Self-adjoint problems

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

The formulas of Appendix A are easily transformed to time-dependent forms via the Fourier-Laplace transform. For the purpose of upcoming extensions to the nonself adjoint case, we introduce the self-adjoint Hamiltonian H = FT = Htt. Furthermore the function 0 considered above is said to belong to the domain of the operator H, i.e.,

bounded operators, the range R(H) and the domain D(H) are identical with the full Hilbert space however, in general applications they will differ as we will see later. 

Thus, every factor on the right-hand-side of (3) is known and (3) represents a well-posed problem for the local error indicators We have developed and implemented a similar error-estimation procedme for non-self-adjoint problems. 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

We will assume that problem (37) is solvable for any right-hand sides (p H there exists an operator A with the domain V A ) = H. All the constants below are supposed to be independent of h. In what follows the space H is equipped with an inner product (, ) and associated norm II. T II = /i x, x ). The writing A = A > 0 means that A is a self-adjoint... 

Let T be a self-adjoint positive definite linear operator in Hilbert space H equipped with an inner product (,) and let / be a given element of the space H. The problem of minimizing the functional... 

A non-self-adjoint boundary-value problem acquires the form ... 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

With this in mind, problem (8) is convenient to be taken in the operator form Ay = y. The operator A so defined is self-adjoint and nonnegative ... 

Clearly, stability is an intrinsic property of schemes regardless of approximations and interrelations between the resulting schemes and relevant differential equations. Because of this, any stability condition should be imposed as the relationship between the operators A and B. More specifically, let a family of schemes specified by the restrictions on the operators A and B be given A = A > 0 or Ay, v) = y, Av) and Ay, y) > 0 for any y, v H, where (, ) is an inner product in H, B > 0 and B B B is non-self-adjoint). The problem statement consists of extracting from that family a set of schemes that are stable with respect to the initial data, having the form... 

Locke, BR Arce, P Park, Y, Applications of Self-Adjoint Operators to Electrophoretic Transport, Enzyme Reactions, and Microwave Heating Problems in Composite Media—II. Electrophoretic Transport in Layered Membranes, Chemical Engineering Science 48, 4007, 1993. 

The Holstein-Primakoff transformation also preserves the commutation relations (70). Due to the square-root operators in Eqs. (78a)-(78d), however, the mutual adjointness of S+ and 5 as well as the self-adjointness of S3 is only guaranteed in the physical subspace 0),..., i- -m) of the transformation [219]. This flaw of the Holstein-Primakoff transformation outside the physical subspace does not present a problem on the quantum-mechanical level of description. This is because the physical subspace again is invariant under the action of any operator which results from the mapping (78) of an arbitrary spin operator A(5i, 2, 3). As has been discussed in Ref. 100, however, the square-root operators may cause serious problems in the semiclassical evaluation of the Holstein-Primakoff transformation. 

Self-adjoint operators. Let H be a self-adjoint2,0 (or hypermaximal2 0) operator, i.e., a symmetric (Hermitian)20 operator for which the eigenvalue problem is completely solvable. H admits of spectral decomposition20... 

Relation to the variational method. As we remarked in Introduction, we base our theory on the variational method in its generalized form.57) It will be convenient to give here a sketch of this relation. If FI is a self-adjoint operator for which we are to solve the eigenvalue problem, the cited variational method consists in making the quantity... 

This is certainly true for the minimal residual method (13)—(14) under conditions (34). Here yn is a solution of problem (13) and p is specified by formula (39). The fastest move method is useless in that case because the operator A is non-self-adjoint in such a setting. 

The stochastic resonance is determined by the longitudinal (with respect to n) modes of the relaxational problem (4.90). Since A is not a self-adjoint operator, it produces, together with the spectrum of eigenvalues ,, two sets of eigenfunctions defined as... 

We will start by setting up a simple 2x2 matrix that (without interaction) displays perfect symmetry between the particle and its antiparticle image. Note that it is well known that the Klein-Gordon and the Dirac equation can be written formally as a standard self-adjoint secular problem (see e.g. [11,12]), based on the simple Hamiltonian matrix (in mass units)... 

In physics, it is customary to formulate the eigenvalue problem of a self-adjoint many-particle Hamiltonian H and the associated boundary conditions which define its spectrum , in terms of the mathematical framework based on the theory of the Hilbert space. Before proceeding, it may hence be... 

The puzzle depended on the simple fact that most physicists using the method of complex scaling had not realized that the associated operator u - the so-called dilatation operator - was an unbounded operator, and that the change of spectra -e.g. the occurrence of complex eigenvalues - was due to a change of the boundary conditions. Some of these features have been clarified in reference A, and in this paper we will discuss how these properties will influence the Hartree-Fock scheme. The existence of the numerical examples finally convinced us that the Hartree-Fock scheme in the complex symmetric case would not automatically reduce to the ordinary Hartree-Fock scheme in the case when the many-electron Hamiltonian became real and self-adjoint. Some aspects of this problem have been briefly discussed at the 1987 Sanibel Symposium, and a preliminary report has been given in a paper4 which will be referred to as reference D. 

Note that the Euler equation can be obtained formally by applying the adjoint operator. A, to both sides of equation (4.1). However, the Euler equation (4.4) is not in general cases equivalent to the original inverse problem (4.1). The main characteristic of the Euler equation is that it provides the minimum of the misfit functional. The Euler equation (4.4) is equivalent to the original equation (4.1) if each of these equations has a unique solution in M. Note that, in this case, the operator A A is always a positive self-adjoint operator, because... 

It is hence evident that the CBP clarifies the structure of the special propagator theory in an excellent way. However, there is still a specific problem connected with the fact that the Liouvillian L is usually not self-adjoint with respect to this binary product ... 

The last member of (3.31) is a typical bivariational expression, but—since the Liouvillian L is self-adjoint—one should expect that the bivariational principle (3.15) should lead to a solution, where Ct is proportional to C. This is usually not the case, unless some specific conditions are satisfied, and one is hence facing about the same problems as discussed before in connection with the quantity l, and one may then resort to the methods developed in Section II. 

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