Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian ??(???Δ)^α/2???q in R ^d , for q?≥?0, α?∈?(0,2). We obtain sharp estimates of the first eigenfunction φ ₁ of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim_{|x| →?∞?} q(x)?=?∞ and comparable on unit balls we obtain that φ ₁(x) is comparable to (|x|?+?1)^???d???α (q(x)?+?1)^???1 and intrinsic ultracontractivity holds iff lim_{|x| →?∞?} q(x)/log|x|?=?∞. Proofs are based on uniform estimates of q-harmonic functions.     

Riesz basis property of system of root functions of second-order differential operator with involution

The properties of the root functions are studied for an arbitrary operator generated in L ₂(?1, 1) by the operation with involution of the form Lu = ?u″(x)+αu″(?x)+q(x)u(x)+ qν(x)u(ν(x)), where α ∈ (?1, 1), ν(x) is an absolutely continuous involution of the segment [?1, 1] and the coefficients q(x) and qν(x) are summable functions on (?1, 1). Necessary and sufficient conditions are obtained for the unconditional basis property in L ₂(?1, 1) for the system of the root functions of the operator.     

Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

For the self-adjoint Schrödinger operator ? defined on ? by the differential operation ?(d/dx)² + q(x) with a distribution potential q(x) uniformly locally belonging to the space W ₂ ^?1, we describe classes of functions whose spectral expansions corresponding to the operator ? absolutely and uniformly converge on the entire line ?. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

A limit theorem for continuous selectors

Any (measurable) function K from Rⁿ to R defines an operator K acting on random variables X by K(X) = K(X₁,..., X_n), where the X_j are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x₁, x₂,..., x_n) ∈ {x₁, x₂,..., x_n}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ω_H in the interval (0, 1) so that, for any random variable X, the iterates H^(N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.     

Uniform,on the entire axis,convergence of the spectral expansion for Schrödinger operator with a potential-distribution

A uniform, on ?, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ? by the differential operation (?d/dx)² + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W ₂ ^?1. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ?.     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

Congruence by Dissection of Topological Discs—An Elementary Approach to Tarski's Circle Squaring Problem

Let G be a group of affine transformations of the plane R ² and let the family F consist of all topological discs in R ² whose boundary is subject to some smoothness condition (general, rectifiable, piecewise C ¹ , piecewise C ² ). Are any two members D,E ∈ F congruent by dissection with respect to G such that all the pieces in the corresponding dissections of D and E belong to F as well? We give an affirmative answer if G contains all affine transformations and F consists of the discs whose boundary is piecewise C ¹ . An example shows that C ¹ cannot be replaced by C ² . Moreover, if G is either the group of equiaffine transformations or the group of similarities, then congruence by dissection of two convex discs D and E turns out to be essentially equivalent to congruence by dissection of the boundaries bd(D ) and bd(E ).     

Some matrix functional equations

We investigate the pair of matrix functional equations G(x)F(y) = G(xy) and G(x)G(y) = F(y/x), featuring the two independent scalar variables x and y and the two N×N matrices F(z) andG(z) (with N an arbitrary positive integer and the elements of these two matrices functions of the scalar variable z). We focus on the simplest class of solutions, i.e., on matrices all of whose elements are analytic functions of the independent variable. While in the scalar (N = 1) case this pair of functional equations only possess altogether trivial constant solutions, in the matrix (N > 1) case there are nontrivial solutions. These solutions satisfy the additional pair of functional equations F(x)G(y) = G(y/x) andF(x)F(y) = F(xy), and an endless hierarchy of other functional equations featuring more than two independent variables.     

Blow-up of <Emphasis Type="Italic">p</Emphasis>-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power

$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$

where Ω is either a bounded domain or the whole space ? ^N , q(x) is a positive and continuous function defined in Ω with 0 < q _? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q ₊ < p ? 1, all the solutions are global. If q _? > p ? 1, there exist global solutions, while for given q _? < p ? 1 < q ₊, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? ^N , the Fujita phenomenon occurs if 1 < q _? ? q ₊ ? p ? 1 + p/N, while if q _? > p ? 1 + p/N, there exist global solutions.

