathcal F_c^{\alpha},}$ we give a lower and an upper bound for the counting number of the eigenvalues of $Q_c^{\alpha},$ when $c>>1.$

Hyperfine-induced spin relaxation of a diffusively moving carrier in low dimensions: Implications for spin transport in organic semiconductors

The hyperfine coupling between the spin of a charge carrier and the nuclear spin bath is a predominant channel for the carrier spin relaxation in many organic semiconductors. We theoretically investigate the hyperfine-induced spin relaxation of a carrier performing a random walk on a d-dimensional regular lattice, in a transport regime typical for organic semiconductors. We show that in d=1 and d=2 the time dependence of the space-integrated spin polarization, P(t), is dominated by a superexponential decay, crossing over to a stretched exponential tail at long times. The faster decay is attributed to multiple self-intersections (returns) of the random walk trajectories, which occur more often in lower dimensions. We also show, analytically and numerically, that the returns lead to sensitivity of P(t) to external electric and magnetic fields, and this sensitivity strongly depends on dimensionality of the system (d=1 vs. d=3). Furthermore, we investigate in detail the coordinate dependence of the time-integrated spin polarization, $\sigma(r)$, which can be probed in the spin transport experiments with spin-polarized electrodes. We demonstrate that, while $\sigma(r)$ is essentially exponential, the effect of multiple self-intersections can be identified in transport measurements from the strong dependence of the spin decay length on the external magnetic and electric fields.

Exponentially sparse representations of Fourier integral operators

We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth. It is known that such an infinite matrix is sparse and well organized, being in fact concentrated along the graph of the corresponding canonical transformation. Here we show that, if the phase and symbol have a regularity of Gevrey type of order s&gt;1 or analytic ( s=1 ), the above decay is in fact sub-exponential or exponential, respectively. We also show by a counterexample that ultra-analytic regularity ( s&lt;1 ) does not give super-exponential decay. This is in sharp contrast to the more favorable case of pseudodifferential operators, or even (generalized) metaplectic operators, which are treated as well.

Approximation of Mixed Order Sobolev Functions on the d-Torus: Asymptotics, Preasymptotics, and d-Dependence

We investigate the approximation of d-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $$s>0$$ on the d-dimensional torus, where the approximation error is measured in the $$L_2$$ -norm. In other words, we study the approximation numbers $$a_n$$ of the Sobolev embeddings $$H^s_\mathrm{mix}(\mathbb {T}^d)\hookrightarrow L_2(\mathbb {T}^d)$$ , with particular emphasis on the dependence on the dimension d. For any fixed smoothness $$s>0$$ , we find two-sided estimates for the approximation numbers as a function in n and d. We observe super-exponential decay of the constants in d, if n, the number of linear samples of f, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by approximations using only a few linear samples (small n). We present some surprising results for the so-called “preasymptotic” decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

Spectral decay of time and frequency limiting operator

For fixed c, the Prolate Spheroidal Wave Functions (PSWFs) ψn,c form a basis with remarkable properties for the space of band-limited functions with bandwidth c. They have been largely studied and used after the seminal work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications rely heavily on the behavior and the decay rate of the eigenvalues (λn(c))n≥0 of the time and frequency limiting operator, which we denote by Qc. Hence, the issue of the accurate estimation of the spectrum of this operator has attracted a considerable interest, both in numerical and theoretical studies. In this work, we give an explicit integral approximation formula for these eigenvalues. This approximation holds true starting from the plunge region where the spectrum of Qc starts to have a fast decay. As a consequence of our explicit approximation formula, we give a precise description of the super-exponential decay rate of the λn(c). Also, we mention that the described approximation scheme provides us with fairly accurate approximations of the λn(c) with low computational load, even for very large values of the parameters c and n. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

An exact formulation of the time-ordered exponential using path-sums

We present the path-sum formulation for the time-ordered exponential of a time-dependent matrix. The path-sum formulation gives the time-ordered exponential as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as self-avoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on a graphs, and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.

