Finite energy space

Finite energy space L2(R) is the vector space of finite energy functions (or say, square integrable) such that. 

The upper bound of the escape probability, corresponding to t oo, can also be analytically calculated. In this limit, the electron is allowed to make many revolutions in the same orbit around the cation before it is scattered into another orbit. Under this condition, the electron motion may be described as diffusion in energy space [23]. The escape probability calculated by using the energy diffusion model is also included in Fig. 3. We see that the simulation results for finite x properly approach the energy diffusion limit. 

This Master Equation is turned into a finite set of coupled linear differential equations by truncating the energy space at some high value that is only visited very infrequently, and graining the energy. 

Defect lines in smectics, unlike those in nematics, often do not continuously shrink with time and spontaneously disappear. Instead, there often seems to be a finite energy barrier that must be overcome if a smectic defect is to disappear. This difference between nematics and smectics is a consequence of the layer-spacing constraint that exists in smectics but not nematics. Because of this constraint, topological defects in smectics cannot be removed without ripping layers, and this requires a finite energy. 

Once the energy spectrum of the finite model space is known, one can use its eigenvalues to compute the macroscopic property of interest, using its exact Statistical Mechanics expression. For instance, the magnetic susceptibility x at a given temperature (T) is computed by using the formula ... 

Such processes can be realized experimentally both in the time domain and in the energy domain. We may send a pulse of light of finite width, that is a wavepacket in momentum and energy spaces, onto the molecular system and monitor the scattered light as a function of frequency, direction, and time. Alternatively we may use a continuous wave (CW) field, a wave of infinite duration (relative to relevant... 

The remaining feature in the finite-resolution spectrum, at 0.800 eV, is more difficult to assign. On the basis of energy spacings it appears to be [08°]. 

Our model of the crystal comprises the ensemble of a reference molecule (RM) and its surrounding molecules (SMs), i.e. by a cluster of molecules partially obeying (being finite) the space group symmetry of the crystal. The steric energy E ) per RM in the crystal, will be ... 

A signal space is simply a collection of signals (functions) that satisfies a certain mathematical structure. The signal spaces with finite energy and finite power structures are particularly interesting in signal processing. 

If due to the finite size of the metallic system, only a finite number of states contribute to an energy band, we get a finite level spacing between neighbouring energy levels within a band. If this energy difference AE of... 

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