Unbounded Similarity Transformations and the Associated Change of Boundary Conditions

Unbounded Similarity Transformations and the Associated Change of Boundary Conditions 

Some General Aspects on the Boundary Conditions Associated with the Eigenvalue Problem 

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 

According to von Neumann (9), an abstract Hilbert space X is a linear space A = x having a binary product x1 x2 , which satisfies the conditions, Eqs. (1.8)—(1.10), and which further contains all its limit points in the norm x = x x 1/2 and is separable. The last assumption means that there exists an enumerable set Jf = xk which is everywhere dense in and which ensures the existence of at least one complete orthonormal set p = (pk spanning the 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 

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