How to Deal with Nonlocality and
Pseudodifferential Operators. An
Example: The Salpeter Equation
A. Lattanzi
Dedicated to Prof. Decio Levi on the occasion of his 70th
birthday
Abstract The spinless (1+1)D free-particle Salpeter equation, a relativistic version
of the Schrödinger equation, is presented focusing the attention on its nonlocality
and its consequences on the structure of the solution.
Keywords Salpeter equation · Nonlocality · Operator method · Evolution
operators
1 The Spinless Salpeter Equation: An Introduction
The spinless time-dependent Salpeter equation is a relativistic version of the timedependent Schrödinger equation [1–9]. It is obtained by replacing the classical
energy–momentum relation with the relevant relativistic relation, possibly including
also some potential interactions. Contrary to the Schrödinger equation as well as to
the other wave equations in relativistic quantum mechanics, i.e. the Klein–Gordon
and the Dirac equations, the Salpeter equation has been the object of few analyses.
This is definitely because of the mathematical complexity due to the presence of a
pseudodifferential operator that implies a nonlocality increasing the difficulties to
work in the coordinate space [10, 11].
Recent literature is revealing a greater interest in the Salpeter equation [12–
29]. The reasons are mainly two: on the one hand, there are the advantages with
respect to the Klein–Gordon and Dirac equations and on the other hand, there
A. Lattanzi ()
H. Niewodnicza´ nski Institute of Nuclear Physics, Polish Academy of Sciences, Kraków, Poland
ENEA Frascati Research Centre, FSN-FUSPHYS-TSM, Frascati (Rome), Rome, Italy
e-mail: ambra.lattanzi@ifj.edu.pl
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_9
101
Pseudodifferential Operators. An
Example: The Salpeter Equation
A. Lattanzi
Dedicated to Prof. Decio Levi on the occasion of his 70th
birthday
Abstract The spinless (1+1)D free-particle Salpeter equation, a relativistic version
of the Schrödinger equation, is presented focusing the attention on its nonlocality
and its consequences on the structure of the solution.
Keywords Salpeter equation · Nonlocality · Operator method · Evolution
operators
1 The Spinless Salpeter Equation: An Introduction
The spinless time-dependent Salpeter equation is a relativistic version of the timedependent Schrödinger equation [1–9]. It is obtained by replacing the classical
energy–momentum relation with the relevant relativistic relation, possibly including
also some potential interactions. Contrary to the Schrödinger equation as well as to
the other wave equations in relativistic quantum mechanics, i.e. the Klein–Gordon
and the Dirac equations, the Salpeter equation has been the object of few analyses.
This is definitely because of the mathematical complexity due to the presence of a
pseudodifferential operator that implies a nonlocality increasing the difficulties to
work in the coordinate space [10, 11].
Recent literature is revealing a greater interest in the Salpeter equation [12–
29]. The reasons are mainly two: on the one hand, there are the advantages with
respect to the Klein–Gordon and Dirac equations and on the other hand, there
A. Lattanzi ()
H. Niewodnicza´ nski Institute of Nuclear Physics, Polish Academy of Sciences, Kraków, Poland
ENEA Frascati Research Centre, FSN-FUSPHYS-TSM, Frascati (Rome), Rome, Italy
e-mail: ambra.lattanzi@ifj.edu.pl
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_9
101