Weyl’s Type Estimates on the Eigenvalues of Critical Schrödinger Operators

We consider on a bounded domain \(\Omega \subset {\mathbb{R}}^N\) , the Schrödinger operator ? Δ ? V supplemented with Dirichlet boundary solutions. The potential V is either the critical inverse square potential V(x) = (N ? 2)²/4|x|^?2 or the critical borderline potential V(x) =  (1/4)dist(x, ?Ω)^?2. We present explicit asymptotic estimates on the eigenvalues of the critical Schrödinger operator in each case, based on recent results on improved Hardy–Sobolev type inequalities.     

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

For a two-dimensional Schrödinger operator H _{α V}  = ?Δ ?αV with the radial potential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N _?(H _{α V} ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N _?(H _{α V} ) = O(α) and for the validity of the Weyl asymptotic law.     

Scattering the Geometry of Weighted Graphs

Given two weighted graphs (X, b_k, m_k), k =?1,2 with b₁ ～ b₂ and m₁ ～ m₂, we prove a weighted L¹-criterion for the existence and completeness of the wave operators W_±(H₂, H₁, I_1,2), where H_k denotes the natural Laplacian in ?²(X, m_k) w.r.t. (X, b_k, m_k) and I_1,2 the trivial identification of ?²(X, m₁) with ?²(X, m₂). In particular, this entails a general criterion for the absolutely continuous spectra of H₁ and H₂ to be equal.     

Zur Chronologie des alten Ägypten

The velocityv of the propagation of discharge along the anode of a self-quenchingG—M-counter is a function of total pressureP, pressure of the quenching gasP _D, radius of the cathoder _a and of the anoder _i andV _ü the difference between working- and starting-potential. For the mixtures argon-methylal, argon-alcohol and helium-alcohol isv=v ₀·exp[k·(V _ü/V _e)^1/2] withv ₀ the velocity at the starting potentialV _e v ₀=(a+b·P _D/P)·V _n ^1/2 ·exp [(c?d·P_D/P·V _n ^?1/2 ] andV _n=V _e·(lnr _a/r _i)^?1.k, a, b, c andd are characteristical constants of the filling gas.     

The electron energy spectrum and superconducting transition temperature of strongly correlated fermions with three-center interactions

The renormalizations of the fermionic spectrum are considered within the framework of the t-J* model taking into account three-center interactions (H₍₃₎) and magnetic fluctuations. Self-consistent spin dynamics equations for strongly correlated fermions with three-center interactions were obtained to calculate quasi-spin correlators. A numerical self-consistent solution to a system of ten equations was obtained to show that, in the nearest-neighbor approximation, simultaneously including H₍₃₎ and magnetic fluctuations at n>n₁ (n₁ ≈ 0.72 for 2t/U = 0.25) caused qualitative changes in the structure of the energy spectrum. A new Van Hove singularity is then induced in the density of states, and an additional maximum appears in the T_c(n) concentration dependence of the temperature of the transition to the superconducting phase with order parameter symmetry of the d _x ²?y² type.     

Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains

On a smooth bounded domain \(\Omega \subset {\bf {\rm R}}^N\) we consider the Schrödinger operators ? Δ ? V, with V being either the critical borderline potential V(x) =  (N ? 2)²/4 |x|^?2 or V(x) =  (1/4) dist(x, ?Ω)^?2, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.     

Limit of Fluctuations of Solutions of Wigner Equation

We consider fluctuations of the solution W _ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W _ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the initial data is singular in the x variable, that is, W _ε (0, x, k) = δ(x)f(k) and \({f\in {\mathcal{S}}(\mathbb{R}^d)}\), then the laws of the rescaled fluctuation \({Z_\varepsilon(t):=\varepsilon^{-1/2}[W_\varepsilon(t,x,k)-\bar{W}(t,x,k)]}\) converge, as ε → 0+, to the solution of the same radiative transport equation but with a random initial data. This complements the result of [6], where the limit of the covariance function has been considered.     

Long Time,Large Scale Limit of the Wigner Transform for a System of Linear Oscillators in One Dimension

We consider the long time, large scale behavior of the Wigner transform W _? (t,x,k) of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile et al. in Phys. Rev. Lett. 96 (2006) to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile et al. in Arch. Rat. Mech. 195(1):171–203 (2009). In the present paper we prove that in the unpinned case there exists γ ₀>0 such that for any γ∈(0,γ ₀] the weak limit of W _? (t/? ^3/2γ ,x/? ^γ ,k), as ??1, satisfies a one dimensional fractional heat equation \(\partial_{t} W(t,x)=-\hat{c}(-\partial_{x}^{2})^{3/4}W(t,x)\) with \(\hat{c}>0\). In the pinned case an analogous result can be claimed for W _? (t/? ^2γ ,x/? ^γ ,k) but the limit satisfies then the usual heat equation.     

