Dispersion Estimates for the Discrete Laguerre Operator

Aleksey Kostenko,Gerald Teschl

doi:10.1007/s11005-016-0831-0

Aleksey Kostenko, Gerald Teschl

Open Access

https://doi.org/10.1007/s11005-016-0831-0

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Abstract

We derive an explicit expression for the kernel of the evolution group $\exp(-\mathrm{i} t H_0)$ of the discrete Laguerre operator $H_0$ (i.e. the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression we show that the norm of the evolution group acting from $\ell^1$ to $\ell^\infty$ is given by $(1+t^2)^{-1/2}$.

Highlights

We are concerned with the one-dimensional discrete Schrodinger equation iψ (t, n) = H0ψ(t, n), (t, n) ∈ R × N0, (1.1)associated with the Laguerre operator ⎛1 1 0 0 · · ·⎞ H0 = ⎜⎜⎜⎜⎜⎝100 ···⎟⎟⎟⎟⎟⎠ (1.2)in 2(N0)
We derive an explicit expression for the kernel of the evolution group exp(−it H0) of the discrete Laguerre operator H0 in terms of Jacobi polynomials
We show that the norm of the evolution group acting from 1 to ∞ is given by (1 + t2)−1/2

Summary

Introduction

It is a special case of a self-adjoint Jacobi operator whose generalized eigenfunctions are precisely the Laguerre polynomials explaining our name. This result in turn is based on the following explicit expression for the kernel of the evolution group e−it H0 given in terms of Jacobi polynomials (see [4,25] for the definition and basic properties): THEOREM 1.2. The proof of Theorems 1.1 and 1.2 is given and it is based on the fact that every element of the kernel of e−it H0 is a Laplace transform of a product of two Laguerre polynomials (Lemma 2.3).

Results

Conclusion