# Neutrino Signatures of One Hundred 2D Axisymmetric Core-Collapse Supernova
Simulations

David Vartanyan1, Adam Burrows2
1Carnegie Observatories, 813 Santa Barbara St, Pasadena, CA 91101, USA; NASA
Hubble Fellow
2Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University,
Princeton, NJ 08544, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present in this paper a public data release of an unprecedentedly-large set
of core-collapse supernova (CCSN) neutrino emission models, comprising one
hundred detailed 2D-axisymmetric radiation-hydrodynamic simulations evolved
out to as late as $\sim$5 seconds post-bounce and spanning a extensive range
of massive-star progenitors. The motivation for this paper is to provide a
physically and numerically uniform benchmark dataset to the broader neutrino
detection community to help it characterize and optimize subsurface facilities
for what is likely to be a once-in-a-lifetime galactic supernova burst event.
With this release we hope to 1) help the international experiment and modeling
communities more efficiently optimize the retrieval of physical information
about the next galactic core-collapse supernova, 2) facilitate the better
understanding of core-collapse theory and modeling among interested
experimentalists, and 3) help further integrate the broader supernova neutrino
community.

###### keywords:

stars - supernovae - general

††pubyear: 2023††pagerange: Neutrino Signatures of One Hundred 2D Axisymmetric
Core-Collapse Supernova Simulations–Neutrino Signatures of One Hundred 2D
Axisymmetric Core-Collapse Supernova Simulations

##  1 Introduction

Core-collapse supernova theory has evolved significantly over the last six
decades (Burrows 2013; Janka 2012; Müller et al. 2017; Ott et al. 2018;
O’Connor & Couch 2018b; Vartanyan et al. 2018; Burrows et al. 2019; Burrows et
al. 2020; Glas et al. 2019; Burrows & Vartanyan 2021; Vartanyan & Burrows
2020; Vartanyan et al. 2022; Tsang et al. 2022; Wang et al. 2022) and has
reached a pivotal juncture in its history. Previously, delayed by the
complexity and computational resources necessary to address the relevant
physics in sufficient and credible multi-dimensional detail, most
sophisticated core-collapse supernova simulations now witness explosions
without artifice. The turbulent neutrino-driven mechanism, at least in broad
outline, is now the consensus mechanism for the vast majority of core-collapse
supernova explosions (Janka, 2012; Burrows, 2013; Burrows et al., 2020;
Burrows & Vartanyan, 2021). This is not to say that there is no need for
significant quantitative improvement. Moreover, there remain issues concerning
various neutrino-matter interactions, the role of neutrino oscillations, and
the actual (and determinative) structures of the massive-star progenitors.
Nevertheless, from a slight conceptual distance, the theory, though complex,
is looking realistic and can boast many detailed predictions.

Given these developments, we present in this paper a public data release of an
unprecedentedly-large set of core-collapse supernova (CCSN) neutrino emission
models, comprising one hundred detailed 2D-axisymmetric radiation-hydrodynamic
simulations evolved out to as late as $\sim$5 seconds post-bounce and spanning
a extensive range of massive-star progenitors111The full dataset can be found
at: http://www.astro.princeton.edu/~burrows/nu-emissions.2d.large/ and
https://dvartany.github.io/data/. Though some 1D simulation suites exist
(Ugliano et al., 2012; Sukhbold et al., 2016; Horiuchi et al., 2018; Suwa et
al., 2019; Kresse et al., 2021), spherical-symmetry is insufficient to capture
the convective nature of the CCSN that significantly affects the emergent
neutrinos (Dessart et al. 2006; Radice et al. 2017. Moreover, these
simulations often use explosion prescriptions rather than sophisticated
neutrino transport, or reduced the dimensionality to 1D to continue to several
seconds post-bounce (e.g. Bollig et al. 2021).

