Some Implications of Endogenous Stabilization Policy

macroeconomics, stabilization policy

Direct Calculation of Energy Eigenvalue Spectra from Time Evolution of Nonstationary States

An arbitrary function in the eigenfunction space of some quantum‐mechanical Hamiltonian may be thought to represent the initial configuration ψ(q,0) of a nonstationary state. The system develops in time according toψ(q,t)=exp(−itH)ψ(q,0)=∑k=0∞(−it)kk!Hkψ(q,0).Defining F(t)≡〈ψ(q,0), ψ(q,t) 〉, and hk≡〈ψ(q,0), HHkψ(q,0) 〉, and taking the Fourier transformG(ω)=∫−∞∞dtexp(−iωt)F(t),we obtainG(ω)=2π∑k=0∞hkk!(ω).In terms of the formal expansion in the energy eingenfunctionsψ(q,0)=∑n=0∞cnϕn(q)+∫0∞dωc(ω)ϕ(ω,q),the Fourier transform representsG(ω)=2π∑n=0∞∣cn∣2δ(ω+ωn)+2π∣c(−ω)∣2,which exhibits, in principle, the entire eigenvalue spectrum. In this paper, a direct method of calculating eigenvalue spectra, based on the foregoing principle, is proposed. Two modifications are required for computational practicability: (i) use of a finite representation for the delta function and truncation of the summation (a); (ii) replacement of the integrals hk by hk(q′) ≡ Hkψ(q,0)]q = q′hk(q′)≡Hkψ(q,0)]q=q′. The modified spectral function is taken to beGN(τ,ω,q′)=2π∑k=0Nhk(q′)k!χτ(k)(ω),with χτ(ω)≡sinτω/πω. The sequence GN(τ,ω,q′) is shown to converge as N→∞ if in the Expansion (b) the coefficients cn and c(ω) decrease with ω as exp(—ω/λ) or faster. Assuming convergence, the spectral function represents a broadened eigenvalue spectrum.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69783/2/JCPSA6-41-11-3412-1.pd

On the Bonding Character of First‐Row Monohydrides

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70973/2/JCPSA6-41-12-4004-2.pd

Two‐point characteristic function for the Kepler–Coulomb problem

Hamilton’s two‐point characteristic function S (q2t2,q1t1) designates the extremum value of the action integral between two space–time points. It is thus a solution of the Hamilton–Jacobi equation in two sets of variables which fulfils the interchange condition S (q1t1,q2t2) =−S (q2t2,q1t1). Such functions can be used in the construction of quantum‐mechanical Green’s functions. For the Kepler–Coulomb problem, rotational invariance implies that the characteristic function depends on three configuration variables, say r1,r2,r12. The existence of an extra constant of the motion, the Runge–Lenz vector, allows a reduction to two independent variables: x≡r1+r2+r12 and y≡r1+r2−r12. A further reduction is made possible by virtue of a scale symmetry connected with Kepler’s third law. The resulting equations are solved by a double Legendre transformation to yield the Kepler–Coulomb characteristic function in implicit functional form. The periodicity of the characteristic function for elliptical orbits can be applied in a novel derivation of Lambert’s theorem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70864/2/JMAPAQ-16-10-2000-1.pd