Dynamical Characteristics of a Non-canonical Scalar-Torsion Model of Dark Energy

In this paper, we analyze the phase-space of a model of dark energy in which a non-canonical scalar field (tachyon) non-minimally coupled to torsion scalar in the framework of teleparallelism. Scalar field potential and non-minimal coupling function are chosen as V(?) = V₀?ⁿ and f(?) = ?^N, respectively. We obtain a critical point that behaves like a stable or saddle point depending on the values of N and n. Additionally we find an unstable critical line. We have shown such a behavior of critical points using numerical computations and phase-space trajectories explicitly.     

Spectra and mean energies of prompt neutrons from <Superscript>238</Superscript>U fission induced by primary neutrons of energy in the region <Emphasis Type="Italic">E</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><20 MeV

The time-of-flight technique is used to measure the ratios R(E, E _n )=N(E, E _n )/N_Cf(E) of the normalized (to unity) spectra N(E, E _n ) of neutrons accompanying the neutron-induced fission of ²³⁸U at primary-neutron energies of E _n =6.0 and 7.0 MeV to the spectrum N_Cf(E) neutrons from the spontaneous fission of ²⁵²Cf. These experimental data and the results of their analysis are discussed together with data that were previously obtained for the neutron-induced fission of ²³⁸U at the primary energies of E _n =2.9, 5.0, 13.2, 14.7, 16.0, and 17.7 MeV.     

Percolation and the electron–electron interaction in an array of antidots

A square lattice of microcontacts with a period of 1 μm in a dense low-mobility two-dimensional electron gas is studied experimentally and numerically. At the variation of the gate voltage V _g , the conductivity of the array varies by five orders of magnitude in the temperature range T from 1.4 to 77 K in good agreement with the formula σ(V _g ) = (V _g ?V _g ^* (T))^β with β = 4. The saturation of σ(T) at low temperatures is absent because of the electron–electron interaction. A random-lattice model with a phenomenological potential in microcontacts reproduces the dependence σ(T, V _g ) and makes it possible to determine the fraction of microcontacts x(V _g , T) with conductances higher than σ. It is found that the dependence x(V _g ) is nonlinear and the critical exponent in the formula σ ∝ ? (x - 1/2) ^t in the range 1.3 < t(T, V _g ) < β.     

Resolvent Expansion and Time Decay of the Wave Functions for Two-Dimensional Magnetic Schrödinger Operators

We consider two-dimensional Schrödinger operators H(B, V) given by Eq. (1.1) below. We prove that, under certain regularity and decay assumptions on B and V, the character of the expansion for the resolvent (H(B, V) ? λ)^?1 as λ → 0 is determined by the flux of the magnetic field B through \({\mathbb{R}^2}\) . Subsequently, we derive the leading term of the asymptotic expansion of the unitary group e ^{?i t H(B, V)} as t → ∞ and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ^{?i t H(0, V)}.     

Thermodynamic study of compounds of the Nd-Ba-Cu-O system

Standard enthalpies of formation for solid solutions of composition Nd_{1 + x} Ba_{2 ? x} Cu₃O _y (x = 0–0.8, y = 6.65–7.24) from oxides were determined by solution calorimetry. The heat capacity of NdBa₂Cu₃O_6.87 phase was measured in the range 5–320 K by low-temperature adiabatic calorimetry. The absolute entropy S ^o(T), the difference of enthalpies H ^o(T)-H ^o(0 K), and the reduced Gibbs energy Φ^o(T) = S ^o(T)–[H ^o(T)–H ^o(0)]/T were calculated on the basis of smoothed dependence C _p (T) in the 0–320 K range. An assessment was made for the heat capacities and the absolute entropies of solid solutions Nd_1+x Ba_2?x Cu₃O _y . The obtained set of thermodynamic parameters can be used for the calculation of phase equilibria in the Nd-Ba-Cu-O system.     