Random sequential adsorption of starlike particles.

Random packing of surfaceless starlike particles built of 3 to 50 line segments was studied using random sequential adsorption algorithm. Numerical simulations allow us to determine saturated packing densities as well as the first two virial expansion coefficients for such objects. Measured kinetics of the packing growth supports the power law known to be valid for particles with a finite surface; however, the dependence of the exponent in this law on the number of star arms is unexpected. The density autocorrelation function shows fast superexponential decay as for disks, but the typical distance between closest stars is much smaller than between disks of the similar size, especially for a small number of arms.

On the number of connected components of random algebraic hypersurfaces

We study the expectation of the number of components $b_0(X)$ of a random algebraic hypersurface $X$ defined by the zero set in projective space $\mathbb{R}P^n$ of a random homogeneous polynomial $f$ of degree $d$. Specifically, we consider "invariant ensembles", that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. The classification due to E. Kostlan shows that specifying an invariant ensemble is equivalent to assigning a weight to each eigenspace of the spherical Laplacian. Fixing $n$, we consider a family of invariant ensembles (choice of eigenspace weights) depending on the degree $d$. Under a rescaling assumption on the eigenspace weights (as $d \rightarrow \infty$), we prove that the order of growth of $\mathbb{E} b_0(X)$ satisfies: $$\mathbb{E} b_{0}(X)=\Theta\left(\left[ \mathbb{E} b_0(X\cap \mathbb{R}P^1) \right]^{n} \right). $$ This relates the average number of components of $X$ to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial $f|_{\mathbb{R}P^1}.$ The proof requires an upper bound for $\mathbb{E} b_0(X)$, which we obtain by counting extrema using Random Matrix Theory methods from recent work of the first author, and it also requires a lower bound, which we obtain by a modification of the barrier method. We also provide a quantitative upper bound for the implied constant in the above asymptotic; for the real Fubini-Study model these estimates reveal super-exponential decay of the leading coefficient (in $d$) of $\mathbb{E} b_0(X)$ (as $n \rightarrow \infty$).

Gabor representations of evolution operators

We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schrödinger-type propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolic-type operators. Besides the class of operators, the main novelty of the paper is the proof of super-exponential (as opposed to super-polynomial) off-diagonal decay for the Gabor matrix representation.

Spin-boson model through a Poisson-driven stochastic process

We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness of the ground state of the spin-boson, and analyze ground state properties. In particular, we prove super-exponential decay of the number of bosons, Gaussian decay of the field operators, derive expressions for the positive integer, fractional and exponential moments of the field operator, and discuss the field fluctuations in the ground state.

Gabor frame decomposition of evolution operators and applications

We compute the Gabor matrix for Schr\"odinger-type evolution operators. Precisely, we analyze the Heat Equation, already presented in \cite{2012arXiv1209.0945C}, giving the exact expression of the Gabor matrix which leads to better numerical evaluations. Then, using asymptotic integration techniques, we obtain an upper bound for the Gabor matrix in one-dimension for the generalized Heat Equation, new in the literature. Using Maple software, we show numeric representations of the coefficients' decay. Finally, we show the super-exponential decay of the coefficients of the Gabor matrix for the Harmonic Repulsor, together with some numerical evaluations. This work is the natural prosecution of the ideas presented in \cite{2012arXiv1209.0945C} and \cite{MR2502369}.

CMV Matrices with Super Exponentially Decaying Verblunsky Coefficients

We investigate several equivalent notions of the Jost solution associated with a unitary CMV matrix and provide a necessary and sufficient conditions for the Jost solution to consist of entire functions of finite growth order in terms of super exponential decay of Verblunsky coefficients. We also establish several one-to-one correspondences between CMV matrices with super-exponentially decaying Verblunsky coefficients and spectral data associated with the first component of the Jost solution.

