Inverse Problems in the Theory of Small Oscillations

Discrete Multichannel Scattering with Step-like Potential

“…This information is, generally speaking, insufficient for reconstruction the whole matrix L. However if the matrix L 1 corresponding to the "central" part of the system is sufficiently sparse and also we know the matrix B(0) = diag{b σ (0)} σ∈C which realizes connections between the channels and the central part of the system, the whole matrix L can be recovered from the scattering data. We refer the reader to Chapter 11 in [15], where statemets of such type are obtained.…”

“…This relation provides existence of Jost solutions on each channel σ ∈ C. It is well known, see e.g. [15,18], that for each σ ∈ C there is a family {e σ (k, θ)} ∞ k=0 of functions holomorphic inside the open disk D, continuous up to the boundary and, for each θ ∈ T and k ≥ 1,…”

“…The functions E(k, ω) and P (k, ω) are holomorphic in a neighborhood of D, except, perhaps zero, where P (k, ω) may have poles. Moreover, it is well known, see [15,18], that the Jost solutions form a fundamental system of solutions of the finite-difference equation (4.1). If |θ| = 1 and |θ| = 1, it follows from (4.5) that e(θ),ē(θ) = e(θ −1 ) are independent solutions of (4.1) and any other solution {x(k, θ)} k≥0 can be expressed as x(k, θ) = m(θ)e(k, θ) + n(θ)e(k, θ −1 ), with m(θ), n(θ) independent of k. Thus, for ω ∈ T + σ the function p σ (k, ω) may be expressed in terms of the corresponding Jost solutions:…”

“…In order to have such reconstruction, one needs to demand additional sparsity conditions of the central part. We refer the reader to the recent book [15], which contains examples of such conditions.…”

Uncertainty principle for discrete Schrödinger evolution on graphs

“…We will show that if a solution of (1.2) decays sufficiently fast on one thread at two different times, then the solution is trivial on the whole thread. To this end we will combine techniques on scattering theory on such graphs, developed in [9,11] and techniques of the growth of entire functions, present e.g. in [8], to follow a similar strategy as it was done in [6] in theorem 2.3, where it was proven that if a solution u(t, n) of the problem ∂ t u(t, n) = i(∆u(t, n) + V (n)u(t, n)), n ∈ Z, t ∈ [0, 1] (1.3)…”