Reduction of spin glasses applied to the Migdal-Kadanoff hierarchical lattice

A reduction procedure to obtain ground states of spin glasses on sparse graphs is developed and tested on the hierarchical lattice associated with the Migdal-Kadanoff approximation for low-dimensional lattices. While more generally applicable, these rules here lead to a complete reduction of the lattice. The stiffness exponent governing the scaling of the defect energy E with system size L, (E) ~L ^y, is obtained as y ₃ = 0.25546(3) by reducing the equivalent of lattices up to L = 2¹⁰⁰ in d = 3, and as y ₄ = 0.76382(4) for up to L = 2³⁵ in d = 4. The reduction rules allow the exact determination of the ground state energy, entropy, and also provide an approximation to the overlap distribution. With these methods, some well-know and some new features of diluted hierarchical lattices are calculated.     

Two singular integrals containing a partial wave of the two-body coulomb T-matrix

The operatorsT _C,l E+i0)[–G ₀(E+i0)]^1–i andT _C,l(E+i)G ₀[–iG ₀(E+i)]ⁱ acting on spaces of Hölder continuous, differentiable and analytic functions are investigated. The results of their action are expressed in terms of explicit singular factors and terms and Hölder (differentiable, analytic) functions. The most singular part of these operators is shown to be determined by a simple functional.     

On the Theory of Gravitational Field in Finsler Spaces

Some structural considerations are made on the Finslerian gravitational field: A Finslerian metrical structure such as g_λχ(x, y) = γ_λχ(x) + h_λχ(x, y) is proposed, where γ_λχ denotes the Riemann metric of Einstein's gravitational field, while hλχ the Finsler metric induced by the Riemann metric h_ij(y) of the internal field; The intrinsic behaviour of the internal variable y, which is expressed as ?ⁱ = K(x, y) y^j in the internal field, is grasped by the Finslerian parallelism δyⁱ (=0), which is reflected in the spatial structure of the external gravitational field by the mapping relation δy^χ = e(x) δyⁱ. The whole metrical Finsler connection D for g_λχ(i.e., Dg_λχ = 0) is determined by taking account of the intrinsic behaviour δy^χ.     

Electron in the field of two plane electromagnetic waves and an inhomogeneous electric field

This work examines the motion of an electron in an electromagnetic field consisting of the field of two plane electromagnetic waves of different frequencies ₁, ₂ and an electric fieldE'=(E₀ + E₃cos₃)e₃. The method of separation of variables is used to construct the solution of the Hamilton-Jacobi equation. The motion of the electron in the external electromagnetic field is described by those solutions of the Dirac equation coinciding with the eigenfunctions of the commutating operators.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 89–94, February, 1982.     

Resonant tunnelling in asymmetrical double barrier structures – complex coordinate approach

The complex coordinate approach of theoretical atomic spectroscopy is applied to the study of resonant tunnelling in (Ga,In)As/(Al,In)As-based asymmetrical double-barrier heterostructures within applied electric field. The method yields complex eigenvalues E = E _r – i/2 of the Hamiltonian matrix evaluated in a normalizable basis, where E _r is the resonant energy and = / is the tunnelling lifetime. The dependence of E _res on the field and barrier asymmetry is discussed.     

Disentangling q-Exponentials: A General Approach

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, E_q(A+B) = E_q0(A)E_q1 (B) _i=2 E_qi. By systematically transforming the q-exponentials into exponentials of series and using the conventional Baker–Campbell–Hausdorff formula, we prove that one can make any choice for the bases ^qi, i=0, 1, 2, ..., of the q-exponentials in the infinite product. An explicit calculation of the operators C _i in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for ₀ = ₁ =1, and ₂ = 2, and ₃ = 3. This confirms and reinforces a result of Sridhar and Jagannathan, on the basis of fourth-order calculations.     

Calculations of TM-polarized nonlinear waves guided by thin dielectric films

We found the dispersion relations for TM-polarized nonlinear waves guided by a dielectric film of thicknessd bounded on one side by a nonlinear uniaxial medium characterized by a dielectric tensor _xx = _y = _c , _zz = _c + _c |E _z |², _c >0 (self-focusing medium),E _z being the electric-field component perpendicular to the surfaces. Numerical calculations are given for the power dependence of the propagation wave vector. For sufficiently larged/ (: wavelength) we have found regions with multiple solutions corresponding to the same power flow. This suggests possible applications to optical devices.     

Far-Infrared Laser Stark Spectroscopy of CH3OD

The high resolution Stark spectra of the singly deuterated methanol isotope, CH₃OD, have been studied using the HCN laser with Stark fields up to approximately 60 000 V/cm. Numerous families of absorption lines have been observed in both parallel and perpendicular polarizations, resulting in the following assignments: with the 311 m line – J _K = 18₁ 18₀ E ₂, _t = 1; and with the 337 m line – J _k = 6₄ 5₃ E ₂, _t = 0 and J _K = 14₆ 13₅ A, _t = 1. Zero-field frequencies for the assigned transitions are in agreement with Fourier transform measurements and those calculated from the available molecular constants.     

