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A class of representations of the canonical commutation relations is studied, each of which is characterized by an expectation functional that is the exponential of a Euclidean‐invariant quadratic form of the test functions. The underlying field operators are realized as the direct product of two Fock representations and the consequences of this realization are analyzed. Compatible Hamiltonians are constructed and an extensive study of the most general quadratic Hamiltonians is presented. In order to include thermodynamic examples, the analysis includes indefinite Hamiltonian spectra as well as the usual definite spectra. Finally, conditions are given for a theory to be local in the sense that all time derivatives of the field operator commute with one another at equal times but unequal spatial arguments.

Recent developments have shown that the pure Lagrange‐Hamiltonian formalisms can be extended to problems (chiefly, classical systems involving dissipative forces) previously regarded as outside such theories. The Hamilton‐Jacobi equations corresponding to certain such systems are given here, and the structure, separability, and solution of these equations are studied. Examples treated here include a particle moving freely and under a constant force in a viscous medium, the damped‐harmonic oscillator in one and three dimensions, and a particle moving in one dimension with quadratic friction in an arbitrary potential. In all cases, the Hamilton‐Jacobi equation separates into time and space components, and the complete solution is obtained.

A theorem for the boundedness and asymptotic stability of a point reactor with an arbitrary feedback is stated and proved. The criteria obtained are shown to be essentially the same as those given by Akcasu and Dolfes. The theorem is applied to a reactor with an arbitrary linear feedback and to a xenon‐controlled reactor with a flux reactivity coefficient whose feedback mechanism involves quadratic non‐linearity. It is also compared to a criterion obtained by Corduneanu in the case when delayed neutrons are ignored and the feedback mechanism is linear.

The quantum‐mechanical problems of N 1‐dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center‐of‐mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl], with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi st...

An exact method for determining the critical percolation probability, pc, for a number of two‐dimensional site and bond problems is described. For the site problem on the plane triangular lattice pc = ½. For the bond problem on the triangular, simple quadratic, and honeycomb lattices, pc=2 sin (118π),12,1−2 sin (118π), respectively. A matching theorem for the mean number of finite clusters on certain two‐dimensional lattices, somewhat analogous to the duality transformation for the partition function of the Ising model, is described.

We diagonalize a many‐fermion Hamiltonian consisting of terms quadratic as well as quartic in the field operators. A dual spectrum of eigenstates is an interesting result. We also derive a formula for obtaining the free energy at finite temperature.

A class of fourth‐order spin correlations whose sum is closely related to the specific heat is calculated exactly by Pfaffian perturbation theory. In the particular case when the four spins lie on a lattice axis, it is shown that the reduced fourth‐order correlation function has the asymptotic form 〈σ0, 0σ1, 1σk, kσk+1, k+1〉−〈σ0, 0σ1, 1〉 〈σk, kσk+1, k+1〉≏A′(e−2βk/k2), where β−1 is the (positive) mean range of order. The decay of correlations with spin separation k is symmetric about the critical point β = 0. The amplitude A′ is 1/π2 at the critical point and is 1/(2π) at all other temperatures. Correlations on ferro‐ and antiferromagnetic triangular and quadratic lattices are discussed in some detail.

Asymptotic expansions are derived for the density operator ρN(x′, x)= ∑ n=1Nψn(x′)ψn(x) for small and large values of the relative distance x′ − x with a WKB expansion for bound‐state wavefunctions in a plane, one‐dimensional potential. In addition to some previously obtained corrections, the particle‐density expansion includes a number of steady and oscillating correction terms to the zero‐order Thomas‐Fermi density which are related to the more recently derived corrections of Payne and of Kohn and Sham. A convenient and accurate approximation to the density operator for all arguments is also obtained from the first two terms of the expansion for large relative distances. The expansions become inadequate in the vicinity of the classical turning point of the highest energy state, as illustrated by numerical comparisons with the exact‐density operator for the quadratic potential. A novel method is described for summing a finite series of oscillating terms such as occur in the density operator.

New criteria characterizing Slater determinants and quasiparticle vacuum states are obtained. These criteria are expressed as quadratic homogeneous equations in the coefficients of the development of the trial wavefunction in a basis of Slater determinants. A redefinition of quasiparticle vacuum states permits the introduction of quasiparticle transformations which are more general than the generalized Bogoliubov transformations.

The S‐matrix approach of Hurst is applied to rederive the partition function for the dimer problem. By using every two lattice sites as a unit cell and a set of six creation and annihilation operators at each cell, the expression for the partition is reduced to a form equivalent to the vacuum expectation value of an S matrix with a quadratic interaction Lagrangian.

The derivation of series expansions appropriate for low temperatures or high applied magnetic fields for the two‐dimensional Ising model of a ferromagnet and antiferromagnet is studied as a field grouping. New results are given for the high field polynomials for the triangular lattice to order 10, the simple quadratic lattice to order 15, and the honeycomb lattice to order 21.

