The chiral determinant and the eta invariant

For { _y },y, a one parameter family of invertible Weyl operators of possibly non-zero index acting on spinors over an even dimensional compact manifoldX, we express the phase of the chiral determinant det _– in terms of the invariant of a Dirac operator acting on spinors over ×X.Supported in part by NSF Grant No. PHY-82-15249Supported in part by NSF Grant PHY 8605978 and the Robert A. Welch Foundation     

Some properties of an “aesthetic” field theory

We continue our study of the Lorentz-invariant field theory based on the equations _jk;l ⁱ =0 and g_ij;k=0. To first order in a perturbation expansion, we find _jk;l ⁱ =0 reduces to the wave equation. In orders higher than the first, we find that _jk;l ⁱ =0 cannot be linearized. We also find that the simple wave-type equation g^ij2g/xⁱx^j=0 is contained in the theory when an appropriate choice is made for the parameters at the origin point.     

On the Lax representation for the nonlinear evolution equations

By the classical differential geometry techniques it is shown that a general partial differential equation of the second order with two independent variables can be represented in the Lax operator form [X ₁ X ₂]=0, whereX _i =/x _i – ⁱ ,i=1,2 and ⁱ are the 3×3 matrices. The problem of the introduction of the spectral parameter in this representation is shortly discussed.Presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.The author is pleased to thank V. K. Mel'nikov for the discussion of this work.     

Existence of solutions for Schrödinger evolution equations

We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationiu/t=(–1/2)u+V(t,x)u,u(0)=u ₀. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u ₀C ¹(,L ²) C ⁰(H ²( ⁿ )) foru ₀H ²( ⁿ ). The conditions are general enough to accommodate moving singularities of type x^–2+(n4) or x^–n/2+(n3).     

Contribution to the thermodynamics of the isomorphous γ ⇆ α-transition in metallic cerium

Using the method of the theory of thermodynamic stability and the form of the lines of the phase equilibrium of the isomorphous -transition in metallic cerium, the authors investigate the behavior of the basic thermodynamic characteristics of this transition in the two-phase region and in a neighborhood of the critical point. Attention is focused primarily on the little-studied adiabatic quantities (T/C_V, (–P/V)_S, (T/V)_S). It is shown that along the phase tie-lines these quantities do not depend on V and S; in the limit of the critical point all adiabatic quantities have nonzero minima, and all isodynamic quantities (T/Cp, (–P/V)_T, (T/V)_P) approach zero according to the same law. The obtained thermodynamic results are compared with existing experimental data and models which can be interpreted thermodynamically. It is concluded that critical phenomena in cerium correspond to critical behavior of the first type.Dnepropetrovsk State University Dedicated to the 300th Anniversary of the Unification of the Ukraine with Russia.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 66–70, May, 1991.     

Differential equations in the spectral parameter

We determine all the potentialsV(x) for the Schrödinger equation (– _x ² +V(x))=k² such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formB(k, _k )ø=(x)ø. For each suchV(x) we determine the algebra of all possible operatorsB and the corresponding functions (x)This research was partially supported by NSF grant DMS 84-03232 and ONR contract NOOO14-84-C-0159     

On the dependence of the coercive field of BaTiO3 single crystals on their thickness

The paper gives the results of measuring the coercive field of single-domain single crystals of BaTiO₃, the thickness of which was lowered by successive etching. The initial decrease in the thicknessd is accompanied by a sudden increase in the coercive fieldE _c (E _c /d –1·5 × × 10⁵ V/cm²); after etching off layers larger than 10^–3 ÷ 2×10^–3cm the coercive field grows much more slowly (E _c /d –7×10³ V/cm²). The high initial growth ofE _c is interpreted by means of Schottky exhaustion layers with non-zero gradient of the electric potential.In conclusion, the authors would like to thank H. Arend, P. Coufová and J. Jarý for providing high-quality single crystals and for much valuable advice during the work and V. Dvoák and K. Pátek for remarks on this paper.     

F-Information, a Unitless Variant of Fisher Information

A new information matrix [F] with elements F _mn = (y _m - a _m )(y _n - a n) ( ln p(y | a)/a _m ) ( ln p(y | a)/a _n ) is analyzed. The PDF p(y | a) is the usual likelihood law. [F] differs from the Fisher information matrix by the presence of the first two factors in the given expectation. These factors make F _mn unitless, in contrast with the Fisher information. This lack of units allows F _mn values from entirely different phenomena to be compared as, for example, Shannon information values can be compared. Each element F _mn defines an error inequality analogous to the Cramer-Rao inequality. In the scalar case F _mn F, for a normal p(y|a) law F = 3, while for an exponential law F = 9. A variational principle F = min (called FMIN) allows an unknown PDF p(x) to be estimated in the presence of weak information. Under certain conditions F obeys a Boltzmann F-theorem F/t 0, indicating that F is a physical entropy. Finally, the trace of [F] may be used as the scalar information quantity in an information-based principle for deriving distribution laws p of physics.     