A uniform suite of neutrino simulation data is essential to ensure that the
neutrino experimental and theoretical communities are adequately equipped to
optimize the retrieval of physical information about the next galactic core-
collapse supernova. Such an event is a major secondary motivation for the
flotilla of subsurface facilities emerging (e.g., Super-Kamiokande, SK, Abe et
al. (2016) or Hyper-Kamiokande, HK, Hyper-Kamiokande Proto-Collaboration et
al. (2018); the Deep Underground Neutrino Experiment, DUNE, Ankowski et al.
(2016); Migenda (2018); the Jiangmen Underground Neutrino Observatory, JUNO,
An et al. (2016); THEIA Sawatzki et al. (2021); and IceCube, Abbasi et al.
(2011); Köpke & IceCube Collaboration (2011)). The motivation for this paper
is to provide a physically and numerically uniform benchmark dataset to this
broader neutrino detection community to help it characterize and optimize
subsurface facilities for this once-in-a-lifetime galactic epiphany.

There exists neutrino-analysis software coupling CCSN simulations with
neutrino signals, such as SNEWPY (Baxter et al., 2021) and SNOwGLoBES
(Scholberg et al., 2021; Nagakura et al., 2021b) within a collaborative
framework (e.g., Huber et al. 2022; Al Kharusi et al. 2020). Moreover, early
and joint detection of neutrinos with gravitational waves would provide
information on the global dynamics of CCSNe, the turbulent radiation-
hydrodynamic instabilities, and the compact remnant properties, providing
constraints on the nuclear equation of state (Vartanyan et al., 2019).

In Sec. 2, we summarize the methods of this paper, including the models
studied and a few simulation properties. In Sec. 3, we discuss various example
CCSN neutrino simulation diagnostics and correlations. One purpose of this
data release is to enable others to find their own diagnostics, signatures,
correlations, and systematics, but we kickstart such an effort with the modest
set of examples. In Sec. 3.1, we compare and calibrate results from analogous
2D and 3D simulations In Sec. 4 we summarize the import of this reference
neutrino dataset.

##  2 Methods

To ascertain their explodability, we originally selected a subset of one
hundred initial massive-star progenitor models at core-collapse from 9 to
26.99 M⊙ and evolved them in 2D (axisymmetry) using our state-of-the-art code
Fornax. The progenitor models between 9 to 11.75 M⊙ were selected from
Sukhbold et al. (2016) and the models between 12 and 26.99 M⊙ were selected
from Sukhbold et al. (2018). Fornax (Skinner et al., 2019) is a multi-
dimensional, multi-group radiation-hydrodynamic code constructed to study
core-collapse supernova explosions and their byproducts. It features an M1
solver (Vaytet et al., 2011) for neutrino transport with detailed neutrino
microphysics and approximate general relativity (Marek et al., 2006). We
employed for this study 12 neutrino energy groups spaced logarithmically
between 1 and 300 MeV for the electron neutrinos and to 100 MeV for the anti-
electron- and “$\nu_{\mu}$”-neutrinos (bundled as $\mu$ and $\tau$ neutrinos
and anti-neutrinos) and the SFHo nuclear equation of state Steiner et al.
(2013), broadly consistent with extant theoretical and experimental
constraints on the nuclear equation of state (Lattimer, 2023; Reed et al.,
2021; Tews et al., 2017). The progenitor models and simulations are initially
non-rotating, though some degree of rotation is naturally induced due to
fallback (Blondin & Mezzacappa, 2007; Coleman & Burrows, 2022).

The models have a resolution of 1024$\times$256 in $r$, $\theta$ with outer
radii extending between 30,000 km for the lower mass stellar progenitors out
to 100,000 km for the most massive progenitors. These models were chosen to be
representative as much as possible of the Salpeter initial mass function. They
were selected to span broadly the distribution in density profiles,
compositional interfaces, compactness, and Ertl parameter $\mu_{4}$/$M_{4}$
(Ertl et al., 2016). Generally, we truncated the black hole forming models at
$\sim$1 second post-bounce, when it was clear they had not exploded, and
continued the exploding models out to $\sim$3$-$5 seconds post-bounce. We dump
and provide the data every millisecond (ms) and angle-average the
luminosities. At the dataset URLs (see footnote and Data Availability
statement), we provide the luminosities, average neutrino energies, and RMS
neutrino energies, as well as the full neutrino spectra, for all three species
categories.

