P-adic Schrödinger-type equation
type: Item
instance of: scholarly article
title: P-adic Schrödinger-type equation
zbMATH Open document ID: 0702.35219
DOI: 10.1007/BF00397056
published in: Letters in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: A Schrödinger-type equation is considered in relation to p-adic quantum mechanics. The differentiation operator is defined by \[ D\psi (x)=\int_{{\mathbb{Q}}_ p}| k|_ p{\tilde \psi}(k)\chi (-kx) dk=\frac{p^ 2}{p+1}\int_{{\mathbb{Q}}_ p}\frac{\psi (x)-\psi (y)}{| x-y|_ p^ 2} dy. \] Here \({\mathbb{Q}}_ p\) is the field of p-adic numbers, \(\psi \in L_ 2({\mathbb{Q}}_ p)\), \(\psi\) (x)\(\in C\), \({\tilde \psi}\) is the Fourier transformation of \(\psi\) (x), i.e. \({\tilde \psi}\)(k)\(=\int \chi (kx)\psi (x)dx\), \(\chi (x)=e^{2\pi i\{x\}p}\) is the additive character on \({\mathbb{Q}}_ p\), \(\{x\}_ p\) is the rational part of the p-adic number and dx is the Haar measure on \({\mathbb{Q}}_ p\). The dynamics for the p-adic free particle is described by the equation \((D_ t-D^ 2_ x)\psi =0,\) called Schrödinger-type equation. The Cauchy problem for this equation is discussed. For the Schrödinger-type equation with a potential \((D_ t-D^ 2_ x+V(| x|_ p))=0\) also a solution is presented. The p-adic quantum harmonic oscillator is defined and for it a new set for vacuum states is presented. The correspondence principle with standard quantum mechanics is also discussed.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81Q99
zbMATH DE Number: 4150953
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: p-adic quantum mechanics
zbMATH Keywords: Schrödinger-type equation
zbMATH Keywords: p-adic free particle
zbMATH Keywords: Cauchy problem
zbMATH Keywords: p-adic quantum harmonic oscillator
author: V. S. Vladimirov
author: Igor V. Volovich
MaRDI profile type: Publication
cites work: Number theory as the ultimate physical theory
cites work: Q3469302
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description: scientific article; zbMATH DE number 4150953