On the Existence of a Maximizer for the Strichartz Inequality

It is shown that a maximizing function u _*L ² does exist for the Strichartz inequality e ^it _x ² u _L ⁶ _t (L ⁶ _x )Su _L ², with S>0 being the sharp constant.     

Determination of the strain potential constant by means of the p-InSb photoconductivity spectra under uniaxial compression

The influence of uniaxial pressure (0 < P < 2600 kg/cm2) on the intrinsic photoconductivity (PC) spectrum of p-InSb at 93 and 15°K is investigated. At 77°K the carrier concentration and mobility in the specimens were, respectively, (1.4–3.2)·10¹⁴ cm^–3 and 7000 cm²/V. sec. It is established that the maximum in the PC(E_m) spectra under compression is shifted towards higher energies. In the low-compression range E_m/P=5·10^–6eV·cm²/kg, while E_m/P=1·10^–6eV·cm²/kg for P > 1000 kg/cm². It is shown that the shift of the maximum of the intrinsic PC spectra with pressure is due to the growC;th in the forbidden bandwidth (E_g), and the change in parameters characterizing carrier diffusion in the specimen bulk (the diffusion coefficient, lifetime, surface recombination velocity) plays no part. The change in E_m/P with pressure is explained by the influence of valence band splitting. The deformation potential constants of the valence band |b|=(1.7±0.3) eV and |d|=(4.4±0.8)eV are calculated on the basis of a comparison between experimentally obtained data and theoretical results.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 159–162, February, 1976.     

Kinetics and mechanism of copper fracture due to deformation in surface-active paths

The general kinetic characteristics of copper fracture in the presence of surface-active bismuth-lead baths during creep and elongation under tension are explained. It is shown that the subcritical stage of crack development controls the process, whereupon the effects of stresses , temperature, strain rate , surface energy at the copper-bath interface _SL, and surface energy at the grain boundaries _b on the rate of crack development l/ are analyzed. The basic conclusions are that: a)l/=(–) ( and being constants here); b) the crack development activating energy ) the reduction of energy _b, achieved by intergranular internal adsorption of 0.5% antimony, lowers the value of about 50 times; d) a 30% increase in surface energy _SL reduces the cracking rate 30 times, according to the relation (where A=6 · 10^–15 cm²); and e) .Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 7–15, May, 1976.     

Hyperbolic initial-boundary-value problem for magnetic flux compression by plane liners

Based on the (relativistic) Maxwell equations with displacement current E/t, the initial-boundary-value problem for the compression of an initially homogeneous magnetic fieldB={0,B(x,t),0} between a fixed liner atx=0 and a detonation-driven liner atx=s(t) is solved analytically. By homogenizing the boundary conditions at the moving boundary, the transient electromagnetic fields are shown to be a superposition of quasistatic elliptic (E/t=0) and hyperbolic (E/t0) wave solutions. The wave equation is solved by a Fourier expansion in time-dependent eigenfunctionsf _n =f _n [nx/s(t)] for the variable region 0xs(t), where the Fourier amplitudes _n (t) are determined by coupled differential equations of second order. It is concluded that the conventional elliptic flux compression theories (E/t=0) hold approximately for nonrelativistic liner speeds , whereas the hyperbolic theory (E/t0) is valid for arbitrary liner speeds .     

The hard-sphere fluid: New exact results with applications

A theorem for convolution integrals is proved and then applied to extend the second zero-separation theorem to the bridge functionb(r) and direct-correlation tail functionsd(r). This theorem allows us to exactly relateb(r)/r andd(r)/ratr=0 for the hard-sphere fluid to the contact value of the radial distribution functiong(r) atr= ⁺. From this we obtain immediately the exact values of b(r)/r and d(r)/r atr=0 through second order in number density . Using our results to compare the exact and Percus-Yevick (PY) bridge function, we find that they differ significantly. After obtaining the bridge function and tail function and their derivatives atr=0 andr= through, we suggest new approximations forb(0) andd(0) as well as an analytical integral-equation theory to improve the PY approximation in the pure hard-sphere fluid. The major deficiency of that approximation has been its poor assessment of the cavity function inside the hard-core region. Our theory remedies this defect in a way that yields ay(r) that is self-consistent with respct to the virial and compressibility relations and also the two zero-separation relations involvingy(r) and its spatial derivative atr=0.     