Rellich type theorems for unbounded domains

We give several generalizations of Rellich's classical uniqueness theorem to unbounded domains. We give a natural half-space generalization for super-exponentially decaying inhomogeneities using real variable techniques. We also prove under super-exponential decay a discrete generalization where the inhomogeneity only needs to vanish in a suitable cone. The more traditional complex variable techniques are used to prove the half-space result again, but with less exponential decay, and a variant with polynomial decay, but with supports exponentially thin at infinity. As an application, we prove the discreteness of non-scattering energies for non-compactly supported potentials with suitable asymptotic behaviours and supports.

Equilibration in long-range quantum spin systems from a BBGKY perspective

The time evolution of -spin reduced density operators is studied for a class of Heisenberg-type quantumspin models with long-range interactions. In the framework of the quantumBogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy, we introduce anunconventional representation, different from the usual cluster expansion, which casts thehierarchy into the form of a second-order recursion. This structure suggests ascaling of the expansion coefficients and the corresponding time scales in powers ofN1/2 with thesystem size N, implying a separation of time scales in the large-system limit. For special parametervalues and initial conditions, we can show analytically that closing the BBGKY hierarchyby neglecting -spin correlations never leads to equilibration, but gives rise to quasi-periodic timeevolution with at most independent frequencies. Moreover, for the same special parameter values and in the large-N limit, we solve the complete recursion relation (the full BBGKY hierarchy),observing a superexponential decay to equilibrium in rescaled timeτ = tN − 1/2.

Exponential estimates and holomorphic extensions for semilinear elliptic pseudodifferential equations

We consider semilinear equations p(x, D)u = F [u], where the linear parts p(x, D) are pseudo-differential operators of Shubin type, with symbols satisfying a global ellipticity condition in ℝ n . For classical solutions u ∈ H s (ℝ n ), s > n/2, we obtain properties of super-exponential decay and holomorphic extension, expressed in terms of Gelfand–Shilov classes.

Tagged-particle motion in glassy systems under shear: Comparison of mode coupling theory and Brownian dynamics simulations

We study the dynamics of a tagged particle in a glassy system under shear. The recently developed integration through transients approach, based on mode coupling theory, is continued to arrive at the equations for the tagged-particle correlators and the mean squared displacements. The equations are solved numerically for a two-dimensional system, including a nonlinear stability analysis of the glass solution, the so called β-analysis. We perform Brownian Dynamics simulations in 2D and compare with theory. After switch on, transient glassy correlation functions show strong fingerprints of the stress overshoot scenario, including, additionally to previously studied superexponential decay, a shoulder-like slowing down after the overshoot. We also find a new type of Taylor dispersion in glassy states which has intriguing similarity to the known low-density case. The theory qualitatively captures most features of the simulations with quantitative deviations concerning the shear-induced time scales. We attribute these deviations to an underestimation of the overshoot scenario in the theory.

Calculation of the convex roof for an open entangled harmonic oscillator system

We explicitly calculate the time dependence of entanglement via the convex roof extension for a system of noninteracting harmonic oscillators. These oscillators interact only indirectly with each other by way of a zero-temperature bath. The initial state of the oscillators is taken to be that of an entangled Schr\odinger-cat state. This type of initial condition leads to superexponential decay of the entanglement when the initial state has the same symmetry as the interaction Hamiltonian.

Decay and regularity for harmonic oscillator-type equations

We prove super-exponential decay estimates and holomorphic extensions for the solutions of harmonic oscillator-type equations. The functional setting for our estimates is given by the Gelfand–Shilov classes . In the one-dimensional case, explicit solutions are given in terms of special functions of hypergeometric confluent type.

Existence and decay of solutions to a semilinear Schrödinger equation with magnetic field

In this paper we study the decay property of solutions of a semilinear Schrödinger equation, $-(∇ - iA)^{2}u+(V-E)u=Q|u|^{p-2}u$, on $\mathbb R^{n}$, where $n \geq 2$ and $2<p <2^{*}$. We give a lower bound estimate of nontrivial solutions at infinity. In two-dimensional case, we give super-exponential decay estimates of solutions at infinity. Moreover, we show the existence of a nontrivial solution under additional assumptions on potentials.