Inverse Problem for Harmonic Oscillator Perturbed by Potential,Characterization

Consider the perturbed harmonic oscillator Ty=-y+x²y+q(x)y in L²(), where the real potential q belongs to the Hilbert space H={q, xq L²()}. The spectrum of T is an increasing sequence of simple eigenvalues _n(q)=1+2n+_n, n 0, such that _n 0 as n. Let _n(x,q) be the corresponding eigenfunctions. Define the norming constants _n(q)=lim_xlog |_n (x,q)/_n (-x,q)|. We show that for some real Hilbert space and some subspace Furthermore, the mapping :q(q)=({_n(q)}₀, {_n(q)}₀) is a real analytic isomorphism between H and is the set of all strictly increasing sequences s={s_n}₀ such that The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -ypy, p L²(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces We obtain their basic properties, using their representation as spaces of analytic functions in the disk.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

On practical calculations with reflection curves for perfect crystals

The application of the Prins function for practical calculations is rather laborious. In the present article it is shown by analysis of some of the properties of this function that in most experiments with perfect crystals one may use functions of a much simpler form. Such a function is proposed, its use is shown on some examples, a method of simplified calculation of the fundamental constants of the Prins function is given, and finally some numerical examples are added as illustration.

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. , . , , , , .

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These curves will be hereafter called Prins functions.

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The author is greatly indebted to M. Malkovská for the experimental data in Chapter VI. C and to V. Horáková for numerical computations of the integrals in Appendix 2.     

On the Effect of Increasing the Radiative-Transition Rate Near a Nanosize Point

A short review of theoretical and experimental studies concerning the photoexcited florescence and Raman scattering of light for a substance in a space containing small material bodies is presented. Calculations of the radiativetransition probability for atoms (molecules) in the vicinity of bodies with a size much smaller than the light wavelength are performed. The probabilities of the singlephoton and doublephoton transitions are shown to increase by factors of 9 and 81 in the vicinity of a nanosize sphere with dielectric constant ||\ 1. The probability of a radiative transition in the vicinity of the vertex of a conic needle bearing up against a plane (both with || 1) increases by factors of (/R _in)² and (/R _in)⁴ for singlephoton and doublephoton transitions, respectively (R _in is the curvature radius for the needle vertex). This theoretical result is suggested as an explanation of the effect of increasing the radiation process intensity in the experiments carried out in the studies cited below.     

ИЗЛУЧЕНИЕ ЭЛЕКТРИЧЕ СКОГО ДИПОЛЯ НАД ГРАН ИЦЕЙ ДВУХ СРЕД

, , . , - - . - . -, - , . , - ; , , .     

Baryon decays and gamma-spectroscopy of hypernuclei

The applicability of (K ^–,^–) - and (K ^–,^– N)-reactions on^13,14C and^14,15N nuclei to the study of -transitions in primary and daughter -hypernuclei is discussed. The intensity of -deexcitation of ¹³C state |S ¹²C(15·11 MeV; 1⁺1): 1/2⁺ has been shown to be comparable with the intensity of baryon decay. Isospin selection rules are used to distinguish excitation energy ranges of primary hypernuclei, where the identification of the secondary -lines is probable.Presented at the symposium Mesons and Light Nuclei, Bechyn, Czechoslovakia, May 27–June 1, 1985.     

КРУЧЕНИЕ ПРИЗМАТИЧЕ СКИХ СТЕРЖНЕЙ ШВЕЛЛЕ РНОГО ПОПЕРЕЧНОГО СЕЧЕНИЯ

- [3], [4], U(x, y) . - U(x, y) -, . B § 10 - . , - , - D, . - .     

Some applications of the discrete fourier transform to problems of crystal lattice deformation II

The method elaborated in [1] is applied to the solution of some problems for a plane lattice and the linear chain. The method can be used to investigate deformations around crystal lattice defects.

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, [1] . .

     

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.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Mean field theory for Coulomb systems

We study a classical charge symmetric system with an external charge distributionq in three dimensions in the limit that the plasma parameter zero. We prove that ifq is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field(x) where is given by-+ 2z sinh() =q This is the mean field equation of Debye and Hückel. The proof uses the sine-Gordon transformation, the Mayer expansion, and a correlation inequality.Work partially supported by NSF Grant MCS 82-02115.     