Quantization of the Maxwell field in linear conformal invariant gauges

It is shown that the generalized Lorentz gauges provide all linear conformal invariant gauges, i.e. gauges such that A =0.     

The equation curl 73-173-173-1(x)=0 in quantum field theory

In one time and arbitrarily many space dimensions we obtain necessary and sufficient conditions for the existence of a local operator solution of the equation _v =W _v . Here the given local fieldsW _v satisfy _µ W _v - _v W _µ=0 and the spectrum of the two point function (,W _µ(x)W _v (y)) is assumed to have a mass gap.Research supported in part by Stiftung Volkswagenwerk and the Alfred P. Sloan Foundation.On leave from II. Institut für Theoretische Physik der Universität Hamburg, Germany.     

TypeN vacuum space-times as special functions in ℂ2

The members of one explicit class of functions in ² are identified with the geodetic shear-free null congruences in Minkowski's space-time. Members of a second explicit class are identified with the type-N vacuum space-times with twist-free rays. These two classes are special subclasses from a larger class of functions associated with the type-N space-times. This larger class is characterized in the following way: If and are holomorphic variables in ², thenu (, , ), a function holomorphic in, belongs to the class provided the function u/ u satisfies the tangential Cauchy-Riemann equation for an antiholomorphic function on the 3-surface whereu (, , ) has real values.This work was supported in part by NSF grant No. MPS74-14191-A01.     

Correlations of Spin States for Icosahedral Double Group

The irreducible bases of the icosahedral double groups I and I_h are explicitly presented in their respective group spaces. Applying these bases to the spin states |j, , we obtain a simple formula for combining the spin states into the symmetry-adapted bases which belong to a given row of given irreducible representations of I and I_h.     

Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems

We consider a ferromagnetic Ising spin system isomorphic to a lattice gas with attractive interactions. Using the Fortuin, Kasteleyn and Ginibre (FKG) inequalities we derive bounds on the decay of correlations between two widely separated sets of particles in terms of the decay of the pair correlation. This leads to bounds on the derivatives of various orders of the free energy with respect to the magnetic fieldh, and reciprocal temperature . In particular, if the pair correlation has an upper bound (uniform in the size of the system) which decays exponentially with distance in some neighborhood of (,h) then the thermodynamic free energy density (,h) andall the correlation functions are infinitely differentiable at (,h). We then show that when only pair interactions are present it is sufficient to obtain such a bound only ath=0 (and only in the infinite volume limit) for systems with suitable boundary conditions. This is the case in the two dimensional square lattice with nearest neighbor interactions for 0<₀, where ₀ ^–1 is the Onsager temperature at which (,h=0) has a singularity. For >₀, (,h)/h is discontinuous ath=0, i.e. ₀=_c, where _c ^–1 is the temperature below which there is spontaneous magnetization.Research supported by AFOSR Contract # F 44620-71-C-0013.     

Method of a singular point and determination of fields of anisotropy of polycrystalline hexagonal magnets

A numerical computation is performed on the magnetization curves and their derivatives for magnets of hexagonal syngony for isotropic polycrystalline specimens as well as materials with sheet texture in the basis plane. It is shown that investigation of the second derivatives ²M/H² for a degree of texture f_T > 0.4 permits obtaining information about the magnitude of the anisotropy field independently of its type. The singularities of M(H) and ²M/H² are examined for magnetization processes of the first kind. In this case the singular point method permits measuring the magnitude of the critical fields, the critical magnetizations, and the anisotropy field on a textured specimen with f_T>0.7, which yields information about the magnitudes of the high order anisotropy constants in the long run.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 39–42, January, 1989.     

Weakly Bound States in Light Hypernuclei

From the viewpoint of critical stability, we discuss the three- and four-body structure of ⁶He, ⁷He, ⁴He, and ⁴H. With the + + N three-body model, ⁶He is found to have a three-layer structure of the matter distribution: core, a skin and neutron halo. Also the level structure of ⁷He with the three-body model of ⁵He + n + n is predicted. This stimulates a new study of neutron-rich and proton-rich hypernuclei. By performing a four-body calculation with both NNN and NNN channels and with both NN and NNN channels, we show that the N-N and -N couplings are very important in critical stability of few-body hypernuclear systems.