L2 Hilbert space

There are two realizations of the abstract Hilbert space which are frequently used in physics the sequential Hilbert space J( 0, and the L2 Hilbert space. The sequential Hilbert space Jf0 consists of all infinite column vectors c = ct with complex elements having a finite norm c, so that... 

The problem treated in this paper is complicated by the fact that the linear operator T is supposed to be linearly defined on a linear space A = F of all complex functions F = F(X) of a real composite variable X = jcl5 x2, , xN, of which the L2 Hilbert space is only a small subspace. We will refer to this space A = F as the definition space of the operator T. Since it contains all complex functions, it is stable under complex conjugation, and, according to Eq. (1.50) one has T F = (TF ), which means that also the complex conjugate operator T is defined on this space. 

The adjoint operator is defined only on the domain D(Tf) inside the L2 Hilbert space. In the important case, when the operator T is complex symmetric within the domain Z)(Tt), so that T+ = T, one has first of all... 

The L2 Hilbert space T is a subspace of the linear space A, and we note that the operator V transforms the elements P(Z) into complex functions of the composite complex variable Z, so that... 

One should also be aware of the fact that for the general complex transformation V, there may exist wave functions VF = F(A ) defined on the real axis which are quadratically integrable, but which cannot be analytically continued in such a way that the symbol L/VP becomes meaningful. In discussing an unbounded operator U and its domain D(U), it may then be practical to introduce the complement C(U) only with respect to the part of the L2 Hilbert space, for which the Operator U may be properly defined. A more detailed discussion of these problems is outside the scope of this review. In the following discussion, we will not further specify the form of the transformation U. 

If U is an unbounded operator, it has a specific domain D(U) in the L2 Hilbert space. We note further that the range R(U) of the operator U is the same as the domain of the inverse operator l/ 1, and vice versa. Hence, one has in a condensed form the relations... 

We previously introduced the symbol C(U) for the complement of the domain D(U) with respect to the L2 Hilbert space, and it is thus evident that the elements in the image set A(U) = UC(U) must be situated outside the L2 Hilbert space and that one has the reciprocity relations... 

Our discussion is complicated by the fact that, in the definition Eq. (2.27) of the transformed operator T, the operator T occurs in the middle. If 4 is an element of the L2 Hilbert space, then the result 4"" = T P is obtained by three successive mappings,... 

In this case the eigenfunction must hence belong to the complement C(U ) and the parent function F must then be situated in the set A U l) outside the L2 Hilbert space. Substituting the relation = UF into the second relation of Eq. (2.36) and multiplying the left by U one obtains... 

In this situation, one can achieve a further simplification by observing that it is not necessary to choose the linearly independent set 0 complex instead one can start from a real set = %, (p2,..., complete orthonormal basis for the L2 Hilbert space. The existence of such bases is well known. This means that in the approximate treatment of the eigenvalue problem for H, the relation, Eq. (2.73), is now replaced by the simpler relation... 

It is hardly necessary to emphasize that the present review is a fairly simple exercise in linear algebra and is intended to familiarized theoretical physicists and chemists working on the quantum theory of matter with the fundamental properties of the unbounded similarity transformations as applied to N-electron systems. Special attention has been given to the change of the spectra and how it is related to the domain of the transformation applied and to the fact that the eigenfunctions may be transformed not only within the L2 Hilbert space, but also out of and into this space (see Fig. 1). 

D(u) in the one-particle L2 Hilbert space with the complement C(u), and - in analogy with (A.2.31) - one gets the following four cases ... 

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