Bilinear operator

Theorem 9.32 ([9, 20]). Vl together with bilinear operations given by Y v,z), the vaeuum vector 1 = 1 0e°, and the eonformal vector uj is a vertex algebra. If in addition L is positive definite, it is a vertex operator algebra. 

Definition 9.19. A vertex algebra is a Z-graded vector space V = neZ V(n) equipped with bilinear operations indexed by n E Z... 

Hence, for the transfer of polarization between two energy-matched spins, the (effective) coupling term must contain both orthogonal bilinear operators, since = 0 if = Q or = 0, that is, if Ia qI = adqI-... 

Note that only contains bilinear operators. However, can... 

To calculate Ha in the representation of Hq we must consider a bilinear operator Jp J for two arbitrary spins i and in this representation. Taking into account that [fp.i, fp.4] = 0 we get ... 

These operators must be seen as a replacement for the uncoupled bilinear operators (j,y [Eq. (2.37)]. For example, the operator is given explicitly by... 

Shaka AT, Lee CJ, Pines A (1988) Iterative schemes for bilinear operators — application to spin decoupling. J Magn Reson 77 274—293... 

As seen from eq. (10.21), the first-order property is given as an expectation value of operators linear in the perturbation. The second-order property contains two contributions, an expectation value over quadratic (or bilinear) operators and a sum over products of matrix elements involving linear operators connecting the ground and excited states. 

The inner product is a symmetric, bilinear operation if a, are real scalars and... 

The isotropic bilinear operator discussed so far is the most widely considered interaction in polynuclear magnetic systems since it accounts for an important part of the physics. However, it is not the whole story. In the very beginning of this chapter, we... 

To check that this sum indeed cyclically permutes the spin functions, we compare the outcome of acting with A234 and acting with the sum of bilinear operators on the wave function P = afiafi — Pa a. 

Repeating this for the other two products of bilinear operators and summing the results of acting on paPa as well, we obtain... 

We then come from spin operators to second quantization operators in a standard manner, expand the square root in Eq. (2.3) in a bilinear operator form, and, finally, perform the Fourier transform. As a result, we obtain the following expression for the spin wave energy ... 

The terms in (la) and (lb) both involve sums of single nuclear spin operators Iz. In contrast, the terms in (lc) involve pairwise sums over the products of the nuclear spin operators of two different nuclei, and are thus bilinear in nuclear spin. If the two different nuclei are still of the same isotope and have the same NMR resonant frequency, then the interactions are homonuclear if not, then heteronuclear. The requirements of the former case may not be met if the two nuclei of the same isotope have different frequencies due to different chemical or Knight shifts or different anisotropic interactions, and the resulting frequency difference exceeds the strength of the terms in (lc). In this case, the interactions behave as if they were heteronuclear. The dipolar interaction is proportional to 1/r3, where r is the distance between the two nuclei. Its angular dependence is described below, after discussing the quadrupolar term. 

In this section analytical expressions for ENDOR transition frequencies and intensities will be given, which allow an adequate description of ENDOR spectra of transition metal complexes. The formalism is based on operator transforms of the spin Hamiltonian under the most general symmetry conditions. The transparent first and second order formulae are expressed as compact quadratic and bilinear forms of simple equations. Second order contributions, and in particular cross-terms between hf interactions of different nuclei, will be discussed for spin systems possessing different symmetries. Finally, methods to determine relative and absolute signs of hf and quadrupole coupling constants will be summarized. 

The operators of physical interest can be expanded as a power series in the bilinear products b bp of the boson operators.4 Special cases include the Hamiltonian H,... 

The (bilinear) expansion in the products of boson operators b]j serves to ensure the correspondance with quantum mechanics. To see this explicitly, say A, B, and C are operators familiar from wave mechanics and let A,B, and C be their corresponding matrix representations. If [A, 3] = C, then [A, B] = C. Now define... 

The same expansion can be done for quantum-mechanical problems with halfinteger spin, except that one needs fermion operators, a,1, and a. The bilinear products... 

The following notation has been introduced in Eq. (4.92) As denote coefficients of terms linear in the Casimir operators, A.s denote coefficients of terms linear in the Majorana operators, Xs denote coefficients of terms quadratic in the Casimir operators, Ks denote coefficients of terms containing the product of one Casimir and one Majorana operator, and Zs denote coefficients of terms quadratic in the Majorana operators. This notation is introduced here to establish a uniform notation that is similar to that of the Dunham expansion, where (Os denote terms linear in the vibrational quantum numbers, jcs denote terms that are quadratic in the vibrational quantum numbers and y s terms which are cubic in the quantum numbers (see Table 0.1). Results showing the improved fit using terms bilinear in the Casimir operators are given in Table 4.8. Terms quadratic in the Majorana operators, Z coefficients, have not been used so far. A computer code, prepared by Oss, Manini, and Lemus Casillas (1993), for diagonalizing the Hamiltonian is available.2... 

Using only terms linear in the Casimir operators. c Using all the terms bilinear in the Casimir operators in Eq. (4.92). 

Adapted from Iachello and Oss (1990). Terms both linear and bilinear in the Casimir operators in Eq. (4.96) have been used in the fit. See Appendix C. States are designated both by normal-mode quantum numbers and by localmode quantum numbers. 

The operator a i) in the Heisenberg algebra, of course, corresponds to the operator constructed in Chapter 8. But our commutator relation (8.14) differs from the standard one, we need to modify operators. In fact, it is more natural to change also the sign of the bilinear form. Hence we dehne... 

The Hamiltonian is bilinear with respect to the creation and annihilation operators. If coi b for all k (i.e., there is no resonance), then we can diagonalize the Hamiltonian as... 

By postulating the correlation operator to be a sum of two-electron operators and assuming the occupied orbitals to be localized, we were able to show that the correlation energy can in fact be approximately expressed in terms of the bilinear expression... 

Quadratic forms and self-adjoint operators.375 Let bilinear form defined in a linear manifold 2)(/) dense in... 

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