Time and band limiting operator and Bethe ansatz

Abstract

The time and band limiting operator is introduced to optimize the reconstruction of a signal from only a partial part of its spectrum. In the discrete case, this operator commutes with the so-called algebraic Heun operator which appears in the context of the quantum integrable systems. The construction of both operators and the proof of their commutativity is recalled. A direct connection between their spectra is obtained. Then, the Bethe ansatz, a well-known method to diagonalize integrable quantum Hamiltonians, is used to diagonalize the Heun operator and to obtain insights on the spectrum of the time and band limiting operator.

Bethe ansatz diagonalization of the Heun-Racah operator

“…For p = N , there exists a formula (see equation (4.21)) which allows to absord these unwanted terms in the sum on the rhs of (4.13). Such a relation is the foundation of the modified Bethe ansatz introduced in [5] for the XXX spin chain with generic boundaries, and later used in different contexts [1,2,6,9,13,18,19]. We explain both cases in detail in the following paragraphs.…”

“…The Heun operators have found many applications in various contexts. For example, they were seen in [35,12,13] to provide an algebraic interpretation for the existence of the commuting operators in the time and band limiting problem [46,42]. They also naturally lead to the commuting tridiagonal operator [27] arising in the computation of the entanglement entropy for 1D free Fermion models [23,24] with couplings given by the recurrence coefficients of orthogonal polynomials of the Askey scheme.…”