Absence of eigenvalues for the periodic Schrödinger operator with singular potential in a rectangular cylinder

We consider the periodic Schrödinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δ _Σ(x,y), where Σ is a periodic system of hypersurfaces. We establish that there are no eigenvalues in the spectrum of this operator, provided that Σ is sufficiently smooth and σ ∈ L _p,loc(Σ), p > d ? 1.     

The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set Ω⊂Rd whose boundary Γ has a finite (d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D₀ on L₂(Γ) by form methods. The operator −D₀ is self-adjoint and generates a contractive C₀-semigroup S=(St)_t>0 on L₂(Γ). We show that the asymptotic behaviour of St as t→∞ is related to properties of the trace of functions in H¹(Ω) which Ω may or may not have.     

Linear fractional composition operators on H2

If ? is an analytic function mapping the unit diskD into itself, the composition operatorC _? is the operator onH ² given byC _?f=fo?. The structure of the composition operatorC _? is usually complex, even if the function ? is fairly simple. In this paper, we consider composition operators whose symbol ? is a linear fractional transformation mapping the disk into itself. That is, we will assume throughout that $$\varphi \left( z \right) = \frac{{az + b}}{{cz + d}}$$ for some complex numbersa, b, c, d such that ? maps the unit diskD into itself. For this restricted class of examples, we address some of the basic questions of interest to operator theorists, including the computation of the adjoint.     

Best constants of harmonic approximation of some convolution classes associated with Laplace operator in d-variables

In this paper,we consider the best EFET(entire functions of the exponential type) approximations of some convolution classes associated with Laplace operator on R d and obtain exact constants in the spaces L1(R2) and L2(Rd).Moreover,the best constants of trigonometric approximations of their analogies on Td are also gained.     

Metric entropy of the integration operator and small ball probabilities for the Brownian sheet

Let T_d : L₂([0, 1]^d) → C([0, 1]^d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k⁻¹ (log k)d− 1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]^d under the C([0, 1]^d)-norm can be estimated from below by exp(−Cɛ⁻²¦ log ɛ¦^2d−1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.     

On the extendable regular integrals of the n-body problem

It is shown that the n-body problem in a d-dimensional space has no C¹-extendable regular integrals if n ? d + 1.     

A dimension series for multivariate splines

For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceC _k ^r (Δ) consisting of allC ^r piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑ _k≥0 dim_? C _k ^r (Δ)λ^k and show, under mild conditions on Δ, that this always has the formP(λ)/(1?λ) ^d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=f _d (Δ), andP′(1)=(r+1)f _d?1 ⁰ (Δ). We discuss how the polynomialP(λ) and bases for the spacesC _k ^r (Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.     

Über ein Turánsches Problem für ℓ-1 radiale, positiv definite Funktionen

Turán’s problem is to determine the greatest possible value of the integral ∫_? ^df(x)dx/ f (0) for positive definite functions f (x), x ∈ ?^d, supported in a given convex centrally symmetric body D ? ?^d. In this note we consider the 2-dimensional Turán problem for positive definite functions of the form f(x) = φ (∥x∥1), x ∈ ?², with φ supported in [0,π].     

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators L _Θ,Ω in L ²(Ω; d ⁿ x) ^m , m, n ∈ ?, associated with divergence form elliptic partial differential expressions L with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains Ω ? ? ⁿ . In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions L which act as $$Lu = - \left( {\sum\limits_{j,k = 1}^n {\partial _j } \left( {\sum\limits_{\beta = 1}^m {a_{j,k}^{\alpha ,\beta } \partial _k u_\beta } } \right)} \right)_{1 \leqslant \alpha \leqslant m} , u = \left( {u_1 , \ldots ,u_m } \right).$$ The (nonlocal) Robin-type boundary conditions are then of the form $$v \cdot ADu + \Theta [u|_{\partial \Omega } ] = 0{\text{ on }}\partial \Omega ,$$ where Θ represents an appropriate operator acting on Sobolev spaces associated with the boundary ?Ω of Ω, ν denotes the outward pointing normal unit vector on ?Ω, and $Du: = \left( {\partial _j u_\alpha } \right)_{_{1 \leqslant j \leqslant n}^{1 \leqslant \alpha \leqslant m} } .$ Assuming Θ ≥ 0 in the scalar case m = 1, we prove Gaussian heat kernel bounds for L _Θ,Ω, by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on ?Ω. We also discuss additional zero-order potential coefficients V and hence operators corresponding to the form sum L _Θ,Ω + V.     

The discrete analogue of a differential operator and its applications

We construct a discrete analogue D _m (hβ) of the differential operator d^2m /dx ^2m + 2ω ²d^2m?2 /dx ^2m?2 + ω ⁴d^2m?4 /dx ^2m?4 for any m ≥ 2. In the case m = 2, we apply in the Hilbert space K ₂(P ₂) the discrete analogue D ₂(hβ) for construction of optimal quadrature formulas and interpolation splines minimizing the seminorm, which are exact for trigonometric functions sin ωx and cos ωx.     