Of the 100 models, we find that 63 explode and 37 fail to do so, ultimately
resulting in black hole formation. All models that fail to do so lie between
12$-$15.38 M⊙, and within this mass range we find $\sim$70% of models do not
explode, all else being equal. Interestingly, CCSN simulations do consistently
find models in this range are less explodable (Burrows et al. 2019; Burrows et
al. 2020; Burrows & Vartanyan 2021; Summa et al. 2016; O’Connor & Couch
2018a).

The models presented comprise both the largest set of 2D-axisymmetric
radiation-hydrodynamic CCSNe simulations published to-date and are out to the
latest times for any such collection. This set is a precursor to an upcoming
release of an analogous set of 3D CCSNe simulations out to very late times.
The latter, however, requires substantially more computer time that perforce
will delay its release. Concerning the general explosion quantities (neutrino
energies and luminosities, explosion timing, and outcome), we expect roughly
similar behavior in 2D and 3D. Of course, for explosion morphologies,
energies, nucleosynthesis, and gravitational emission wave, one requires 3D
simulations. However, for general comparative and diagnostic purposes, we
suggest that 2D simulations are a robust compromise between explosion breadth
(we can do many simulations out to later times in a relatively short period of
time) and simulation fidelity. The 2D models highlighted here (though not
these neutrino data) have been published in other contexts in two earlier
papers (Wang et al., 2022; Tsang et al., 2022); these provided both
analytical/semi-analytical and machine-learning predictions of explosion
outcome.

##  3 Results: Example Trends and Correlations

The main unifying principle guiding this release is that compactness (O’Connor
& Ott, 2011) provides a natural ordering of many physical quantities. As we
will show, these include the neutrino luminosities, mean neutrino energies,
and infall accretion rates. Compactness provides a stronger monotonic
prediction of these properties than does the progenitor mass. These ideas are
not new. However, we illustrate them here using the largest set ever assembled
of detailed, two-dimensional, radiation-hydrodynamic CCSN models. In Table 1,
we summarize the 100 models, indicate which explode, and show total runtimes,
the compactness (at 1.75 M⊙), and the explosion times.

We emphasize that these correlations do not indicate compactness is a good
metric of explosion outcome, as has often been assumed (e.g., Sukhbold et al.
2018). In fact, compactness is not a monotonic indicator of explosion at all
$-$ models with low-compactness may not explode by the neutrino mechanism,
while models with high compactness may explode (Burrows et al., 2019; Burrows
et al., 2020; Wang et al., 2022). Curiously, we find here and previously in
print (Vartanyan et al., 2018; Burrows et al., 2020; Tsang et al., 2022; Wang
et al., 2022) that regions of intermediate compactness, which correspond to
models between 12 and 15 M⊙, are less likely to explode by the neutrino
mechanism (Burrows et al., 2019; Burrows et al., 2020).

In an earlier study (Tsang et al., 2022), we found that a random-forest
machine-learning approach was able to identify regions of intermediate
compactness less likely to explode with $\sim$80$\%$ accuracy, and that this
corresponded with analytical and semi-analytical estimates of explosion
outcome (Wang et al., 2022). We note that using the Silicon-Oxygen (Si/O)
interface as an explosion metric performed better at predicting explosion
outcome, with a $\sim$90$\%$ accuracy (see also Fryer 1999; Vartanyan et al.
2018; Summa et al. 2016; O’Connor & Couch 2018b).

###  3.1 2D to 3D Comparison

In Fig. 1, we plot for a series of 2D models the neutrino luminosities and
average energies, respectively, with their 3D counterparts. We emphasize that,
generally, there is less short-timescale variability in angle-averaged 3D
neutrino quantities than in the corresponding angle-averaged 2D neutrino
quantities. We associate this with the artificial axisymmetric constraint in
2D. Note also that, in 3D, we expect typical angular variations by viewing
angle of $\sim$15$\%$ in the neutrino quantities (Vartanyan & Burrows, 2020).
At later times (beyond 1$-$2 seconds post-bounce), we see deviations in the
electron neutrino and anti-neutrino luminosities for the more massive
progenitors, with 2D simulations yielding higher luminosities as a result of
the sustained higher accretion in 2D.

