Characteristic-value problem

The characteristic-value problem - more often referred to as the eigenvalue problem - is of extreme importance in many areas of physics. Not only is it the very basis of quantum mechanics, but it is employed in many other applications. Given a Hermitian operator a, if their exists a function (or functtofts) g such that... 

Characteristic value problems. The way in which the displacement symmetry can be used to eliminate one variable is well known. One writes. 

Generalization of the characteristic value problem. The characteristic value problem can be formulated as the quest for the irreducible linear manifolds which are invariant under an operator. The principal result of the spectral theory of normal operators can be formulated, from this point of view, as the statement that all irreducible linear manifolds of normal operators are one-dimensional. Similarly, one can ask for irreducible closed linear manifolds which are invariant under a set of operators. Since a closed linear manifold which is invariant under a set of operators is also invariant under the group or algebra generated by these operators, one is naturally led in this way to a linear manifold which belongs to an irreducible representation of a group or an algebra. 

Among all the non-numerical approximation methods, the effectiveness of the variational methods is perhaps most surprising [11]. This method serves to determine characteristic values of linear operators. Since the time variable can be eliminated from almost every reactor equation by transforming it into a characteristic value problem, the variational method should have wide applications in reactor theory. Its use has been limited, so far, because Boltzmann s operator is not self adjoint or normal. Whether this limitation is a necessary one, remains to be seen. The reason for the great accuracy of the variational principle in simple problems of quantum mechanics is that any function which is positive everywhere and has a single maximum can be so well approximated by any other similar function. Thus... 

In general. Equation (1) will not admit of any time-independent solutions except the trivial one 0 = 0. To cast (1) into the form of the more usual characteristic value problem, we introduce a fictitious neutron multiplicity. 

Thus the problem of determining the critical flux distribution and the value of the static criticality constant reduces to the solution of the characteristic value problem... 

The explicit energy and angular dependence in the characteristic value problem (4a) can be eliminated, as first suggested by G. Placzek and G. VoIkofF [1] by introducing as variable, S (x), the number of fission neutrons produced per unit time at x ... 

Equation (11) has the form of a characteristic value problem whereas (10) does not— i.e., the kernel H characterizes the medium, so to speak, in all of its non-multiplying aspects it is the multiplying or chain character of the reaction in the medium which introduces the homogeneous linear term on the right-hand side of (11), and thus makes the basic reactor equation for all reactors a characteristic value problem. Much of the discussion to be given over the next two days will be based on Equation (11) or a variant thereof. 

Homogeneous Algebraic Equations and the Characteristic-Value Problem 121... 

Homogeneous Algebraic Equations and THE Characteristic-Value Problem... 

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