Abstract: It is shown that if a function of x and t satisfies the Korteweg‐de Vries equation and is periodic in x, then its Fourier components satisfy a Hamiltonian system of ordinary differential equations. The associated Poisson bracket is a bilinear antisymmetric operator on functionals. On a suitably restricted space of functionals, this operator satisfies the Jacobi identity. It is shown that any two of the integral invariants discussed in Paper II of this series have a zero Poisson bracket.

Abstract: where L and B are linear and F is a non-linear operator defined in some Banach space ~ . The quantity # is a real parameter. Assume that F(tt, 0)= 0 so that u = 0 is always an equilibrium. In many physical situations the null solution is a stable equilibrium for/~ less than some critical value/to but becomes unstable when tt is increased beyond/~o. One is then interested in knowing conditions under which other stable equilibria bifurcate from the trivial solution at criticality. Such questions arise, for example, in convection problems in fluid dynamics; buckling problems in elasticity; criticality problems in nuclear reactor design, etc. For the purposes of this paper the principle of linearized stability is assumed to hold: that is, fi is a stable equilibrium if all the eigenvalues of the "derivative" operator L p B + F~(t~; 2) (1.2)

Abstract: Relativistic energies are computed for a series of j states corresponding to 1s22sm2pn (0?m?2; 0?n?6) configurations by solution of the hartree-fock-dirac equations The j states calculated are those states whose eigenfunctions within a configuration correspond to a unique and maximum eigenvalue of the operator j2 In addition, average relativistic energies are obtained for all configurations considered The two sets of results are compared, and in certain cases (for the 1s22p, 1s22p5, 1s22s22p, 1s22s22p5 configurations) permit a determination of multiplet splittings (e(2p12/-2p32/)) The results of these splittings are in excellent agreement with available experimental data The present results also tend to confirm the assumption that the relativistic energy contributions for the j averaged ls states are the same for all states arising from the same configuration This makes it possible to evaluate the relativistic energies of all states belonging to the configurations of interest to this paper These in turn serve to re-evaluate more correctly recently obtained correlation energies for the same states

Abstract: New experimental measurements at high resolution on the ν2 band of CS2 at about 390 cm−1 are reported and analyzed, together with previous data on this molecule, to obtain sufficient vibrational and rotational constants for the calculation of the force constants in the general quartic force field. The well‐known Fermi resonance between the vibrational states | υ1, υ2, l2, υ3 〉 and | υ1 − 1, υ2 + 2, l2, υ3 〉 is analyzed taking into account the dependence of the matrix element of the Fermi resonance operator on the rotational quantum number J and on the three vibrational quantum numbers. The anharmonic force constants were determined by least‐squares adjustment to the observed spectroscopic data.

Abstract: We present a short proof of the known theorem that every operator, not of the formλI + compact, whereλ ≢ 0, is a commutator.

Abstract: The localization problem in relativistic quantum mechanics consists in finding (i) the operator representative X k of position and/or its eigenstates (called localized states), (ii) their properties and (iii) the representatives and properties of variables related to position such, as time, proper time and velocity. We devote this review mainly to position and briefly to variables like velocity. Time and proper time will be discussed only when relevant to position.

Abstract: The problem of wave motion in a stochastic medium is treated as an application of stochastic operator theory and as a generalization of Papers I and II (and previous work by the author) to the case of partial differential equations and random fields without monochromaticity assumptions and closure approximations. Connections to the theory of partial coherence are considered. The stochastic Green's function for the two‐point correlation of the solution process can be determined so the correlation can be obtained. Spectral spreading in a ``hot'' medium is easily demonstrable and can be calculated.

Abstract: In the paper we establish estimates for the strong and the weak moduli of continuity of the operator of best approximation in the space of continuous functions.

Abstract: A previously presented [1] equation of motion of a reduced density matrix for a system of spins coupled to a second system of spins, the latter being strongly relaxed on account of their coupling to a thermal reservoir, is rederived without recourse to coarse-grained methods. It is shown that the procedure developed here, in contrast to the earlier approach [1], can be extended to corrections of fourth and higher orders in the strength of the coupling between the two groups of spins. The fourth-order terms are presented and applied to the 2A2B3X spin system, the expression derived being compared, for the first-order limit, with the result of an exact calculation. It is shown that the approach [1] becomes invalid in cases for which there is more than one eigenoperator of the Liouville operator appropriate to the rapidly relaxing spins whose eigenvalues are zero or very small. This is shown to occur when the rapidly relaxing group of spins possesses nuclear permutation symmetry and the relaxation of differe...

Abstract: We analyse the possibility of constructing a model for current amplitudes satisfying i) planar duality, ii) complete factorization, iii) the divergence conditions of CVC and current algebra, and iv) the absence of unphysical singularities. The consistency of the points i), ii) and iii) requires the contribution of spurious states in the factorization of the two-current amplitudes. The point iv) is related to the off-shell extrapolation of the amplitudes and also shows the necessity of spurious states. We therefore suggest that spurious states must have a physical interpretation in terms of hadrons. We then consider in detail a specific model for the vector-current operator. The relevant current amplitudes are shown to have a good behaviour in the limit of zero-current momenta; the single-current amplitudes are dominated by the external line insertions and the two-current amplitudes satisfy the Ward-Takahashi divergence conditions. The two-current amplitudes, however, have a bad behaviour for arbitrary momenta, and also show a violation of CVC due to the presence of spurious-pole contributions.

Abstract: where for each r € HO, oo) A(r) is an operator in a Hilbert space H and & acts on ^-valued functions on fO, co). Restricting the domain of <£ appropriately, we can regard 3? as an operator in f) = £2 (0, °° ; H}. Our purpose is to develop an eigenfunction expansion theory associated with the differential operator <£. If dim H=l, i.e. jfiT=C, then & is an ordinary second-order differential operator and A(r) is simply an operator of multiplication by a function g(r). For real-valued g(r) a rather complete eigenfunction expansion theory has been worked out by Weyl ^10], Stone Q8j, Titchmarsh Q9j, Kodaira £4], Q5] and others. But when H is an infinite-dimensional Hilbert space, it seems that no complete theory, comparable with the one for ordinary differential operators, has been presented. Rofe-Beketov Q7] considers the case where A(f) is a bounded selfadjoint operator-valued function on QO, oo) which is continuous in the uniform operator topology. He shows that there exist a non-negative definite, bounded opera tor -valued, interval function p(/), /CR? and a bounded operator-valued function a)(r, ^) on Q03 oo)5 satisfying

Abstract: This paper discusses the asymptotic behavior of C2 solutions u=u(t, x1,. .., xv) of the inequality (1) ILul co. The operator L is a second order hyperbolic operator with variable coefficients. The main results establish the maximum rate of decay of nonzero solutions of (1). This rate depends on the asymptotic behavior of k1, k2, and the time derivatives of the coefficients of L.