Новый метод измерени я магнитокалорическ ого эффекта у ферромагни тных полупроводнико в

, .     

A matrix method for estimating the Liapunov exponent of one-dimensional systems

Let : [0, 1][0, 1] be a piecewise monotonie expanding map. Then admits an absolutely continuous invariant measure. A result of Kosyakin and Sandler shows that can be approximated by a sequence of absolutely continuous measures _n invariant under piecewise linear Markov maps _itn. Each _itn is constructed on the inverse images of the turning points of . The easily computable measures _n are used to estimate the Liapunov exponent of . The idea of using Markov maps for estimating the Liapunov exponent is applied to both expanding and nonexpanding maps.     

Quantum Pushdown Automata

Quantum automata are mathematical models for quantum computing. We analyze the existing quantum pushdown automata, propose a q quantum pushdown automata (qQPDA), and partially clarify their connections. We emphasize some advantages of our qQPDA over others. We demonstrate the equivalence between qQPDA and another QPDA. We indicate that qQPDA are at least as powerful as the QPDA of Moore and Crutchfield with accepting words by empty stack. We introduce the quantum languages accepted by qQPDA and prove that every -q quantum context-free language is also an -q quantum context-free language for any (0, 1) and (0, 1).     

Double quantum transitions in paramagnetic resonance

The theory of double quantum transitions of the M=±2 type, with regard to inhomogeneously broadened spin systems is studied in this paper with the approximation ²T₂T₃ 1. We suppose that the inhomogeneous broadening is formed by an inhomogeneous crystal field. The obtained results describe the magnitude of absorption as a function of the h.f. power and also describe the shape of the absorption curve. It is demonstrated that in inhomogeneously broadened spin systems the absorption curve of double quantum transitions has the form of the difference of two different Lorentz's curves and that at the saturation ( ²T₂T₁ 1) the absorption increases with the cube of the h.f. field intensity. The shape of the curves is expressed by means of phenomenological relaxation constants of the system.

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M=±2 ²T₂T₃ 1. , - . . , ( ²T₂T₁ 1) . .

     

On the Long Time Behavior of Infinitely Extended Systems of Particles Interacting via Kac Potentials

We analyze the long time behavior of an infinitely extended system of particles in one dimension, evolving according to the Newton laws and interacting via a non-negative superstable Kac potential (x)=(x), (0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order . We next study the motion of a fast particle interacting with a background of slow particles, and we prove that its velocity remains almost unchanged for a very long time (at least proportional to ^–1 times the velocity itself). Finally we shortly discuss the so called Vlasov limit, when time and space are scaled by a factor .     

Analyticity properties of the surface free energy of the Ising model

For an Ising ferromagnet with nearest-neighbour interactions of strengthK and surface magnetic fieldh, the surface free energy in the presence of a positively (or negatively) magnetized zero-field bulk phase is shown to be analytic inh for Reh<K–/, where =2.96 ... and is the inverse temperature. This puts the lower boundK–/ on the values ofh at which wetting and layering transitions can take place.     

МОДЕЛИРОВАНИЕ ЭЛЕКТ РОННО-ОПТИЧЕСКИХ СИС ТЕМ С ОСЕВОЙ СИММЕТРИЕЙ П РИ ПОМОЩИ СЕТИ СОПРОТИВЛЕНИЙ

The paper explains the theory of modelling electrostatic fields by a resistance network. The conditions, which the resistance network must satisfy, are derived and the question of modelling electrodes of different shapes is solved. The finished network and the results obtained on it when modelling a jet for a linear h-f accelerator of electrons are described. Particular attention is paid to the influence of a space charge, the modelling of which is an advantage of this method.

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1- , 1964., , .

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. .     

Duality for a Lorentz quantum group

We give the algebra _q ^/* dual to the matrix Lorentz quantum group _q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form _q ^/* U _q (sl(2, )) U _q (sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.     

Power-law falloff in the kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas

We give a rigorous proof of power-law falloff in the Kosterlitz-Thouless phase of a two-dimensional Coulomb gas in the sense that there exists a critical inverse temperaturegb and a constant >0 such that for all> and all external charges R we have , whereG (x) is the two-point external charges correlation function,=dist(, Z), and for 0$$ " align="middle" border="0"> . In the case of a hard-core or standard Coulomb gas with activityz, we may choose=(z) such that(z)24 asz0.