Inverse Spectral Problem for Band Operators and Their Sparsity Criterion In Terms of Inverse Problem Data

On the Links between Miura Transformations of Bogoyavlensky Lattices and Inverse Spectral Problems for Band Operators

“…Now, consider in brief the inverse spectral problem for the operators M ; for its full description, see e.g. [26]. First, for λ ∈ Ω(M ), where Ω(M ) is the resolvent set of M, we define the following functions named the Weyl solutions of M [26,30,31]:…”

An Inverse Spectral Problem for a Special Class of Band Matrices and Bogoyavlensky Lattice

“…We identify the matrix M with the operator defined as the closure of the operator acting on the dense set of finite vectors from l 2 [0, ∞), where its action is described via matrix calculus (and keep the same notation M for this operator). Now, consider in brief the inverse spectral problem for the operators M; for its full description see e. g. [25]. First, for λ ∈ Ω(M), where Ω(M) is the resolvent set of M, we define the following functions named the Weyl solutions of M [29,30,25]:…”

“…Now, consider in brief the inverse spectral problem for the operators M; for its full description see e. g. [25]. First, for λ ∈ Ω(M), where Ω(M) is the resolvent set of M, we define the following functions named the Weyl solutions of M [29,30,25]:…”

“…The elements of M can be recovered from S(M(λ, M)) in the recurrent manner starting from a i+q,i , see [25], and the latter are obtained from the following formulas:…”

“…For the complete proof of sparsity criterion for the matrices M in terms of S(M(λ, M)) see [25], Theorem 2. Another property of the moments of S(M(λ, L1))(t) and S(M(λ, L2))(t) can be established directly using ( 4),( 6) and ( 17)-( 18), namely:…”