Theory of nonadiabatic transition for general twostate curve crossing problems. I. Nonadiabatic tunneling caseThe squeezed state approach of the semiclassical limit of the timedependent Schrödinger equationWe show rigorously, within the two-state approximation, that in the semicJassicallimit h->O a nonadiabatic transition induced by an analytic time-dependent Hamiltonian is localized to the vicinity of a complex crossing of the two adiabatic potential curves, with transition amplitude independent of the nonadiabatic coupling and given by a simple formula of A. M. Dykhne.

S U M M A R YExamples of Grafenberg-array data showing anomalous P-waves which typically arrive 3-5s after the direct P-wave and which have a slowness 0.7-0.8 sdeg-' smaller than direct P are presented. This additional phase is most frequently observed for events located in the NE portion of the southern Kurile Island subduction zone 73"-80" from Grafenberg, but systematically disappears for events in the SW portion of this zone.Because of the magnitude of the slowness difference, these observations cannot be attributed to a complex source rupture process nor to multipathing through the descending slab. Likewise, they may not be accounted for by near-receiver structure because these phases are not seen for all Kurile events. If present they appear at all stations of the array but they follow direct P too closely to be a multiple from the Moho. Therefore, we conclude they are very likely caused by lower mantle velocity structure.The most likely explanation is the presence of a P velocity jump of about 3 per cent approximately 290 km above the core-mantle boundary, since such a reflector in the lowermost mantle not only gives a good fit of traveltimes and slowness but is also able to model the waveform and the amplitudes of this additional P phase. The distribution of bounce points on this reflector for the Kurile events indicates a lateral extension of this velocity anomaly under northern Siberia of about 150 km by at least 200 km. The best fitting S-wave model has a reflector in the same depth, but the velocity contrast seems to be only about 2 per cent suggesting a different behaviour of the P and S velocity in D".Few events from other regions in this distance range are suitable for a definitive analysis of this kind. From among this group some observations indicate a lower mantle anomaly under the Lomonosow Ridge and under northern Greenland; but since the lower mantle under western Siberia, northern Novaya Zemlya, the Azores Islands region and the USSR-Afghanistan border region does not produce an additional phase in the Grafenberg recordings it is very unlikely that such a velocity anomaly in the lowermost mantle is a global feature.

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The quantum mechanical propagator of a massive particle in a linear gravitational potential derived already in 1927 by Earle H. Kennard [2, 3] contains a phase that scales with the third power of the time T during which the particle experiences the corresponding force. Since in conventional atom interferometers the internal atomic states are all exposed to the same acceleration a, this T 3 -phase cancels out and the interferometer phase scales as T 2 . In contrast, by applying an external magnetic field we prepare two different accelerations a 1 and a 2 for two internal states of the atom, which translate themselves into two different cubic phases and the resulting interferometer phase scales as T 3 . We present the theoretical background for, and summarize our progress towards experimentally realizing such a novel atom interferometer.

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Synthetic seismograms based upon first-order perturbation theory are analysed to test the validity of assumptions which form the basis of current velocity inversion procedures. It is found that the lowest order geometrical optics approximation, namely that measured normal mode eigenfrequencies reflect the average structure underlying the source-receiver great circle path, becomes less valid near nodes in the source radiation pattern and near the surface wave foci at the source and its antipode. These failures are a consequence of singlet interference within an isolated normal mode multiplet. The technique of determing frequency by fitting a single resonance peak to a multiplet yields results which agree well with the first-order theory for slow and fast paths where excitation is dominated by one pair of singlets but on intermediate paths where singlet interference is more of a problem, agreement is not as good. Inversion of small data sets is particularly sensitive to frequency fluctuations near radiation nodes, while larger sets are influenced more by antipodal deviations from geometrical optics. The latter leads to inversions which fail to recover the short wavelength structure of the starting model. Basing inversions directly upon first-order theory shows promise of improving recovery of short wavelengths.

S U M M A R YObservations of underside P-wave reflections from discontinuities deep within the upper mantle which precede the P'P' phase have provided important constraints on the existence and physical properties of these dynamically critical features. In this study, we extend earlier work by examining for the first time, broad-band recordings of those earthquakes for which one would expect to observe P'P' and its associated precursors P'dP'. An exhaustive search of 20 yr of Carnegie data and 11 yr of Grafenberg data uncovered a number of fine P'P' observations, but in only one or perhaps two instances are there evidence of a precursor from near 670 km depth.These null results are consistent with what has been reported in earlier studies and provide us with an opportunity to estimate the variability of P velocity near 670 km or, alternatively, the scale length of discontinuity topography which acts to defocus the precursor. This has been done by using the reflectivity and Gaussian beam methods to compute synthetic seismograms for a suite of 1-D and 2-D earth models which vary slightly from PREM. We find that the null data may be satisfied either by (i) reducing the PREM impedance jump at 670 km from 8.5 to 4-5 per cent; (ii) smoothing this 8.5 per cent jump linearly over a zone 15-30km thick or (iii) imposing deformations in the 670 km discontinuity as small as 10 km in amplitude and 300 km in wavelength. For the one case of an observed precursor, the relative size of the underside reflection and main P'P' phase suggests that, in this instance, the precursor's amplitude is correctly predicted by PREM.

Differential and total cross sections for elastic scattering and resonant–near-resonant charge exchange have been computed for H+–H(1s) and H+–D(1s) collisions in the energy range 0 to ~0.1 eV, using a rigorous quantum mechanical formulation of slow collision theory in which all spurious couplings of the perturbed stationary states (PSS) theory are removed. Orbiting and shape resonances are described and compared in the two systems, and in addition the distinctive features of the HD+ case, such as the appearance of several Feshbach resonances in the region below the D+–H(1s) threshold, are identified and discussed. Most of the effects of the isotopic H(1s)–D(1s) splitting disappear at collision energies near the end of this range.