###  3.2 A Few Example Compactness Correlations

The compactness parameter characterizes the core structure and is defined as
(O’Connor & Ott, 2011):

$\xi_{M}=\frac{M/M_{\odot}}{R(M)/1000\,\mathrm{km}}\,,$ (1)

where the subscript $M$ denotes the interior mass coordinate at which the
compactness parameter is evaluated. For our purposes, we evaluate the
compactness parameter $\xi_{1.75}$ at $M$ = 1.75 M⊙, generally encompassing
the Si/O interface for many of the progenitor models. The compactness is often
used as an ab-initio explosion condition because it depends only on the
progenitor properties. While higher compactness is correlated with higher
luminosities, accretion rates, and remnant masses (O’Connor & Ott, 2013),
compactness does not readily lend itself as an explosion condition, and
suggestions that explosion is inhibited above a certain compactness parameter
are likely false (Burrows et al., 2020; Wang et al., 2022). In all diagnostics
indicated below, we provide ordering by both compactness and by progenitor
ZAMS mass.

In Fig. 2, we plot the mean shock radii as a function of time after bounce and
as a function of both progenitor mass and compactness. Note that all models
stall at $\sim$60 ms and at similar shock radii, with lower mass models
reviving into explosion earlier. For models that do explode, we note a
distribution of explosion time that is nearly monotonic with compactness and
ranges from $\sim$120 ms post-bounce to $\sim$600 ms post-bounce. This range
in explosion times corresponds to the timing of Si/O interface accretion $-$
for instance, model 26.99 M⊙ has an Si/O interface near 2 M⊙ at core-collapse,
and hence accretes the Si/O interface and explodes significantly later. Note
that among the models we present here between 12 and 15 M⊙ with compactness
$\sim$0.6 many do not explode. Models with both lower and higher compactness
do explode. The location of the Si/O interface often best explains the
explosion properties (Wang et al. 2022; Tsang et al. 2022); models between
12$-$15 M⊙ have weak Si/O interfaces further out whose accretion seems
insufficient to inaugurate shock revival. We also note that models which
explode later, and accrete for longer, do so generally with higher explosion
energies.

In Fig. 3, we plot the neutrino luminosity at 500 km redshifted (with both
velocity and lapse corrections) to the observer frame as a function of time
after bounce for all species (the bundled ‘heavy’-neutrino species’ luminosity
is divided by 4 to render the luminosity per species). Note that, though we
see an approximate correlation with progenitor mass, we see a stronger
monotonic correlation with compactness. We see sustained neutrino luminosities
of $\sim$1052 erg s-1 out to $\sim$4-5 seconds after bounce for all species.
Just several milliseconds prior to the $\nu_{e}$ breakout burst, there is a
smaller $\nu_{e}$ luminosity peak due to $\nu_{e}$s produced by the
neutronization of the core during collapse (Thompson et al., 2003; Burrows,
2013; Wallace et al., 2016). Typically, the peak electron-neutrino
luminosities in the pre-breakout burst are $\sim$1053 erg s-1. Just before
bounce, as the density and temperature of the core increase, the competition
between the increase in the opacity and the electron-capture rates leads to a
“mini-peak” (“pre-breakout neutronization peak”) lasting $\sim$milliseconds.
The full electron-neutrino breakout which follows is due to electron capture
off protons liberated by the nascent shock, lasts $\sim$10 ms, and has typical
peak luminosities of $\sim$4.5$\times$1053 erg s-1. Note that the peak heights
and durations for both the breakout and pre-breakout burst depend weakly on
the compactness, with higher compactness yielding a slightly longer duration
and higher peak.