Bifurcation in a coupled logistic map. Some analytic and numerical results

We study the bifurcation pattern, two- and four-cycle generation, and supertrack functions in the case of the coupled logistic system given byX _n+1=x _n (1–2y _n ) +y _n ,Y _n+1=y _n(1-y _n ), which is of immense importance in various biophysical processes. We deduce analytic formulas for the two -and four-cycle fixed points and cross-check them numerically. The agreement is quite good. Next the bifurcation pattern is explained with the help of analytically derived supertrack functions. To discuss the stability of the system in the various zones defined by the parameter values (, ), the Lyapunov exponents are evaluated, showing a nice transition from the stable to the unstable region. An interesting phenomena occurs at=4, where the logistic itself is chaotic. We then show that near the fixed point an analytic solution can be obtained for the renormalization group equation. In the special case=1,=4 a neat analytic formula can be deduced for then-times iterated values of (x _i ,y _i ).     

Internal Lifschitz singularities of disordered finite-difference Schrödinger operators

The integrated density of states has C-like singularities, ln|k(E)–k(E _c )|=–|E–E _c |^–v/2 _c (E), with _c >0, a milder function at the edges of the spectral gaps which appear when the distribution function of the potentiald has a sufficiently large gap. The behaviour of _c nearE _c is determined by the local continuity properties ofd near the relevant edge: _c (E)=O(1) ifd has an atom and =O(ln|E–E _c |) if is (absolutely) continuous and power bounded.     

Densité des vecteurs propres généralisés d'une classe d'opérateurs compacts non auto-adjoints et applications

We consider a closed densely defined linear operatorT in a Hilbert spaceE, and assume the existence of ₀ (T) such thatK = (T - ₀ I)^-1 is compact and the existence ofp>0 such thats _n (K)=o((n ^–1/p)), whereS _n (K) denotes the sequence of non-zero eigenvalues of the compact hermitian operator . In this work, sufficient conditions (announced in [1]) are introduced to assure that the closed subspace ofE spanned by the generalized eigenvectors ofT coincides withE. These conditions are in particular verified by a family of non-self-adjoint operators arising in reggeon's field theory.     

Transport of light radiation in a spatially finite three-dimensional scattering medium

We propose a new method for solving radiation transport problems, which permits including in analytic form for the case of normal incidence the effect of spatial finiteness of the scattering medium. The formation of the light field accompanying changes in the optical parameters and optical dimensions of the medium is analyzed.this paper, we examine the simplest case of a geometry of a scattering medium in the form of a parallelipiped with optical length _x, height _y, and width _z. The analysis is performed for the case =1, _y = _z with the latter varying in the range 0.1 to . The results obtained show that the light field depends strongly on the optical dimensions of the medium. The limiting values of the optical dimensions (_y = _z), beginning with which the spatial finiteness of the medium can be neglected, are determined.Translated from Izvestiya Vysshikh Uchebnykh Zavednii, Fizika, No. 8, pp. 82–85, August, 1982.     

A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application

Given a positive definite, bounded linear operator A on the Hilbert space ₀≔l ²(E), we consider a reproducing kernel Hilbert space ₊ with a reproducing kernel A(x,y). Here E is any countable set and A(x,y), x,y∊ E, is the representation of A w.r.t. the usual basis of ₀. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space ₋ with a kernel function B(x,y), which is the representation of the inverse of A in a sense, so that ₋⊃ ₀⊃ ₊ becomes a rigged Hilbert space. We investigate the ratios of determinants of some partial matrices of A and B. We also get a variational principle on the limit ratios of these values. We apply this relation to show the Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I+A)⁻¹ on the set E. 2000 Mathematics Subject Classification: Primary: 46E22 Secondary: 60K35     

A mathematical approach to the effective Hamiltonian in perturbed periodic problems

We describe a rigorous mathematical reduction of the spectral study for a class of periodic problems with perturbations which gives a justification of the method of effective Hamiltonians in solid state physics. We study the partial differential operators of the formP=P(hy, y, D _y +A(hy)) onR ⁿ (whenh>0 is small enough), whereP(x, y, ) is elliptic, periodic iny with respect to some lattice , and admits smooth bounded coefficients in (x, y).A(x) is a magnetic potential with bounded derivatives. We show that the spectral study ofP near any fixed energy level can be reduced to the study of a finite system ofh-pseudodifferential operatorsE(x, hD _x , h), acting on some Hilbert space depending on . We then apply it to the study of the Schrödinger operator when the electric potential is periodic, and to some quasiperiodic potentials with vanishing magnetic field.     

