# Vacuum distribution, norm and spectral properties for sums of monotone position operators

## TL;DR

This paper analyzes the spectral and distributional properties of sums of monotone position operators on Fock space, revealing their vacuum distribution as a monotone binomial and providing bounds for spectral support.

## Contribution

It establishes the spectrum and norm of sums of monotone position operators, characterizes their vacuum distribution as a monotone binomial, and derives recurrence formulas and bounds for spectral support.

## Key findings

- Spectrum of partial sums coincides with vacuum distribution support.
- Vacuum distribution for sums is a monotone binomial distribution.
- Provides recurrence formulas and bounds for spectral support.

## Abstract

We investigate the spectrum for partial sums of m position (or gaussian) operators on monotone Fock space based on $\ell^2(\mathbb{N})$. In the basic case of the first consecutive operators, we prove it coincides with the support of the vacuum distribution. Thus, the right endpoint of the support gives their norm. In the general case, we get the last property for norm still holds. As the single position operator has the vacuum symmetric Bernoulli law, and the whole of them is a monotone independent family of random variables, the vacuum distribution for partial sums of $n$ operators can be seen as the monotone binomial with $n$ trials. It is a discrete measure supported on a finite set, and we exhibit recurrence formulas to compute its atoms and probability function as well. Moreover, lower and upper bounds for the right endpoints of the supports are given.