Liouville eigenvalues

If R(k) = E equals a Sturm-Liouville eigenvalue Ek then the normalized eigenfunction uk(r) must be proportional to i/rk(Ek,r) (note that the dependence on the right-hand boundary is not explicitly indicated in ukand Ek), i.e.,... 

Brown [47] shaped up those semiqualitative considerations into a rigorous Sturm-Liouville eigenvalue problem by deriving the micromagnetic kinetic equation... 

The boundary value problem posed by the differential equation (2.166) and the two boundary conditions (2.168) and (2.169) leads to the class of Sturm-Liouville eigenvalue problems for which a series of general theorems are valid. As we will soon show the solution function F only satisfies the boundary conditions with certain discrete values /q of the separation parameter. These special values /q are called eigenvalues of the boundary value problem, and the accompanying solution functions Fi are known as eigenfunctions. The most important rules from the theory of Sturm-Liouville eigenvalue problems are, cf. e.g. K. Janich [2.33] ... 

But from the definition of the Sturm-Liouville eigenvalue problem (Eq.11.48), the LHS of Eq. 11.51 gives... 

V. Ledoux, M. Van Daele and G. Vanden Berghe, Efficient Computation of High Index Sturm-Liouville Eigenvalues for Problems in Physies, Computer Physics Communications, 2009, 180(2), 241-250. 

X here represents various variables and the equation is therefore a partial differential equation. L[ j represents a linear, homogeneous, self-adjoint differential expression of second order, ip is the desired function, p x) the density function and A the eigenvalue parameter of this Sturm-Liouville eigenvalue problem. ... 

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

The difference eigenvalue problem for X can be viewed as the Sturm-Liouville difference problem ... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

The formal similarity between Eq. (10) and the time-dependent Schrodinger equation is striking, and we shall indeed develop methods which are very reminiscent of quantum mechanics. In particular, we may calculate the eigenfunctions and eigenvalues of the unperturbed Liouville operator L0. We look for solutions of ... 

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

In the Liouville formulation one obtains the well-known connection between x and the imaginary part T of the complex resonance eigenvalue... 

In the renormalized composite Liouville space the superoperator F , defined by equation (121), commutes with all the (Hermitian) superoperators Hf, R, and X. Therefore, upon the proper rearrangement of the basis set in this space, one can obtain a factorization of the equation (137) of motion into blocks which are connected with individual eigenvalues of the superoperator F°. This resembles the analogous procedure in the case of static NMR spectra, i.e. those for non-exchanging spin systems (Section II.E.2). The equations for the free induction decay M ID and for the lineshape of an unsaturated steady-state spectrum, in terms of quantities from composite Liouville space, are therefore obtained for exchanging spin systems in a way which is analogous to that for non-exchanging systems (Section II.F). 

In equation (143), the subscript (—1) denotes the subspace of the composite Liouville space which is concerned with the eigenvalue — 1 of the superoperator Ff (the subspace of single-quantum transitions). The vector fx is normalized according to equation (135) and the q, (t) function has the same meaning as in equation (50). The coefficient C is given by [equation (139)1 ... 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

Given that the total hamiltonian may be written as Hw = P2/2M + hw(R), the adiabatic eigenfunctions a R) are the solutions of the eigenvalue problem, hw(R) ot R) = Ea(R) a R). In this adiabatic basis the quantum-classical Liouville operator has matrix elements [12],... 

Sturm-Liouville differential equations, resulting from a separation of variables have been known since the middle of the 19th century. Separation constants, subject to boundary conditions, yield sets of characteristic, or eigenvalue, solutions. 

These lemmas and their counterpart for v (labeling the eigenvalue as /io) establish the existence of the rest points on the boundary of C+ x C+. As before, we label these rest points Eq,Ei,E2- As with the gradostat, the condition for coexistence is tied to the question of invasiveness. Now, however, the conditions take the form of comparison with the eigenvalues of certain Sturm-Liouville problems rather than with the stability modulus of matrices, as was the case in Chapter 6. We describe just enough of this to show the parameters on which the result depends. 

Think of nii as a parameter and let Xinxi) be the largest eigenvalue of the Sturm-Liouville problem just displayed. The eigenvalue X m2) is a strictly increasing function of m2 satisfying... 

The roots h = Hi of this equation are the eigenvalues of the problem, which depend on the Biot number. As Fig. 2.29 shows, there is an infinite series of eigenvalues Hi < /r2 < /U3. .. which is in full agreement with the Sturm-Liouville theory. Only the following eigenfunctions... 

The Sturm-Liouville representation (136) is a formal solution as a knowledge of all eigenfunctions )( 4>), and corresponding eigenvalues Xp is required. However, this representation is very useful because it allows one to obtain a formal solution for the longitudinal complex susceptibility = x i ) According to... 

According to (1.22) and (1.26), the eigenvalues v of the Liouvillian L are distributed symmetrically around the point v = 0, and this implies that, even if the Hamiltonian H in physics is bounded from below, H > a 1, the Liouvillian L is as a rule unbounded. Except for this difference, practically all the Hilbert-space methods developed to solve the Hamiltonian eigenvalue problem in exact or approximate form may be applied also to the Liouvillian eigenvalue problem. In the time-dependent case, the L2 methods developed to solve the Schrodinger equation are now also applicable to solve the Liouville equation (1.7). 

The effective eigenvalue may be defined in the context of the Sturm-Liouville equation as [78]... 

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