Abstract
We analyze special classes of bi-orthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with generalized Riesz systems. In this way, the notion of the first/second type sequences is introduced and studied. We also discuss their relevance in some concrete quantum mechanical system driven by manifestly non self-adjoint Hamiltonians.
Highlights
The employing of non self-adjoint operators for the description of experimentally observable data goes back to the early days of quantum mechanics
It was conjectured [10] that the reality of eigenvalues of H is a consequence of its PT -symmetry: PT H = HPT, where the space parity operator P and the complex conjugation operator T are defined as follows: (Pf )(x) = f (−x) and (T f )(x) = f (x)
PT -symmetric Hamiltonians can be interpreted as self-adjoint ones for a suitable choice of indefinite inner product
Summary
The employing of non self-adjoint operators for the description of experimentally observable data goes back to the early days of quantum mechanics. A sequence {φn} in a Hilbert space H is called a generalized Riesz system (GRS) if there exists a self-adjoint operator Q in H and an orthonormal basis (ONB) {en} such that en ∈ D(eQ/2) ∩ D(e−Q/2) and φn = eQ/2en. One of benefits is the fact that a first type sequence generates a C-symmetry operator C = eQJ where Q is the same operator as in (2) The latter allows one to construct the Hilbert space (H−Q, ·, · −Q) involving {φn} as ONB, directly as the completion of D(C) with respect to “CPT -norm”:. For a second type sequence, each operator C generated by {φn} gives rise to the Hilbert space (H−Q, ·, · −Q) with non-densely defined symmetric operator H. An operator A is called positive [nonnegative] if (Af, f ) > 0 [(Af, f ) ≥ 0] for non-zero f ∈ D(A)
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