6 research outputs found
On the Spectra of Real and Complex Lam\'e Operators
We study Lam\'e operators of the form with and a
half-period of . For rectangular period lattices, we can choose
and such that the potential is real, periodic and regular. It is
known after Ince that the spectrum of the corresponding Lam\'e operator has a
band structure with not more than gaps. In the first part of the paper, we
prove that the opened gaps are precisely the first ones. In the second
part, we study the Lam\'e spectrum for a generic period lattice when the
potential is complex-valued. We concentrate on the case, when the
spectrum consists of two regular analytic arcs, one of which extends to
infinity, and briefly discuss the case, paying particular attention to
the rhombic lattices
Generalized local interactions in 1D: solutions of quantum many-body systems describing distinguishable particles
As is well-known, there exists a four parameter family of local interactions
in 1D. We interpret these parameters as coupling constants of delta-type
interactions which include different kinds of momentum dependent terms, and we
determine all cases leading to many-body systems of distinguishable particles
which are exactly solvable by the coordinate Bethe Ansatz. We find two such
families of systems, one with two independent coupling constants deforming the
well-known delta interaction model to non-identical particles, and the other
with a particular one-parameter combination of the delta- and (so-called)
delta-prime interaction. We also find that the model of non-identical particles
gives rise to a somewhat unusual solution of the Yang-Baxter relations. For the
other model we write down explicit formulas for all eigenfunctions.Comment: 23 pages v2: references adde
Interactive Programs in Dependent Type Theory
. We propose a representation of interactive systems in dependen