In Fig. 4, we plot the electron neutrino luminosity during the plateau phase
(out to $\sim$several tenths of a second) as a function of time after bounce
for all exploding models, and overplot both the peak luminosity and the time
at which the electron neutrino luminosity drops to 60$\%$ of the peak value.
We see a strong correlation between peak luminosity and plateau duration, and
for both with compactness as well, suggesting a ‘Phillips-like’ relation for
CCSN neutrino light curves. Non-exploding models would continue to accrete for
longer times. Note that the duration of the plateau also corresponds strongly
with the explosion time $-$ explosion reverses accretion, resulting in a
noticeable drop in luminosity, often by a factor of several. We emphasize this
in Fig. 5, where we show the electron neutrino luminosity plateau duration
(defined from neutrino breakout until the luminosity drops below 3$\times$1052
erg s-1) for all exploding models, and highlight the strong correlation with
plateau duration and progenitor compactness. In multi-D, we generically expect
sustained accretion along certain directions and explosion/ejection along
others (Vartanyan et al., 2018; Burrows & Vartanyan, 2021).

In Fig. 6, we show the mean neutrino energies for all species at 500 km (again
redshifted to the observer frame). Note that we see a better correlation with
compactness than with progenitor mass. In addition, note that models here
which do not explode sustain accretion for longer and, hence, have slightly
higher mean neutrino energies. In Fig. 7, we repeat Fig. 6, but for the RMS
neutrino energy. We draw the same inferences. In Figs. 8 and 9, we provide
sample spectra for the 9.25- and 26.99-M⊙ models for all three species.

###  3.3 Temporal Fluctuations

With Fig. 10, we briefly explore the fluctuations of neutrino luminosities and
mean energies with time. Depicted is the fractional root-mean square (RMS)
variation of the neutrino luminosity around a boxcar running time-average of 5
ms. We clearly see a hierarchy with both compactness (more compact models vary
more). Furthermore, we see similar variation in time for electron neutrino and
electron anti-neutrino quantities, followed by less variation for the bundled
‘heavy’ neutrinos. Note that we found a similar hierarchy earlier (Vartanyan
et al. 2019, see also Kuroda et al. 2020), looking also at the spatial
variation of the neutrino diagnostics by species. We see less variation in the
neutrino luminosity than the mean neutrino energy.

###  3.4 PNS Mass and Integrated Neutrino Energy Loss

In Fig. 11, we plot the radiated neutrino energy (integrating the total
luminosity for all species) (Nagakura et al., 2021a, b; Nagakura & Vartanyan,
2022) out to 2.5 s post-bounce for all exploding models. We note an excellent
correlation between compactness, baryonic PNS mass, and the integrated energy
radiated. More compact models sustain longer accretion that yield higher PNS
masses and radiate greater neutrino energy. Observing the radiated neutrino
energy will provide a strong constraint on the PNS mass for CCSN detections
(see the quadratic fitting in Nagakura & Vartanyan 2022).

##  4 Conclusions

Traditionally, the supernova neutrino detection community has relied over the
decades for the calibration and benchmarking of their subsurface experiments
on a heterogeneous collection of supernova explosion models (e.g., Sawatzki et
al. 2021 and Ekanger et al. 2022, and references within their Table 1). The
latter have emerged haphazardly and rarely comprise a uniform set that might
enable the study of physical parameters of the (theoretical) explosion. In
addition, no systematic studies have emerged of the neutrino discriminants of
microphysical parameters of the neutrino-matter interaction and/or the nuclear
equation of state that do not explode models artificially. The construction
using the sophisticated numerical tools now available of such a comprehensive
forward model set would indeed be a daunting enterprise. There have been large
model sets available (e.g., Ugliano et al. 2012; Horiuchi et al. 2018; Suwa et
al. 2019; Kresse et al. 2021) employing ad hoc methods to explode 1D models
that would not otherwise explode under the current multi-D neutrino-
powered/turbulence paradigm of supernova explosions (Burrows et al., 1995;
Janka, 2012; Burrows et al., 2020; Burrows & Vartanyan, 2021). These studies
nevertheless span a useful range of truly physical parameters (such as
progenitor mass). However, again these models were not the product of a self-
consistent multi-D treatment. With this paper we have attempted to bridge the
gap between uniform model sets artificially exploded and self-consistent
multi-D (though 2D) datasets that span a wide range of at least one physical
quantity (in this case compactness and/or progenitor mass).

