ABSTRACT
In chapters 3-10, we shall give a theoretical description of scattering by using spectral operator analysis and theory of infinite-dimensional Hilbert vector spaces. The first issue to be addressed here is the introduction of the notion of a spectrum of an operator in a general way, which will be applicable to both discrete and continuous variables. As is well-known [7], in an examination of stationary states of a given physical system, the main task is reduced to searching for the solutions of the time-independent Schro¨dinger equation
HˆΨ = EΨ. (3.1)
Since this is a second-order differential equation, it is clear that the boundary conditions imposed to (3.1) will play a central role in selection of the proper, physical solutions. In the case of closed i.e. bound states, the eigen-values E are negative and discrete, whereas for scattering states, energy E becomes positive and takes its continuous values. In order to correctly formulate the eigen-value problem of type (3.1), we will resort to the theory of Hilbert vector spaces. In so doing, we assume that all the dynamical properties of quantum mechanical systems are contained in the structure of the self-adjoint linear Hamilton operator Hˆ from the separable Hilbert space H, defined in the field of complex numbers C. Here, an abstract Hilbert vector space H is understood as a unitary, complete and separable complex-valued vector space i.e. a linear space of normalizable solutions of the Schro¨dinger equation (3.1), with positive-definite scalar products 〈ψ|ψ〉 ≥ 0. Unitarity means that the space H is equipped with a definition of the scalar (inner) product as a conjugate bilinear, Hermitean symmetrical1 and scalar function in H ×H i.e. with its values (images) in field C. Space H is complete, since it contains all the limiting values of its convergent Cauchy series/sequences. Fundamental or the
ION-ATOM
Cauchy series are those sequences that possess strong convergence i.e. convergence in the norm || ◦ || ≡ 〈◦|◦〉1/2. Separability signifies that H is a countable infinite-dimensional space. Stated more precisely, space H is separable if it contains a countable set S ≡ {ψk}∞k=1, which is everywhere dense in H i.e. for each ψ ∈ H and an arbitrary positive number ² > 0, there exists at least one element ψk such that ||ψk −ψ|| ≤ ². In other words, any vector ψ ∈ H can be approximated with an element from S to an arbitrary accuracy. The separability property of H always holds in quantum mechanical systems of particles with non-zero masses. Contrary to this, however, in field theory where one studies massless particles, systems with infinitely many degrees of freedom are routinely encountered, and the separability condition for H is not fulfilled. In such a case, one deals with non-separable Hilbert spaces H and this represents one of the severest obstacles to formulating scattering theory. We stress that the axiom of completeness and separability of H is of decisive importance for quantum scattering theory, not only in regard to its conceptual foundation of the basic principles, but also with respect to numerical applications that enable comparisons between theory and experiment. Namely, completeness of H assures the existence of at least one complete ortho-normal set {φk}, which can be used as a basis for a so-called expansion theorem
ψ = ∞∑ k=1
ckφk 〈φm|φn〉 = δnm (3.2)
where δnm is the Kronecker δ−symbol: δnn = 1 and δnm = 0 (n 6= m). Expansion coefficients ck are given by the scalar product
ck = 〈φk|ψ〉 (3.3) and the infinite sum in (3.2) is convergent in the norm as a consequence of completeness and separability of H. Set {φk}nk=1 (n <∞) can be easily found by applying e.g. the Gramm-Schmidt successive orthogonalization to a certain non-orthogonal basis {ψk}n<∞k=1 according to [34]
χk+1 = ψk+1 − ∑k
φ1 = ψ1 ||ψ1|| φk>1 =
χk ||χk||
. (3.4) An entirely analogous procedure of the Gramm-Schmidt othogonalization can also be formulated in the case when our starting point {ψk}∞k=1 contains countable infinite number of elements. Further, separability of H guarantees that, even if the infinite sum (3.2) is truncated at a given finite n < ∞ (n ∈ N) and the coefficients ck (1 ≤ k ≤ n) are approximated by rational complex numbers, the norm of each element ψ ∈ H can be numerically computed with arbitrary accuracy ² through some powerful algorithms. Hence, separability of the Hilbert space is the basis of all the numerical applications of quantum
resolvents
mechanics. Except for a few publications (see e.g. Ref. [7]), this statement is virtually non-existent in the literature, but obviously deserves full attention. The conditions of completeness and separability are automatically fulfilled for finite-dimensional vector spaces.
