A "quasiclassical" approximation to the quantum spectrum of the Schrödinger equation is obtained from the trace of a quasiclassical evolution operator for the "hydrodynamical" version of the theory, in which the dynamical evolution takes place in the extended phase space [q(t), p(t), M (t)] = [qi, ∂iS, ∂i∂jS]. The quasiclassical evolution operator is multiplicative along the classical flow, the corresponding quasiclassical zeta function is entire for nice hyperbolic flows, and its eigenvalue spectrum contains the spectrum of the semiclassical zeta function. The advantage of the quasiclassical zeta function is that it has a larger analyticity domain than the original semiclassical zeta function; the disadvantage is that it contains eigenvalues extraneous to the quantum problem. Numerical investigations indicate that the presence of these extraneous eigenvalues renders the original Gutzwiller-Voros semiclassical zeta function preferable in practice to the quasiclassical zeta function presented here. The cumulant expansion of the exact quantum mechanical scattering kernel and the cycle expansion of the corresponding semiclassical zeta function part ways at a threshold given by the topological entropy; beyond this threshold quantum mechanics cannot resolve fine details of the classical chaotic dynamics.

An explicit expression for the generating functional of two-flavor low-energy QCD with external sources in the presence of nonvanishing nucleon densities was derived recently (J. A. Oller, Phys. Rev. C 65 (2002) 025204). Within this approach we derive power counting rules for the calculation of in-medium pion properties. We develop the so-called standard rules for residual nucleon energies of the order of the pion mass and a modified scheme (nonstandard counting) for vanishing residual nucleon energies. We also establish the different scales for the range of applicability of this perturbative expansion, which are √ 6π f π 0.7 GeV for standard and 6π 2 f 2 π /2m N 0.27 GeV for nonstandard counting, respectively. We have performed a systematic analysis of n-point in-medium Green functions up to and including next-to-leading order when the standard rules apply. These include the in-medium contributions to quark condensates, pion propagators, pion masses, and couplings of the axial-vector, vector, and pseudoscalar currents to pions. In particular, we find a mass shift for negatively charged pions in heavy nuclei, M π − = (18 ± 5) MeV, that agrees with recent determinations from deeply bound pionic 207 Pb. We have also established the absence of in-medium renormalization in the π 0 → γ γ decay amplitude up to the same order. The study of ππ scattering requires the use of the nonstandard counting and the calculation is done at leading order. Even at that order we establish new contributions not considered so far. We also point toward further possible improvements of this scheme and touch upon its relation to more conventional many-body approaches. C 2002 Elsevier Science (USA)

We present a simple formalism for the evaluation of the Casimir energy for two spheres and a sphere and a plane, in case of a scalar fluctuating field, valid at any separations. We compare the exact results with various approximation schemes and establish when such schemes become useful. The formalism can be easily extended to any number of spheres and/or planes in three or arbitrary dimensions, with a variety of boundary conditions or nonoverlapping potentials/nonideal reflectors.

An extension of the Gutzwiller trace formula is given that includes diffraction effects due to hard wall scatterers or other singularities. The new trace formula involves periodic orbits which have arcs on the surface of singularity and which correspond to creping waves. A new family of resonances in the two disk scattering system can be well described which is completely missing if only the traditional periodic orbits are used.Comment: 4 pages, latex/REVTEX, 2 PS figure

The scattering problems of a scalar point particle from an assembly of 1 < n < ∞ nonoverlapping and disconnected hard disks, fixed in the two-dimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Their simplicity allows for a detailed study of the quantum mechanics, semiclassics and classics of the scattering. Here, we investigate the connection between the spectral properties of the quantum-mechanical scattering matrix and its semiclassical equivalent based on the semiclassical zeta-function of Gutzwiller and Voros. We construct the scattering matrix and its determinant for any non-overlapping n-disk system (with n < ∞) and rewrite the determinant in such a way that it separates into the product over n determinants of 1-disk scattering matrices -representing the incoherent part of the scattering from the n-disk system -and the ratio of two mutually complex conjugate determinants of the genuine multiscattering matrix M which is of Korringa-Kohn-Rostoker-type and which represents the coherent multidisk aspect of the n-disk scattering. Our quantum-mechanical calculation is well-defined at every step, as the on-shell T-matrix and the multiscattering kernel M−1 are shown to be trace-class. The multiscattering determinant can be organized in terms of the cumulant expansion which is the defining prescription for the determinant over an infinite, but trace-class matrix. The quantum cumulants are then expanded by traces which, in turn, split into quantum itineraries or cycles. These can be organized by a simple symbolic dynamics. The semiclassical reduction of the coherent multiscattering part takes place on the level of the quantum cycles. We show that the semiclassical analog of the mth quantum cumulant is the mth curvature term of the semiclassical zeta function. In this way quantum mechanics naturally imposes the curvature regularization structured by the topological (not the geometrical) length of the pertinent periodic orbits onto the semiclassical zeta function. However, since the cumulant limit m → ∞ and the semiclassical limit,h → 0 or (wave number) k → ∞, do not commute in general, the semiclassical analog of the quantum multiscattering determinant is a curvature expanded semiclassical zeta function which is truncated in the curvature order. We relate the order of this truncation to the topological entropy of the corresponding classical system. We show this explicitly for the 3-disk scattering system and discuss the consequences of this truncation for the semiclassical predictions of the scattering resonances. We show that, under the above mentioned truncations in the curvature order, unitarity in n-disk scattering problems is preserved even at the semiclassical level. Finally, with the help of cluster phase shifts, it is shown A Traces and determinants of infinite dimensional matrices 76 A.1 Trace class and Hilbert-Schmidt class 76 A.2 Determinants det(1+A) of trace-class operators A 77 A.3 Von Koch matrices 80 A.4 Regularization 81 B Exact quantization of the n-d...