Such a uniform suite of neutrino signals from state-of-the-art CCSN
simulations will be, we believe, an essential resource to ensure that the
international experiment and modeling communities can most efficiently
optimize the retrieval of physical information about the next galactic core-
collapse supernova. We must emphasize that our models, though sophisticated,
should not be viewed as the final word. Far from it. They 1) do not
incorporate the possible effects of neutrino oscillations in the core itself;
2) do not address initial core rotation of significance; 3) rely on a
extensive initial progenitor model suite that is a work in progress for the
stellar evolution community; and 4) importantly are performed in axisymmetry.
Nevertheless, these one hundred multi-group radiation/hydrodynamic models
systematically span an important range of stellar masses, were conducted to
significantly later times than is traditional, and were evolved using the same
code and microphysics throughout. The result is hoped to be a large collection
of models of uniform provenance that will facilitate into the near future the
better understanding of core-collapse theory and modeling among interested
experimentalists and the further integration of the broader supernova neutrino
community.

Table 1: Explosion Properties

Model |  Run Time |  $\xi_{1.75}$ |  Explosion Time
---|---|---|---
(M⊙) |  (s, pb) |  |  (s, pb)
9.0 |  3.22 |  6.8e-5 |  0.084
9.25 |  2.51 |  0.003 |  0.078
9.5 |  2.73 |  0.009 |  0.07
9.75 |  3.36 |  0.013 |  0.072
10.0 |  2.77 |  0.017 |  0.124
10.25 |  4.48 |  0.026 |  0.08
10.5 |  4.4 |  0.041 |  0.064
10.75 |  4.37 |  0.087 |  0.068
11.0 |  4.83 |  0.125 |  0.1
11.25 |  4.19 |  0.131 |  0.096
11.5 |  4.33 |  0.137 |  0.064
11.75 |  4.82 |  0.17 |  0.122
12.00-BH |  0.94 |  0.301 |  -
12.03-BH |  0.73 |  0.395 |  -
12.07-BH |  1.02 |  0.332 |  -
12.1-BH |  3.91 |  0.297 |  -
12.13 |  4.67 |  0.249 |  0.114
12.15 |  4.65 |  0.237 |  0.128
12.18-BH |  0.99 |  0.342 |  -
12.20-BH |  0.92 |  0.322 |  -
12.25 |  4.63 |  0.236 |  0.134
12.33-BH |  0.96 |  0.344 |  -
12.40-BH |  0.94 |  0.381 |  -
12.45-BH |  0.94 |  0.376 |  -
12.50-BH |  0.96 |  0.347 |  -
12.54-BH |  0.92 |  0.371 |  -
12.60-BH |  0.91 |  0.439 |  -
12.63 |  4.29 |  0.378 |  0.174
12.70 |  4.35 |  0.366 |  0.208
12.72-BH |  0.93 |  0.372 |  -
12.75 |  3.95 |  0.39 |  0.194
12.80-BH |  0.93 |  0.368 |  -
12.85-BH |  0.95 |  0.464 |  -
12.90-BH |  0.93 |  0.362 |  -
12.93 |  4.19 |  0.475 |  0.254
12.97-BH |  0.93 |  0.411 |  -
13.00-BH |  0.95 |  0.485 |  -
13.05-BH |  0.92 |  0.51 |  -
13.11 |  4.42 |  0.41 |  0.156
13.25-BH |  0.94 |  0.527 |  -
13.27-BH |  0.92 |  0.453 |  -
13.32-BH |  0.95 |  0.542 |  -
13.40-BH |  0.95 |  0.532 |  -
13.45 |  4.31 |  0.427 |  0.156
13.50-BH |  0.98 |  0.375 |  -
13.60-BH |  0.94 |  0.525 |  -
13.75 |  4.22 |  0.502 |  0.138
13.82-BH |  0.91 |  0.529 |  -
13.90-BH |  0.93 |  0.422 |  -
13.96 |  4.32 |  0.5 |  0.174

