Linear operators in spaces with indefinite metric and their applications

Relativistic supersymmetric quantum mechanics based on Klein–Gordon equation

“…We shall propose a new type of a factorization of our Hamiltonians, the feasibility of which will be achieved via an additional partitioning of our Hilbert space(s). We shall emphasize the intimate relationship of the properties of our new factorization to the above-mentioned existence of the level crossings, connected also to the numerical concept of the Jordan blocks in non-Hermitian matrices [17].…”

“…Then, equation (9) would merely define the two identical vectors at both signs (±) at n = 0. Such a feature is characteristic for the non-diagonalizable (usually called Jordan-block) limits of non-Hermitian operators [17]. In the spectra of differential operators these points are also known as "Bender-Wu singularities" [24], as the points of an "unavoided level-crossing" [8] or simply as "exceptional points" [18].…”

Coupling of definitizable operators in Krein spaces

“…For a complete exposition on the subject (and the proofs of the results below) see the books by Azizov and Iokhvidov [1] and Bognár [2]. A vector space K with a Hermitian sesquilinear form [ .…”

Similarity of Indefinite Sturm-Liouville Operators with Singular Potential to a Self-Adjoint Operator

“…In this subsection basic facts from the spectral theory of J-nonnegative operators are collected (the reader can find more details in [3,33]). Consider a Hilbert space H with a scalar product (·, ·) H .…”

“…Let [33,3] for the original definition). The form [·, ·] is called an inner product in the Krein space K and the operator J is called a fundamental symmetry in the Krein space K. Evidently, the form [·, ·] is indefinite on H if and only if H + = {0} and H − = {0}.…”