2019 A new class of Volterra-type integral equations from relativistic quantum physics
Matthias Lienert, Roderich Tumulka
J. Integral Equations Applications 31(4): 535-569 (2019). DOI: 10.1216/JIE-2019-31-4-535


We study a new kind of linear integral equations for a relativistic quantum-mechanical two-particle wave function $\psi (x_1,x_2)$, where $x_1,x_2$ are spacetime points. In the case of retarded interaction, these integral equations are of Volterra-type in the in the time variables, i.e., they involve a time integration from 0 to $t_i = x_i^0, i=1,2$. They are interesting not only in view of their applications in physics, but also because of the following mathematical features: (a) time and space variables are more interrelated than in normal time-dependent problems, (b) the integral kernels are singular, and the structure of these singularities is non-trivial, (c) they feature time delay. We formulate a number of examples of such equations for scalar wave functions and prove existence and uniqueness of solutions for them. We also point out open mathematical problems.


Download Citation

Matthias Lienert. Roderich Tumulka. "A new class of Volterra-type integral equations from relativistic quantum physics." J. Integral Equations Applications 31 (4) 535 - 569, 2019. https://doi.org/10.1216/JIE-2019-31-4-535


Published: 2019
First available in Project Euclid: 6 February 2020

zbMATH: 07169460
MathSciNet: MR4060439
Digital Object Identifier: 10.1216/JIE-2019-31-4-535

Primary: 45D05
Secondary: 45E99 , 81Q40

Keywords: Bethe-Salpeter equation. , Multi-dimensional Volterra integral equations , multi-time wave functions , relativistic quantum mechanics , Singular integral equations , time delay

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium


This article is only available to subscribers.
It is not available for individual sale.

Vol.31 • No. 4 • 2019
Back to Top