P-adic Schrödinger-type equation
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.
scientific article; zbMATH DE number 4150953
scientific article