Model |  Run Time |  $\xi_{1.75}$ |  Explosion Time
---|---|---|---
(M⊙) |  (s, pb) |  |  (s, pb)
14.01 |  4.24 |  0.486 |  0.174
14.13-BH |  1.0 |  0.436 |  -
14.25-BH |  0.98 |  0.418 |  -
14.40-BH |  0.89 |  0.552 |  -
14.41-BH |  0.92 |  0.493 |  -
14.43 |  4.56 |  0.345 |  0.094
14.44-BH |  0.86 |  0.529 |  -
14.70-BH |  0.92 |  0.566 |  -
14.87-BH |  0.92 |  0.59 |  -
15.00-BH |  1.1 |  0.592 |  -
15.01 |  4.69 |  0.289 |  0.118
15.04-BH |  0.92 |  0.617 |  -
15.05 |  3.8 |  0.29 |  0.118
15.38-BH |  1.11 |  0.626 |  -
16.43 |  4.15 |  0.577 |  0.168
16.65 |  4.0 |  0.738 |  0.202
16.99 |  3.74 |  0.731 |  0.208
17.00 |  3.88 |  0.744 |  0.204
17.07 |  4.01 |  0.749 |  0.332
17.10 |  4.01 |  0.668 |  0.212
17.40 |  4.5 |  0.285 |  0.156
17.48 |  4.6 |  0.222 |  0.16
17.50 |  3.87 |  0.797 |  0.272
17.51 |  4.41 |  0.389 |  0.148
17.83 |  4.4 |  0.389 |  0.148
18.04 |  3.63 |  0.824 |  0.262
18.05 |  4.32 |  0.416 |  0.174
18.09 |  4.28 |  0.306 |  0.164
18.10 |  3.85 |  0.801 |  0.274
18.50 |  3.84 |  0.809 |  0.566
19.02 |  3.89 |  0.433 |  0.15
19.56 |  2.01 |  0.851 |  0.39
19.83 |  1.63 |  0.859 |  0.152
19.99 |  1.98 |  0.869 |  0.374
20.08 |  4.02 |  0.716 |  0.208
20.09 |  1.41 |  0.853 |  0.428
20.18 |  3.62 |  0.842 |  0.436
20.37 |  3.62 |  0.822 |  0.288
21.00 |  1.54 |  0.841 |  0.402
21.68 |  1.52 |  0.84 |  0.164
22.00 |  3.83 |  0.776 |  0.37
22.30 |  3.76 |  0.82 |  0.192
22.82 |  4.07 |  0.738 |  0.324
23.00 |  3.98 |  0.744 |  0.146
23.04 |  3.84 |  0.75 |  0.352
23.43 |  4.03 |  0.701 |  0.382
24.00 |  3.88 |  0.772 |  0.248
25.00 |  3.71 |  0.802 |  0.206
26.00 |  3.87 |  0.786 |  0.242
26.99 |  3.67 |  0.811 |  0.57

Table of our 2D simulation results. We include the model name (indicating
black hole formers with -BH), the run time in seconds post-bounce, the
compactness $\xi_{1.75}$ at 1.75 M⊙, and the explosion time in seconds post-
bounce.

Figure 1: 2D and 3D comparison of the neutrino luminosity at 500 km in the
observer frame for all three species for exploding models 11- and 23-M⊙.

Figure 2: Mean shock radii (in km) as a function of time after bounce (in
seconds) for the 100 models explored here as a function of both progenitor
mass (left) and compactness (right). Note that, when ordering by compactness,
we see generally that low- and high-compactness models explode, whereas some
intermediate ones fail to do so (see also Wang et al. 2022; Tsang et al.
2022).

Figure 3: Neutrino luminosity in the observer frame for the three ensemble
species at 500 km as a function of time after bounce for our set of 100 2D
axisymmetric simulations, colored by progenitor mass (left), and by
compactness at 1.75 M⊙ (right). Figure 4: Electron neutrino luminosity for all
exploding models. We overplotted as stars where the plateau luminosity is its
maximum, and in circles where the luminosity is 60$\%$ of peak. Note the
strong correlation between peak luminosity, plateau duration, and compactness.
Figure 5: The plateau duration (in seconds) for the electron neutrino
luminosity for all exploding models as a function of compactness and colored
by progenitor mass. The plateau duration is defined from the end of breakout
until when the electron neutrino luminosity drops below 3$\times$1052 erg s-1.
Note the strong correlation between compactness and plateau duration; more
compact models sustain longer accretion and higher neutrino luminosities.

