Discrete spectrum, of eigenvalues

The net effect of these restrictions is that in some cases, E may have any value then we speak of a continuous spectrum of eigenvalues of E. In other cases, E may be restricted to certain particular values then we have a discrete spectrum of eigenvalues. In these latter cases we say that E is quantized. Ordinarily, for each boundary condition we impose, we introduce a quantization of some observable. 

The function g(r — r ), the so-called linear memory function55> describes the bonding of units into a chain, ip(r) is the external field into which the macromolecule is placed. In a limited volume, the spectrum of eigenvalues in Eq. (3.3) is discrete therefore, for sufficiently high N... 

To support our belief that inertia is the main reason for the deviations from the predictions of the AEP, let us study the dynamical properties of the system of Eq. (4.3). To shed light on that, let us consider again, as in Section IV, the ideal case D = 0. The spectrum of the dgenvalues of the operator of Eq. (4.3) has been studied by Schenzle and Brand and by Graham and Schenzle, who showed that this consists of the superposition of two clearly distinct contributions (1) the discrete spectrum of the eigenvalues... 

Bound states of the discrete spectrum of the atomic (molecular) Hamiltonian, Ha M, in which case E is the total energy, which is one of the real eigenvalues of the TISE, say E . (Bold letters symbolize operators.)... 

In quantum mechanics, the eigenvalues of an operator represent the only numerical values that can be observed if the physical property corresponding to that operator is measured. Operators for which the eigenvalue spectrum (i.e., the list of eigenvalues) is discrete thus possess discrete spectra when probed experimentally. 

The discreteness in this new theory refers to the discrete nature of all measurements. Each measurement fixes a particle s position Xn and time Both and tn are allowed to take on any value in the spectrum of continuous eigenvalues of the operators (xn)op and (tn)op. Notice also that in this discrete theory there is no Hamiltonian and no Lagrangian only Action. 

In (10-124) and (10-125), n and m refer to the eigenvalues of a complete set of commuting observables so Snm stands for a delta function m those observables in the set that have a continuous spectrum, and a Kronecker 8 in those that have a discrete spectrum. 

Case ii). C) is automatically satisfied if B), B0) hold and, moreover, the spectrum of IIt) at least belotu Aq consists of finite discrete eigenvalues with finite multiplicities. To show this we may assume U0 and i cl) to be positive definite (see the remark of 7.1). Then the quadratic form ((i70 + /) is an increas-... 

Consider a quantum mechanical many-body system, e.g., a gas. There is an enormous number of coordinates q of all molecules, and the state of the system is described by a wave function p(q, t) in an enormous Hilbert space H. The evolution is governed by a Hamilton operator H whose spectrum is discrete but inordinately dense. The number of eigenvalues of if in a range of the order of... 

It is clear that if F bejongs to the domain D(U), then the function F = t/ P also belongs to L2 and E = E is a persistent discrete eigenvalue. On the other hand, if P belongs to the complement C(U), the function F = U V is outside L2 and it cannot be an eigenfunction to H hence E becomes a lost eigenvalue, one that does not necessarily belong to the spectrum of H. 

Fig. 2. The change of the energy spectrum under complex scaling. (A) The spectrum of the original Hamiltonian (B) the exact spectrum of the dilated Hamiltonian (C) the approximate spectrum with all the eigenvalues discrete as obtained by the bivariational principle and a truncated finite basis.

Figure 10. k as a function of the intensity of the multiplicative noise, k is defined as (0) with (jc(t))/(ac(0)). The two arrows on the left denote the point where the discrete branch of the eigenvalue spectrum disappears (see Schenzle and Brand ). The two arrows on the right denote the phase transition threshold. [Taken from S. Faetti et al., Z. Ffiys., B47, 353 (1982).]... 

The density of states concept was introduced in Section 1.3. For any operator A characterized by the eigenvalue spectrum nJ we can define a density function Pa (a such that the number of eigenvalues that satisfy a [Pg.82]

For any atomic multipole transition, the excited state can be described in terms of the dual representation of corresponding SU(2) algebra, describing the azimuthal quantum phase of the angular momentum. In particular, the exponential of the phase operator and phase states can be constructed. The quantum phase variable has a discrete spectrum with (2j + 1) different eigenvalues. 

The radiation phase of multipole photons has discrete spectrum in the interval (0,2n). In the classical limit of high-intensity coherent field, the eigenvalues of the radiation phase are distributed uniformly over (0, 2ji). 

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