A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L ₂(T ^ν ×T ^ν ), where T ^ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrödinger operator ? = H₀ + H₁ + H₂, where H₀ is the operator of multiplication by a function and H₁ and H₂ are partial integral operators. We prove several theorems concerning the essential spectrum of ?. We study the discrete and essential spectra of the Hamiltonians H^t and h arising in the Hubbard model on the three-dimensional lattice.     

A Unique Theory of Gravity and Matter

The author has previously suggested that the ground state for 4-dimensional quantumgravity can be represented as a condensation of non-linear gravitons connected by Dirac strings.In this note we suggest that the low-lying excitations of this state can be described by a quasi-topological action of the form ∫d ¹³ z F₄∧F₅∧F₄ , corresponding to a trilinear coupling of solitonic 8-branes and 7-branes. It is shownthat when the excitations associated with F₅ are neglected, the effective action can beinterpreted as a theory of conformal gravity in four dimensions. This in turn suggests that ordinarygravity as well supersymmetric matter and phenomenological gauge symmetries arise from thespontaneous breaking of topological invariance. The possibly deep mathematical significance ofthis theory is also noted. 1999 Elsevier Science Ltd.     

Schauder estimates for a class of second order elliptic operators on a cube

We consider a class of second order elliptic operators on a d-dimensional cube S_d. We prove that if the coefficients are of class C^k+δ(S_d), with k=0,1 and δ∈(0,1), then the corresponding elliptic problem admits a unique solution u belonging to C^k+2+δ(S_d) and satisfying non-standard boundary conditions involving only second order derivatives.     

Spectrum bottom and largest vacuity

The paper considers a scalar second-order elliptic operator with a non-negative random potential term V(x) = ∑ _{X ∈?} W(x?X) ≥ 0 corresponding to a Poisson cloud ? = {X} of “soft obstacles.” The operator acts on functions vanishing outside a large cubic open “box”rQ ⁰ = (?½r, ½r) ^d ?? ^d , d≥ 2. The paper develops a method of estimating from below the spectrum bottom of the operator through the volume of the largest connected set that can be made of smaller “blocks” containing relatively few obstacles. In the case of constant coefficients, the principal eigenvalue λ_{?, V} (r?) of (?? + V) in r? ⁰ is shown to satisfy, with high probability, the estimate where λ_?,* is the infimum of principal value of operator with zero potential term V≡ 0 under the Dirichlet condition on the boundary of a regular set of volume not exceeding one.     

Optimal error estimate and superconvergence of the DG method for first-order hyperbolic problems

We consider the original discontinuous Galerkin method for the first-order hyperbolic problems in d-dimensional space. We show that, when the method uses polynomials of degree k, the L₂-error estimate is of order k+1 provided the triangulation is made of rectangular elements satisfying certain conditions. Further, we show the O(h^2k+1)-order superconvergence for the error on average on some suitably chosen subdomains (including the whole domain) and their outflow faces. Moreover, we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k+1.     

Equivalence of measures for some class of Gaussian random fields

We consider two Gaussian measures P₁ and P₂ on (C(G), $\text{B}$) with zero expectations and covariance functions R₁(x, y) and R₂(x, y) respectively, where R_ν(x, y) is the Green's function of the Dirichlet problem for some uniformly strongly elliptic differential operator A^(ν) of order $\text{2m, m ≥ [}\text{d}\text{2}\text{] + 1}$, on a bounded domain G in $\text{R}$^d (ν = 1, 2). It is shown that if the order of A⁽²⁾ ? A⁽¹⁾ is at most $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are equivalent, while if the order is greater than $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are not always equivalent.     

Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators

We investigate the convergence of a nonlinear approximation method introduced by Ammar et?al. (J. Non-Newtonian Fluid Mech. 139:153–176, 2006) for the numerical solution of high-dimensional Fokker–Planck equations featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson’s equation on a rectangular domain in ?², subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelièvre and Maday (Const. Approx. 30:621–651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173–187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein–Uhlenbeck operator of the kind that appears in Fokker–Planck equations that arise in bead–spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D=D ₁×?×D _N contained in ? ^Nd , where each set D _i , i=1,…,N, is a bounded open ball in ? ^d , d=2,3.     

An ?-dependent formulation of the Kadomtsev-Petviashvili hierarchy

We briefly review a recursive construction of ?-dependent solutions of the Kadomtsev-Petviashvili hierarchy. We give recurrence relations for the coefficients X_n of an ?-expansion of the operator X = X ₀ + ?X ₁ + ? ² X ₂ + ... for which the dressing operator W is expressed in the exponential form W = e^X/?. The wave function ?? associated with W turns out to have the WKB (Wentzel-Kramers-Brillouin) form ?? = e^S/kh, and the coefficients S_n of the ?-expansion S = S ₀ + ?S ₁ + ? ² S ₂ + ... are also determined by a set of recurrence relations. We use this WKB form to show that the associated tau function has an ?-expansion of the form log ?? = ?^?2 F ₀ + ?^?1 F ₁ + F ₂ + ....     

An inverse scattering problem for a perturbed Hill's operator

We consider the inverse scattering problem for the operator L=?d²/dx²+p(x)+q(x), x ∈ R¹. The perturbation potential q is expressed in terms of the periodic potential p and the scattering data. We also obtain identities for the eigenfunctions of the unperturbed Hill's operator L₀=?d²/dx²+p(x).