Gamow Vectors and Borel Summability in a Class of Quantum Systems

We analyze the detailed time dependence of the wave function ψ(x,t) for one dimensional Hamiltonians \(H=-\partial_{x}^{2}+V(x)\) where V (for example modeling barriers or wells) and ψ(x,0) are compactly supported.We show that the dispersive part of ψ(x,t) is the Borel sum of its asymptotic series in powers of t ^?1/2, t→∞. The remainder, the difference between ψ and the Borel sum, i.e., the exponential part of the transseries of ψ, is a convergent expansion of the form \(\sum_{k=0}^{\infty}g_{k}\Gamma_{k}(x)e^{-\gamma_{k} t}\), where Γ _k are the Gamow vectors of H, and iγ _k are the associated resonances; generically, all g _k are nonzero. For large k, γ _k ～const?klog?k+k ² π ² i/4. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way.The decomposition allows for calculating ψ for moderate and large t, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions.The analytic structure of ψ is perhaps surprising: in general (even in simple examples such as square wells), ψ(x,t) turns out to be C ^∞ in t but nowhere analytic on ?⁺. In fact, ψ is t-analytic in a sector in the lower half plane and has the whole of ?⁺ a natural boundary. In the dual space, we analyze the resurgent structure of ψ.     

<Emphasis Type="Italic">nd</Emphasis> scattering at low energies in the two-body potential model

The S-wave phase shift δ(E) for the spin-doublet nd scattering at low energy E is calculated in the framework of the two-body approach. The effective-range-theory formula k cot δ = (1+k²/k ₀ ² )^?1(?1/α+C₂k²+C₄k⁴) is used to obtain approximate analytical results with different potentials. The corresponding coefficients C₂ and C₄ are obtained from our previous calculations of the asymptotic normalization parameter function C _t ² (aκ), where κ is the triton wave number and a is the doublet nd scattering length. The model reasonably describes δ(E), the results being quite sensitive to the choice of the effective nd potential.     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

We study the asymptotic structure of the first K largest eigenvalues λ _k,V and the corresponding eigenfunctions ψ(?;λ _k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ? ^ν , provided the distribution function F(?) of i.i.d. potential ξ(?) satisfies condition ?log(1?F(t))=o(t ³) and some additional regularity conditions as t→∞. For z∈V, denote by λ ⁰(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ _V +ξ(z)δ _z in l ²(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ ⁰(?) in V. We first show that the eigenvalues λ _k,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ _k,V ?B _V )a _V , where the normalizing constants a _V >0 and B _V are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(?;λ _k,V ) is shown to be asymptotically completely localized (as V↑?) at the sites z _k,V ∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.     

Forced snaking: Localized structures in the real Ginzburg-Landau equation with spatially periodic parametric forcing

We study spatial localization in the real subcritical Ginzburg-Landau equation u _t = m ₀ u + Q(x)u + u _xx + d|u|² u ?|u|⁴ u with spatially periodic forcing Q(x). When d>0 and Q ≡ 0 this equation exhibits bistability between the trivial state u = 0 and a homogeneous nontrivial state u = u ₀ with stationary localized structures which accumulate at the Maxwell point m ₀ = ?3d ²/16. When spatial forcing is included its wavelength is imprinted on u ₀ creating conditions favorable to front pinning and hence spatial localization. We use numerical continuation to show that under appropriate conditions such forcing generates a sequence of localized states organized within a snakes-and-ladders structure centered on the Maxwell point, and refer to this phenomenon as forced snaking. We determine the stability properties of these states and show that longer lengthscale forcing leads to stationary trains consisting of a finite number of strongly localized, weakly interacting pulses exhibiting foliated snaking.     

Analytic Structure of Many-Body Coulombic Wave Functions

We investigate the analytic structure of solutions of non-relativistic Schrödinger equations describing Coulombic many-particle systems. We prove the following: Let ψ(x) with \({{\bf x} = (x_{1},\dots, x_{N})\in \mathbb {R}^{3N}}\) denote an N-electron wavefunction of such a system with one nucleus fixed at the origin. Then in a neighbourhood of a coalescence point, for which x ₁ = 0 and the other electron coordinates do not coincide, and differ from 0, ψ can be represented locally as ψ(x) = ψ ⁽¹⁾(x) + |x ₁|ψ ⁽²⁾(x) with ψ ⁽¹⁾, ψ ⁽²⁾ real analytic. A similar representation holds near two-electron coalescence points. The Kustaanheimo-Stiefel transform and analytic hypoellipticity play an essential role in the proof.     

Investigation of the optical constants of arsenic chalcogenide films in the wavelength range 0.5–2.5 μm

Thin films of chalcogenide glasses deposited on quartz glass substrates by thermal evaporation in vacuum have been investigated. The dependences n(λ) and k(λ) for films of different composition have been determined from the transmission spectra. Expressions of the n = A + BL + CL ² + Dλ ² + Eλ ⁴ type (L = (λ ² ? 0.028)^?1 and A, B, C, D, and E are constants) for calculating the refractive indices of As₂Se₃, AsSe₄, AsS₄, and AsS_16.2Se_16.2 films in the wavelength range from 0.5 to 2.5 μm are reported.