Spectrum and numerical range for the Friedrichs model of special form

This study investigates a variant of the Friedrichs operator H subject to a perturbation of rank three, acting within a complex Hilbert space <i>L</i>₂[-<i>&pi;</i> ; <i>&pi;</i> ] . The considered operator is characterized by its linearity, boundedness, and self-adjoint nature, and it models the Hamiltonian associated with a pair of interacting particles confined to a one-dimensional discrete lattice. The research reveals that although the rightmost point of the spectrum of <i>H</i> does not lie within the numerical range of the operator, it appears as a limit point. Depending on whether the so-called "special integral" is finite or divergent, the numerical range of the main operator <i>H</i> is examined through the lens of three related Friedrichs-type operators, each modified by a rank-one perturbation. A central role in this investigation is played by the specification of parameter-dependent functions.