     

On multiple zeros of a partial theta function

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ? is a variable and q ∈ ?, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

Asymptotic formula for the convolution of a generalized divisor function

An asymptotic formula is obtained for the sum of terms σ _it (n)σ_-it (N - n) (t is real) over 0 < n < N with a remainder estimated by O _ε((1+|t|)^1+ε N ^3/4+ε) for any ε > 0. As a consequence, Porter’s result on a power scale for the average number of steps in the Euclidean algorithm is improved.     

On Templeman averages and variation functions

Let S be a countable semigroup acting in a measure-preserving fashion (g ? T _g ) on a measure space (Ω, A, µ). For a finite subset A of S, let |A| denote its cardinality. Let (A _k ) _k=1 ^∞ be a sequence of subsets of S satisfying conditions related to those in the ergodic theorem for semi-group actions of A. A. Tempelman. For A-measureable functions f on the measure space (Ω, A, μ) we form for k ≥ 1 the Templeman averages \(\pi _k (f)(x) = \left| {A_k } \right|^{ - 1} \sum\nolimits_{g \in A_k } {T_g f(x)}\) and set V _q f(x) = (Σ _k≥1|π _k+1(f)(x) ? π _k (f)(x)|q)^1/q when q ∈ (1, 2]. We show that there exists C > 0 such that for all f in L ¹(Ω, A, µ) we have µ({x ∈ Ω: V _q f(x) > λ}) ≤ C(∫_Ω | f | dµ/λ). Finally, some concrete examples are constructed.     

Quantum quaternion spheres

For the quantum symplectic group SP _q (2n), we describe the C ^?-algebra of continuous functions on the quotient space S P _q (2n)/S P _q (2n?2) as an universal C ^?-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the K-groups of this C ^?-algebra in terms of generators of the C ^?-algebra.     

Seifert manifolds and (1, 1)-knots

The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $

has the fundamental group cyclically presented by G _n ((x ₁ ^q ...x _n ^{q l} x _n ^?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.

     

Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval

The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with Jacobi weight ?^(α,β)(x) = (1 ? x)^α(1 + x)^β α ≥ β > ?1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L _q ^(α,β) , 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\), is attained.     

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator A _{ε, ε} > 0, on L ₂(R ^d ;C ⁿ ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA _ε ^1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA _ε ^1/2 ) converges to cos(τ(A ⁰)^1/2) in the norm of operators acting from the Sobolev space H ^s (R ^d ;C ⁿ ) (with a suitable s) to L ₂(R ^d ;C ⁿ ). Here A ₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? _τ ² u _ε (x, τ) = ?A _ε u _ε (x, τ).     

Partial Gaussian Bounds for Degenerate Differential Operators

Let S be the semigroup on \(L_2({{\bf R}}^d)\) generated by a degenerate elliptic operator, formally equal to \(- \sum \partial_k \, c_{kl} \, \partial_l\), where the coefficients c _kl are real bounded measurable and the matrix C(x)?=?(c _kl (x)) is symmetric and positive semi-definite for all x?∈?R ^d . Let Ω???R ^d be a bounded Lipschitz domain and μ?>?0. Suppose that C(x)?≥?μ I for all x?∈?Ω. We show that the operator P _Ω S _t P _Ω has a kernel satisfying Gaussian bounds and Gaussian Hölder bounds, where P _Ω is the projection of \(L_2({{\bf R}}^d)\) onto L ₂(Ω). Similar results are for the operators u ? χ S _t (χ u), where \(\chi \in C_{\rm b}^\infty({{\bf R}}^d)\) and C(x)?≥?μI for all \(x \in {\mathop{\rm supp}} \chi\).     

Homogenization of a nonstationary model equation of electrodynamics

In L ₂(?₃;?₃), we consider a self-adjoint operator ? _ε , ε > 0, generated by the differential expression curl η(x/ε)^?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? _ε ^1/2 ) and ? _ε ^?1/2 sin(τ? _ε ^1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?⁰)^1/2) and (?₀)^?1/2 sin(τ(?₀)^1/2), respectively, in the norm of the operators acting from the Sobolev space H ^s (with a suitable s) to ?₂. Here ?⁰ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? _τ ² v _ε = ?? _ε v _ε , div v _ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).