Three Identical One-Dimensional Quantum Particles with Point Interaction as a Solvable Model: II. Negative Essential Spectrum

This is the second part of our work dealing with spectral analysis for the Hamiltonian of three identical one-dimensional quantum particles. The Hamiltonian is represented as a sum of Laplacian and a singular delta-potential with the symmetric support consisting of six half-lines (leads) with the same origin. Contrary to the first part discussing the discrete spectrum and the eigenfunctions, the second part is devoted to study of the negative essential spectrum and the corresponding generalized eigenfunctions. It turns out that, instead of the Kontorovich–Lebedev integral representation exploited for the eigenfunctions of the discrete spectrum, alternative integral representations of the Watson–Bessel type for the generalized eigenfunctions of the essential spectrum are applied. As in the first part, the symmetry of the support of the corresponding singular potential is made use of. The operator is decomposed to the fibers by means of the Fourier transform on the group of symmetry. Further analysis is connected with the investigation of the problem for the fibers and leads to the study of spectral properties of a functional equation similar to that for the Maryland model. The results obtained here enable one to determine explicit formulas for the generalized eigenfunction and to study their behavior at far distances by means of reduction to the Sommerfeld integral representations. The far field asymptotics of the generalized eigenfunctions are interpreted as surface waves localized near the leads of the support of the singular potential.