Borel summability in the disorder parameter of the averaged Green's function for Gaussian disorder

In this note we prove Borel summability in the disorder parameter of the averaged Green's function <G(E,x,y>) _y of tight binding models $$H_V = - \Delta + V$$ with Gaussian disorder $$d\lambda (V) = (2\pi \gamma )^{ - 1/2} \exp ( - V^2 /2\gamma )dV$$ forγ→0 and fixed large |E|. Using this, we can reconstruct the density of states ?(E)γ from the Borel sums of <G(E,x,x>) _y with ImE↗0 and ImE↘0.     

TM-polarized nonlinear waves guided by asymmetric dielectric layered structures

We found the field structure, exact dispersion relations and power flow ofp-polarized nonlinear guided and surface waves travelling along a three-component layered structure consisting of a film of thicknessd with dielectric constant _b bounded at the negativez-side by a linear medium with dielectric constant _a and at the positivez-side by a nonlinear uniaxial substrate characterized by the diagonal dielectric tensor ₁₁ = ₂₂ = + (|E ₁|² + |E ₂|²), ₃₃ = , <0 (self-defocusing medium),E ₁ andE ₂ being the components of the electric field in thex andy-direction, respectively. It is shown that for sufficiently smalld/ (: wavelength) the nonlinear wave may exist only at power flows exceeding some certain minimum values. For sufficiently larged/ to some values of the power flow there correspond two distinct values of the propagation constant. In this case with increasing of the power flow the number of waveguide modes is decreasing and for higher-order modes the film-waveguide exhibits an optical-power limiter from the above behaviour.     

Positron annihilation studies on the electronic structure of Pd−Ag−H system

Angular Correlation of Annihilation Radiation (ACAR) is shown to be useful to examine the electronic structure of -phase Pd_1–y Ag _y H _x system. Hydrogen absorption by Pd_1–y Ag _y alloys results in the increase of both nearly free andd-localized electron numbers in compliance with the KKR-CPA calculation outcomes. The investigation of -phase systems, PdH _x hydrides as well as Pd_1–y Ag _y H _x materials, failed because of a high concentration of lattice defects.     

Molecular beam electric resonance without A- and B-fields

We report on a novel phenomenon observed during the passage of a pulsed NO beam seeded in He through a combined homogeneous, static electric and RF field, denoted as C-field in a standard molecular beam electric resonance (MBER) experiment. Although we refrain from the state selective A- and B-fields, which are considered crucial for a MBER experiment, the transmitted intensity exhibits as a function of the RF frequency conspicuous dips at resonance frequencies that depend strictly linearly on the static field strength E ₀. Their spectral width is by a factor of 4 smaller than the time of flight broadening. Both, the resonance frequencies and their linear field dependence can be precisely predicted applying a simple expression for the Stark effect to a transition of a single rotational state (J = 3/2) of the electronic ground state . However, this formula is valid only in the high field limit (E ₀ > 1000 kV/m) while the employed field (E ₀ = 1.47 kV/m) was in the extreme low field domain where the large type doubling and hyperfine coupling lead to a purely quadratic Stark effect. We assume that the phenomenon is due to a yet unknown collective rather than to an isolated particle process.Received: 9 May 2003, Published online: 12 August 2003PACS: 33.20.Bx Radio-frequency and microwave spectra - 32.60. + i Zeeman and Stark effects Author for correspondence: A. González Ureña     

Heavy-fermion behavior and spin-glass freezing in Si-stabilized amorphous alloys based on UPt3

U-Pt-Si and U-(Pt, Pd)-Si alloys have been prepared in the amorphous state by splat-cooling. The average nearest-neighbor distance is the same as the U-Pt distance in UPt₃. The low-temperature behavior of these alloys (with composition (U_0.25Pt_0.75)_1–xSi_x, x=0.18, 0.20, 0.25; and (U_0.25(Pt_1–yPd_y)_{0.75 0.8}Si_0.2,y=0.05 and 0.1) has been studied by specific heatC, magnetization and resistivity measurements in the temperature range 1.5 K to 20 K and for magnetic fields up to 6 T. Some samples were investigated down to 0.1 K and the specific heat for one sample was measured in fields up to 22 T. With ₀ =C/T| _T0 between 0.25 and 0.35 J/moleUK², the alloys can be classified as heavy-fermion materials. As opposed to UPt₃, the alloys exhibit spin-glass behavior with a concomitant decrease of ₀. The magnetization exhibits typical spin-glass features (e.g., an irreversibility line). An inverse relationship between ₀ andT _max (temperature ofC/T maximum which is proportional to the spin-glass freezing temperature), irrespective of the Pd content, is observed. The behavior can be qualitatively described by a coexistence of heavy-fermion behavior and spin-glass freezing.