A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions

Under the assumption that $V\in L^2([0,\pi ];dx)$, we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators $?d^2/d{x}^2+V$ in $L^2([0,\pi ];dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $?d^2/d{x}^2+V$ in $L^p([0,\pi ];dx)$, $p\in (1,\infty )$.     

Asymptotic expansions for Laplace transforms of Markov processes

Let ${\mu }^{?}$ be the probability measures on $D[0,T]$ of suitable Markov processes ${\{{\xi }_t^{?}\}}_{0\le t\le T}$ (possibly with small jumps) depending on a small parameter $?>0$, where $D[0,T]$ denotes the space of all functions on $[0,T]$ which are right continuous with left limits. In this paper we investigate asymptotic expansions for the Laplace transforms ${\int }_{D[0,T]}\exp ?\{{?}^{?1}F(x)\}{\mu }^{?}(dx)$ as $?\to 0$ for smooth functionals F on $D[0,T]$. This study not only recovers several well-known results, but more importantly provides new expansions for jump Markov processes. Besides several standard tools such as exponential change of measures and Taylor's expansions, the novelty of the proof is to implement the expectation asymptotic expansions on normal deviations which were recently derived in [13].     

On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.     

Pathological solutions to the Euler–Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems

In this paper, we prove that if $L(x,u,v)\in C^3(R^3\to R)$, $L_{vv}>0$ and $L\ge \alpha |v|+\beta$, $\alpha >0$, then all problems (1), (2) admit solutions in the class $W^{1,1}[a,b]$, which are in fact $C^3$-regular provided there are no pathological solutions to the Euler equation (5). Here $u\in C^3[c,d[$ is called a pathological solution to equation (5) if the equation holds in $[c,d[$, $|\dot{u}(x)|\to \infty$ as $x\to d$, and ${6u6}_{C[c,d]}<\infty$. We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.     

On the norms of quaternionic harmonic projection operators

As a consequence of integral bounds for three classes of quaternionic spherical harmonics, we prove some bounds from below for the $(L^p,L^2)$ norm of quaternionic harmonic projectors, for $p\in [1,2]$.     

Sharp threshold of blow-up and scattering for the fractional Hartree equation

We consider the fractional Hartree equation in the $L^2$-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $M{[u_0]}^{\frac{s?s_c}{s_c}}E[u_0]<M{[Q]}^{\frac{s?s_c}{s_c}}E[Q]$ and $M{[u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2<M{[Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, then the solution $u(t)$ is globally well-posed and scatters; if $M{[u_0]}^{\frac{s?s_c}{s_c}}E[u_0]<M{[Q]}^{\frac{s?s_c}{s_c}}E[Q]$ and $M{[u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2>M{[Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation.     

On the evaluation of prolate spheroidal wave functions and associated quadrature rules

As demonstrated by Slepian et al. in a sequence of classical papers (see Slepian (1983) [33], Slepian and Pollak (1961) [34], Landau and Pollak (1961) [18], Slepian and Pollak (1964) [35], Slepian (1965) [36]), prolate spheroidal wave functions (PSWFs) provide a natural and efficient tool for computing with bandlimited functions defined on an interval. Recently, PSWFs have been becoming increasingly popular in various areas in which such functions occur – this includes physics (e.g. wave phenomena, fluid dynamics), engineering (signal processing, filter design), etc.To use PSWFs as a computational tool, one needs fast and accurate numerical algorithms for the evaluation of PSWFs and related quantities, as well as for the construction of corresponding quadrature rules, interpolation formulas, etc. During the last 15 years, substantial progress has been made in the design of such algorithms – see, for example, Xiao et al. (2001) [40] (see also Bowkamp (1947) [6], Slepian and Pollak (1961) [34], Landau and Pollak (1961) [18], Slepian and Pollak (1964) [35] for some classical results).The complexity of many of the existing algorithms, however, is at least quadratic in the band limit c. For example, the evaluation of the nth eigenvalue of the prolate integral operator requires $O(c^2+n^2)$ operations (see e.g. Xiao et al. (2001) [40]); the construction of accurate quadrature rules for the integration (and associated interpolation) of bandlimited functions with band limit c requires $O(c^3)$ operations (see e.g. Cheng et al. (1999) [8]). Therefore, while the existing algorithms are satisfactory for moderate values of c (e.g. $c⩽{10}^3$), they tend to be relatively slow when c is large (e.g. $c⩾{10}^4$).In this paper, we describe several numerical algorithms for the evaluation of PSWFs and related quantities, and design a class of PSWF-based quadratures for the integration of bandlimited functions. While the analysis is somewhat involved and will be published separately (currently, it can be found in Osipov and Rokhlin (2012) [27]), the resulting numerical algorithms are quite simple and efficient in practice. For example, the evaluation of the nth eigenvalue of the prolate integral operator requires $O(n+c\cdot \log c)$ operations; the construction of accurate quadrature rules for the integration (and associated interpolation) of bandlimited functions with band limit c requires $O(c)$ operations. All algorithms described in this paper produce results essentially to machine precision. Our results are illustrated via several numerical experiments.     

On vector fields with simply connected trajectories and one invariant line

We study polynomial vector fields X on ${\mathbb{C}}^2$ which have simply connected trajectories and satisfy $dP(X)=a?P$, for a constant $a\in {\mathbb{C}}^{?}$ and a primitive polynomial $P\in \mathbb{C}[x,y]$. We determine X, up to an algebraic change of coordinates. In particular, we obtain that X is complete.     

Fermat-type configurations of lines in P3 and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals I in $\mathbb{K}[x,y,z,w]$ for which the containment $I^{(3)}?I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb{P}}^3$, which resemble Fermat configurations of points in ${\mathbb{P}}^2$, see [14]. All examples exhibiting the failure of the containment $I^{(3)}?I^2$ constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.