We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operato...

Spectra of observables in the q-oscillator and q-analogue of the Fourier transform Abstract. Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa + − qa + a = 1) are studied when q > 1. These operators are symmetric but not self-adjoint. They have one-parameter family of self-adjoint extensions. These extensions are derived explicit...

Given, on the Hilbert space H0, the self-adjoint operator B and the skew-adjoint operators C1 and C2, we consider, on the Hilbert space H ≃ D(B)⊕H0, the skew-adjoint operator

We discuss abstract Birman—Schwinger principles to study spectra of self-adjoint operators subject small non-self-adjoint perturbations in a factorised form. In particular, we extend and part improve classical result by Kato which ensures that the spectrum does not change under perturbations. As an application, revisit known results for Schrödinger Dirac Euclidean spaces establish new three-dim...

We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, latter being not necessarily involutive. discuss extension theory, refined polar and singular-value decompositions, antilinear eigenfunction expansions. The study is motivated by physical symmetries quantum mechanics with non-self-adjoint operators.

A class of non-self-adjoint fourth order differential operators with general separated boundary conditions in Weyl’s limit circle case is studied. The dissipation property of the considered operators in L2[a,b) is proven by analysis and by using the characteristic determinant, the completeness of the system of eigenfunctions and associated functions of these dissipative operators also be proven...