Estimating the ground state energy of the Schrödinger equation for convex potentials

In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε$\epsilon$. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d$d$. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$ and ε⁻¹${\epsilon }^{-1}$, while the number of qubits is polynomial in d$d$ and logε⁻¹$log{\epsilon }^{-1}$. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1)$1-\delta ,\delta \in (0,1)$, with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$, ε⁻¹${\epsilon }^{-1}$ and δ⁻¹${\delta }^{-1}$, while the number of qubits is polynomial in d$d$, logε⁻¹$log{\epsilon }^{-1}$ and logδ⁻¹$log{\delta }^{-1}$.     

Two remarks on frequent hypercyclicity

We show that if T:X→X$T:X\to X$ is a continuous linear operator on an F$F$-space X≠{0}$X\ne \left\{0\right\}$, then the set of frequently hypercyclic vectors of T$T$ is of first category in X$X$, and this answers a question of A. Bonilla and K.-G. Grosse-Erdmann. We also show that if T:X→X$T:X\to X$ is a bounded linear operator on a Banach space X≠{0}$X\ne \left\{0\right\}$ and if T$T$ is frequently hypercyclic (or, more generally, syndetically transitive), then the T^∗$T^{\ast }$-orbit of every non-zero element of X^∗$X^{\ast }$ is bounded away from 0, and in particular T^∗$T^{\ast }$ is not hypercyclic.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

An upper bound for a valence of a face in a parallelohedral tiling

Consider a face-to-face parallelohedral tiling of R^d${\mathbb{R}}^d$ and a (d−k)$(d-k)$-dimensional face F$F$ of the tiling. We prove that the valence of F$F$ (i.e. the number of tiles containing F$F$ as a face) is not greater than 2^k$2^k$. If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay k$k$-cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.     

Multidimensional extension of the Morse–Hedlund theorem

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence x$x$ over a finite alphabet is ultimately periodic if and only if, for some n$n$, the number of different factors of length n$n$ appearing in x$x$ is less than n+1$n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let d≥2$d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of Z^d${\mathbb{Z}}^d$ definable by a first order formula in the Presburger arithmetic 〈Z;<,+〉$〈\mathbb{Z};<,+〉$. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse–Hedlund theorem to an arbitrary dimension d$d$ and characterize sets of Z^d${\mathbb{Z}}^d$ definable in 〈Z;<,+〉$〈\mathbb{Z};<,+〉$ in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often.     

Cone isomorphisms and almost surjective operators

Let E$E$ be a Banach lattice and F$F$ a Banach space. A bounded linear operator T:E→F$T:E\to F$ is an isomorphism on the positive cone of E$E$ if and only if T^∗$T^{\ast }$ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T:F→E$T:F\to E$ is almost surjective if and only if T^∗$T^{\ast }$ is an isomorphism on the positive cone of F^∗$F^{\ast }$.     

Partial rigidity of degenerate CR embeddings into spheres

In this paper, we study degenerate CR embeddings f$f$ of a strictly pseudoconvex hypersurface M⊂Cⁿ⁺¹$M\subset {\mathbb{C}}^{n+1}$ into a sphere S$\mathbb{S}$ in a higher dimensional complex space C^N+1${\mathbb{C}}^{N+1}$. The degeneracy of the mapping f$f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f$f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d$d$ of the second fundamental form and all of its covariant derivatives is <n$<n$ (here, n$n$ is the CR dimension of M$M$), then f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$. The converse of this statement is also true, as is easy to see. When the total rank d$d$ exceeds n$n$, it is no longer true, in general, that f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n$n$, then partial rigidity may still persist, but there is a “defect” k$k$ that arises from the ranks exceeding n$n$ such that f(M)$f\left(M\right)$ is only contained in a complex plane of dimension n+d+k+1$n+d+k+1$. Moreover, this defect occurs in general, as is illustrated by examples.     

Constructing cubature formulae of degree 5 with few points

This paper is devoted to construct a family of fifth degree cubature formulae for n$n$-cube with symmetric measure and n$n$-dimensional spherically symmetrical region. The formula forn$n$-cube contains at most n²+5n+3$n^2+5n+3$ points and for n$n$-dimensional spherically symmetrical region contains only n²+3n+3$n^2+3n+3$ points. Moreover, the numbers can be reduced to n²+3n+1$n^2+3n+1$ and n²+n+1$n^2+n+1$ if n=7$n=7$ respectively, the latter of which is minimal.     

On probabilistic results for the discrepancy of a hybrid-Monte Carlo sequence

In many applications it has been observed that hybrid-Monte Carlo sequences perform better than Monte Carlo and quasi-Monte Carlo sequences, especially in difficult problems. For a mixed s$s$-dimensional sequence m$m$, whose elements are vectors obtained by concatenating d$d$-dimensional vectors from a low-discrepancy sequence q$q$ with (s−d)$(s-d)$-dimensional random vectors, probabilistic upper bounds for its star discrepancy have been provided. In a paper of G. Ökten, B. Tuffin and V. Burago [G. Ökten, B. Tuffin, V. Burago, J. Complexity 22 (2006), 435–458] it was shown that for arbitrary ε>0$\epsilon >0$ the difference of the star discrepancies of the first N$N$ points of m$m$ and q$q$ is bounded by ε$\epsilon$ with probability at least 1−2exp(−ε²N/2)$1-2exp(-{\epsilon }^{2}N/2)$ for N$N$ sufficiently large. The authors did not study how large N$N$ actually has to be and if and how this actually depends on the parameters s$s$ and ε$\epsilon$. In this note we derive a lower bound for N$N$, which significantly depends on s$s$ and ε$\epsilon$. Furthermore, we provide a probabilistic bound for the difference of the star discrepancies of the first N$N$ points of m$m$ and q$q$, which holds without any restrictions on N$N$. In this sense it improves on the bound of Ökten, Tuffin and Burago and is more helpful in practice, especially for small sample sizes N$N$. We compare this bound to other known bounds.     

Heat-kernel estimates for random walk among random conductances with heavy tail

We study models of discrete-time, symmetric, Z^d${\mathbb{Z}}^d$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances _ωxy∈[0,1]${\omega }_{xy}\in [0,1]$, with polynomial tail near 0 with exponent γ>0$\gamma >0$. We first prove for all d≥5$d\ge 5$ that the return probability shows an anomalous decay (non-Gaussian) that approaches (up to sub-polynomial terms) a random constant times n⁻²$n^{-2}$ when we push the power γ$\gamma$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay n^−d/2$n^{-d/2}$ for large values of the parameter γ$\gamma$.     

The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R^d${\mathbb{R}}^d$ on L^p(X)$L^p(X)$-spaces are convergent for d?3$d?3$ and p>d/(d-1)$p>d/(d-1)$.     

Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic p$p$-Laplace equation, p∈(1,2)$p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1)$r\in (0,1)$ on a bounded open domain Λ⊂R^d$\Lambda \subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p$p$ and r$r$ respectively (in the Hilbert spaces L²(Λ)$L^2\left(\Lambda \right)$, H⁻¹(Λ)$H^{-1}\left(\Lambda \right)$ respectively). The highly singular limit case p=1$p=1$ is treated with the help of stochastic evolution variational inequalities, where P$\mathbb{P}$-a.s. convergence, uniformly in time, is established.     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.