An arXiv preprint presents a mathematical framework for singular q-Dirac operators that connects a dissipative operator to a selfadjoint dilation, scattering theory and a functional model. It also gives theorem-based statements about discrete spectra and complete systems of eigenvectors and associated vectors, but only under the model's stated assumptions.
This is operator theory, not an empirical test. The modeled objects are maximal dissipative singular q-Dirac operators in L2q(q Z; C2), treated as extensions of a minimal symmetric operator in the Weyl limit-point case. Here, “dissipative” is a label for the operator class being analyzed, not a report of measurements from a physical system. The manuscript says it has no associated data.
Turning dissipation into a scattering problem
The first theorem addresses the boundary condition: it states that T_h is dissipative in the relevant function space. The paper then identifies M_h as a selfadjoint dilation of T_h. In plain terms, the dilation places the original operator inside a broader selfadjoint construction; the paper uses that construction as the setting for its later spectral and scattering analysis.
The boundary parameter also marks different operator regimes. The manuscript describes T_h as accumulative when Im h is less than zero and self-adjoint when Im h equals zero or h is infinity. For the scattering step, the authors apply the Lax–Phillips scheme to the unitary group U_h, using incoming and outgoing subspaces.
Within those spectral representations, S_h is identified as the scattering matrix of the group associated with M_h. The functional-model result identifies the characteristic function of T_h — the function used in that model to represent the operator — with S_h(λ), linking the paper’s scattering and functional-model descriptions.
The spectrum under the theorem’s conditions
For h with positive imaginary part, apart from a possible exceptional value h0, the theorem places the spectrum of T_h in the open upper half-plane. It describes a purely discrete spectrum: countably many isolated eigenvalues of finite multiplicity, with limit points at infinity, and reports a complete system of eigenvectors and associated vectors in L2q(q Z; C2).
When m∞ is meromorphic in the complex plane, the analogous result for Im h less than zero, except possibly h1, places the accumulative operator’s spectrum in the open lower half-plane. The theorem again states pure discreteness, countably many isolated finite-multiplicity eigenvalues with limit points at infinity, and completeness of the eigenvectors and associated vectors.
What remains conditional
These spectral conclusions are conditional on a specific model. The assumptions require q to be positive and less than 1; the coefficient functions p and r must be real-valued, continuous at zero or q-regular there, and nonzero on q Z. The upper- and lower-half-plane statements also allow possible exceptional parameters h0 and h1, while the discrete-spectrum discussion assumes that m∞ is meromorphic throughout the complex plane.
The preprint does not turn those theorems into empirical evidence about physical or clinical outcomes. It reports no empirical sample or associated dataset, and it does not establish meromorphicity of m∞ for every allowable coefficient choice.
Paper data and sources
Original title: Dissipative eigenvalue problems for a singular quantum Dirac system
Authors: Bilender Pasaoglu Allahverdiev, Yelda Aygar
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text