A study is made of a particular system of coupled oscillators to determine how the amount of energy exchange and the recurrence time depend on the parameters of the system, and the accuracy with which these are predicted by the perturbation theory developed in part I of this series. The system consists of N + 1 particles, connected by springs which have a quadratic force term, the strength of which, λi, can vary between different particles (``defects''). The equations of motion for the perfect chain (λi = λ) were solved on a computer for the cases N = 4, λ = ¼, ½, ¾; and N = 9, λ = ½; and supplemented by the previous calculations of Fermi, Pasta, and Ulam, in which N = 32, λ = ¼; 1. The ``energy'' in the linear modes,Ek(t)=12(ȧk2+ωk2ak2) are determined from these computations and compared with the perturbation theory in the first paper of this series. As was found by Fermi, Pasta, and Ulam, when only E1 is initially excited, then only the first few modes acquire an appreciable amount of energy, after which nearly all the energy returns to the first mode in a time τλ. The second‐order perturbation theory is found to give an accurate estimate (within 15%) of both τλ and the amount of energy exchange for all the above cases, except N = 32, λ = 1, which requires higher‐order analysis. It is shown that the nonergodic behavior of this system does not result simply from the incommensurability of the uncoupled frequencies {ωk}, but also from the particular form of mode interaction and the initial conditions used in all calculations, both of which affect the coupled frequency spectrum {Ωk}. To alter the mode interaction a preliminary study is made of ``imperfect'' chains (variable λi), which directly couples low and high modes. It is found that either a large, or else systematic, variation of λi is necessary to appreciably alter the nonergodic behavior of this system. A theoretical examination of the dependence of the coupled frequencies {Ωk} on the initial conditions shows that the ergodic behavior will, in general, be strongly dependent upon the initial conditions.

The partition function for the square lattice completely filled with dimers is analyzed for a finite n × m rectangular lattice with edges and for the corresponding lattice with periodic boundary conditions. The total free energy is calculated asymptotically for fixed ξ = n/m up to terms o(1/n2−δ) for any δ > 0. The bulk terms proportional to nm, the surface terms proportional to (n + m) which vanish with periodic boundary conditions, and the constant terms which reveal a parity and shape dependence are expressed explicitly using dilogarithms and elliptic theta functions.

A compact method is presented for the evaluation of certain statistical averages involving a quadratic Hamiltonian generator.

Involutional matrices M(a, b, c) with three arbitrary parameters are introduced, based on a matrix representation M(R) of the linear homogeneous transformation R ∈ GL(2). Symmetry properties, eigenvalues, and recursion formulas for the representation M(R) are obtained and specialized to the involutional matrices M(a, b, c). A set of special involutional matrices A(ξ), B(ξ), C(ξ), and E(ξ) with one arbitrary parameter ξ are introduced as special cases of M(a, b, c). Their relations are discussed.

A conceptual theory of thin, elastic shells is developed. A general form for the strain energy is postulated in terms of parameters which characterize, in the sense of Euclid, the geometry of the shell. The strain energy permits a formal definition of generalized stresses and then a statement of equilibrium. The physical interpretation of the stresses is obtained from the boundary conditions. The energy coefficients, which are not derivable within the scope of this theory, are related to material constants by using classical results derived from the three‐dimensional theory of elasticity. The result is a complete, self‐contained theory of shells which can be used for prediction. The methods used depend on a combination of Cartesian tensors and exterior differential forms.

The derivation of series expansions appropriate for low temperatures or high applied magnetic fields for the two‐dimensional Ising model of a ferromagnet and antiferromagnet is studied as a temperature grouping. New results are given for the ferromagnetic polynomials for the triangular lattice to order 16, for the ferromagnetic and antiferromagnetic polynomials for the simple quadratic lattice to order 11, and for the honeycomb lattice to order 16.

The partition function of the one‐dimensional Heisenberg model is considered. Hamiltonian of the system H=−12 ∑ [J⊥(σlxσl+1x+σlyσl+1y)+J∥σlzσl+1z]−mH ∑ σlz is expressed in terms of Fermi operators. The term which contains J∥, the quartic term and a part of quadratic term in Fermi operators, have been regarded as perturbation, keeping the symmetry with respect to the magnetic field. Linked‐cluster expansion in an appropriate form for this case has been developed and the partition function has been obtained up to the third order in J∥. Numerical values of energy, specific heat, and susceptibility up to second order in J∥ are shown. The ground‐state energy is EN|J⊥|=−2π−2π2(J∥|J⊥|)−16π3(16−π2144)(J∥J⊥)2+O[(J∥J⊥)3].E/N |J∥| for the antiferromagnetic case J∥ = −|J∥| = J is −0.8899. Agreement with the exact value, −0.8863, is quite satisfactory.

A modified admittance is introduced to give Kubo's admittance at nonzero frequencies and to give the isothermal static admittance at zero frequency within the scope of the Kubo linear‐response theory. The method is demonstrated by exact calculations of the frequency‐dependent perpendicular susceptibility at zero field and its modified susceptibility of the regular Ising model. The results appear as linear combinations of the equilibrium spin‐spin correlation functions of the lattice. The results are valid for all dimensions and all frequencies and temperatures. A (q + 1) × (q + 1) matrix a(q) describes the linear combinations explicitly, where q is the coordination number of the lattice. The properties of this matrix are extensively discussed as a special case of a matrix A(q)(ξ), which satisfies a simple quadratic equation of the form [A(q)(ξ)]2 = (1 + ξ)q for arbitrary values of ξ. Fisher's algebraic transformation of the spin‐spin correlation functions for the regular Ising lattice is derived from the ...

In a recent paper [J. Math. Phys. 8, 423 (1967)], Dyson and Lenard gave a lower bound of the N‐particle Hamiltonian, Coulomb interaction, which was quadratic in N. An essential step in their proof is the fact that the one‐particle Hamiltonian with Yukawa potential −r−1e−μr is nonnegative if μ is not less than some value. The aim of this paper is to improve this positivity result and thereby improve the lower bound of Dyson and Lenard.