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Cauchy problem for a Schrödinger-type equation related to the Riemann zeta function

Bensaid Mohamed

Abstract

We study the Cauchy problem in the space $H^1(Σ)$ for a nonlinear damped Schrödinger equation of the form \[ i u_t + Δu + i λu \, ζ(|u|+1) = 0, \quad u(0,x) = u_0, \] where $ζ$ denotes the Riemann zeta function. This equation involves a damping term inspired by a fundamental object from number theory. We first establish the uniqueness of solutions in the sense of distributions. Then, by considering a regularized problem, we prove the existence of a global solution in $H^1(Σ)$, using uniform energy estimates and compactness arguments. Finally, we show that the limiting solution indeed satisfies the original equation in the weak sense,In the last part of this work, we provide an explicit estimate for the mass associated with the equation \begin{equation}\tag{E0}\label{5454} i v_t + Δv + iλ\frac{v}{|v|} = 0, \qquad v(0,x)=v_0.

Cauchy problem for a Schrödinger-type equation related to the Riemann zeta function

Abstract

We study the Cauchy problem in the space for a nonlinear damped Schrödinger equation of the form where denotes the Riemann zeta function. This equation involves a damping term inspired by a fundamental object from number theory. We first establish the uniqueness of solutions in the sense of distributions. Then, by considering a regularized problem, we prove the existence of a global solution in , using uniform energy estimates and compactness arguments. Finally, we show that the limiting solution indeed satisfies the original equation in the weak sense,In the last part of this work, we provide an explicit estimate for the mass associated with the equation \begin{equation}\tag{E0}\label{5454} i v_t + Δv + iλ\frac{v}{|v|} = 0, \qquad v(0,x)=v_0.

Paper Structure

This paper contains 4 sections, 19 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Construction of the Solution in $H^1(\Sigma)$
  3. NLSZeta with Logarithmic Perturbation
  4. Estimates for the Riemann Zeta Function

Key Result

Proposition 1

Let $u_1,u_2 \in L^\infty(\mathbb{R}_+;H^1(\Sigma))$ be two (distributional) solutions of E with the same initial data. If $\lambda>0$, then $u_1 \equiv u_2$.

Theorems & Definitions (42)

  • Proposition 1: Uniqueness
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 1
  • Remark 2
  • Lemma 4
  • proof
  • ...and 32 more