How the μ-deformed Segal-Bargmann space gets two measures

Stephen Bruce Sontz

Banach Center Publications (2010)

  • Volume: 89, Issue: 1, page 265-274
  • ISSN: 0137-6934

Abstract

top
This note explains how the two measures used to define the μ-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald functions arise in this theory. Also we comment on why it is plausible that only one measure will not work. We follow Bargmann's approach by imposing a condition sufficient for the μ-deformed creation and annihilation operators to be adjoints of each other. While this note uses elementary techniques, it reveals in a new way basic aspects of the structure of the μ-deformed Segal-Bargmann space.

How to cite

top

Stephen Bruce Sontz. "How the μ-deformed Segal-Bargmann space gets two measures." Banach Center Publications 89.1 (2010): 265-274. <http://eudml.org/doc/282441>.

@article{StephenBruceSontz2010,
abstract = {This note explains how the two measures used to define the μ-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald functions arise in this theory. Also we comment on why it is plausible that only one measure will not work. We follow Bargmann's approach by imposing a condition sufficient for the μ-deformed creation and annihilation operators to be adjoints of each other. While this note uses elementary techniques, it reveals in a new way basic aspects of the structure of the μ-deformed Segal-Bargmann space.},
author = {Stephen Bruce Sontz},
journal = {Banach Center Publications},
keywords = {Segal-Bargmann analysis; -deformed quantum mechanics},
language = {eng},
number = {1},
pages = {265-274},
title = {How the μ-deformed Segal-Bargmann space gets two measures},
url = {http://eudml.org/doc/282441},
volume = {89},
year = {2010},
}

TY - JOUR
AU - Stephen Bruce Sontz
TI - How the μ-deformed Segal-Bargmann space gets two measures
JO - Banach Center Publications
PY - 2010
VL - 89
IS - 1
SP - 265
EP - 274
AB - This note explains how the two measures used to define the μ-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald functions arise in this theory. Also we comment on why it is plausible that only one measure will not work. We follow Bargmann's approach by imposing a condition sufficient for the μ-deformed creation and annihilation operators to be adjoints of each other. While this note uses elementary techniques, it reveals in a new way basic aspects of the structure of the μ-deformed Segal-Bargmann space.
LA - eng
KW - Segal-Bargmann analysis; -deformed quantum mechanics
UR - http://eudml.org/doc/282441
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.