Energy Growth in Schrödinger's Equation with Markovian Forcing

Schrödinger's equation is considered on a one-dimensional torus with time dependent potential v(θ,t)=λV(θ)X(t), where V(θ) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant λ is sufficiently small, the average kinetic energy grows as the square-root of time. More generally, the H^s norm of the wave function is shown to behave as t^(s/4A)

Semiclassical low energy scattering for one-dimensional Schr\"odinger operators with exponentially decaying potentials

We consider semiclassical Schr\"odinger operators on the real line of the form $$H(\hbar)=-\hbar^2 \frac{d^2}{dx^2}+V(\cdot;\hbar)$$ with $\hbar>0$ small. The potential $V$ is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions $f_\pm(\cdot,E;\hbar)$ with error terms that are uniformly controlled for small $E$ and $\hbar$, and construct the scattering matrix as well as the semiclassical spectral measure associated to $H(\hbar)$. This is crucial in order to obtain decay bounds for the corresponding wave and Schr\"odinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta $\ell$ where the role of the small parameter $\hbar$ is played by $\ell^{-1}$. It follows from the results in this paper and \cite{DSS2}, that the decay bounds obtained in \cite{DSS1}, \cite{DS} for individual angular momenta $\ell$ can be summed to yield the sharp $t^{-3}$ decay for data without symmetry assumptions.Comment: 44 pages, minor modifications in order to match the published version, will appear in Annales Henri Poincar

Dispersive estimates for Schrodinger operators in dimensions one and three

We prove L^1 --> L^\infty estimates for linear Schroedinger equations in dimensions one and three. The potentials are only required to satisfy some mild decay assumptions. No regularity on the potentials is assumed.Comment: 20 pages. Corrected typos and improved explanatory remarks at the en

Electronic spectra of polyatomic molecules with resolved individual rotational transitions

The density of rotational transitions for a polyatomic molecule is so large that in general many such transitions are hidden under the Doppler profile, this being a fundamental limit of conventional high resolution electronic spectroscopy. We present here the first Doppler-free cw two-photon spectrum of a polyatomic molecule. In the case of benzene, 400 lines are observed of which 300 are due to single rotational transitions, their spacing being weil below the Doppler profile. The resolution so achieved is 1.5 X 10'. Benzene is a prototype planar molecule taken to have D •• symmetry in the ground as weil as in the first excited state. From our ultra-high resolution results it is found that benzene in the excited SI state i8 a symmetrical rotor to a high degree. A negative inertial defect is found for the excited state. The origin of this inertial defect is discused

Computing Macro-Effects and Welfare Costs of Temperature Volatility: A Structural Approach

We produce novel empirical evidence on the relevance of temperature volatility shocks for the dynamics of productivity, macroeconomic aggregates and asset prices. Using two centuries of UK temperature data, we document that the relationship between temperature volatility and the macroeconomy varies over time. First, the sign of the causality from temperature volatility to TFP growth is negative in the post-war period (i.e., 1950–2015) and positive before (i.e., 1800–1950). Second, over the pre-1950 (post-1950) period temperature volatility shocks positively (negatively) affect TFP growth. In the post-1950 period, temperature volatility shocks are also found to undermine equity valuations and other main macroeconomic aggregates. More importantly, temperature volatility shocks are priced in the cross section of returns and command a positive premium. We rationalize these findings within a production economy featuring long-run productivity and temperature volatility risk. In the model temperature volatility shocks generate non-negligible welfare costs. Such costs decrease (increase) when coupled with immediate technology adaptation (capital depreciation)

Energy Growth in Schrödinger's Equation with Markovian Forcing

Schrödinger's equation is considered on a one-dimensional torus with time dependent potential v(θ,t)=λV(θ)X(t), where V(θ) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant λ is sufficiently small, the average kinetic energy grows as the square-root of time. More generally, the H^s norm of the wave function is shown to behave as t^(s/4A)