Figure 6: Average neutrino energy in the observer frame for the three ensemble
species at 500 km as a function of time after bounce (in seconds) as a
function of progenitor mass (left) and compactness (right).

Figure 7: Same as Fig. 6, but for the RMS neutrino energy.

Figure 8: Neutrino spectra for all species as a function of time after bounce
for the 9.25 M⊙ progenitor. We have splined our energy group gridding for
these plots.

Figure 9: Same as Fig. 8, but for the 26.99 M⊙ progenitor.

Figure 10: Fractional RMS fluctuation of the neutrino luminosity and the
average neutrino energy around a boxcar mean of 5 ms . Note the we see a
hierarchy with both compactness (more compact models evince greater
fluctuation) and species (electron neutrinos have the most variation, followed
by electron anti-neutrinos, and then the bundled ‘heavy’ neutrinos). We also
see greater fluctuation in the neutrino luminosity than in the mean neutrino
energy. Figure 11: Baryonic PNS masses (M⊙) versus the total emitted neutrino
energy ( 1052 erg) out to 2.5 s post-bounce for all exploding models. Note
that we see a strong correlation between the integrated energy loss, the
compactness, and the baryonic PNS mass.

## Data Availability

The data underlying this article are publicly available at
https://dvartany.github.io/data/ and
http://www.astro.princeton.edu/~burrows/nu-emissions.2d.large/.

## Acknowledgements

We thank Benny Tsang, Tianshu Wang, Hiroki Nagakura, Matthew Coleman, and
Christopher White for valuable discussion throughout the course of this
project. DV acknowledges support from the NASA Hubble Fellowship Program grant
HST-HF2-51520. AB acknowledges support from the U. S. Department of Energy
Office of Science and the Office of Advanced Scientific Computing Research via
the Scientific Discovery through Advanced Computing (SciDAC4) program and
Grant DE-SC0018297 (subaward 00009650) and support from the U. S. National
Science Foundation (NSF) under Grants AST-1714267 and PHY-1804048 (the latter
via the Max-Planck/Princeton Center (MPPC) for Plasma Physics). The three-
dimensional simulations were performed on Blue Waters under the sustained-
petascale computing project, which was supported by the National Science
Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois.
Blue Waters was a joint effort of the University of Illinois at
Urbana–Champaign and its National Center for Supercomputing Applications. We
also acknowledge access to the Frontera cluster (under awards AST20020 and
AST21003), and this research is part of the Frontera computing project at the
Texas Advanced Computing Center (Stanzione et al., 2020). Frontera is made
possible by NSF award OAC-1818253. Additionally, a generous award of computer
time was provided by the INCITE program, enabling this research to use
resources of the Argonne Leadership Computing Facility, a DOE Office of
Science User Facility supported under Contract DE-AC02-06CH11357. Finally, the
authors acknowledge computational resources provided by the high-performance
computer center at Princeton University, which is jointly supported by the
Princeton Institute for Computational Science and Engineering (PICSciE) and
the Princeton University Office of Information Technology, and our continuing
allocation at the National Energy Research Scientific Computing Center
(NERSC), which is supported by the Office of Science of the U. S. Department
of Energy under contract DE-AC03-76SF00098.

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# Multivariable pseudospectrum in $C^{*}$ algebras

Alexander Cerjan<EMAIL_ADDRESS>Center for Integrated Nanotechnologies,
Sandia National Laboratories, Albuquerque, New Mexico 87185, USA  ,  Vasile
Lauric<EMAIL_ADDRESS>Department of Mathematics, 316 Jackson-Davis
Hall, 1617 S Martin Luther King Jr. Blvd, Tallahassee, Florida 32307, USA  and
Terry A. Loring<EMAIL_ADDRESS>Department of Mathematics and Statistics,
University of New Mexico, Albuquerque, New Mexico, 87123, USA

###### Abstract.

We look at various forms of spectrum and associated pseudospectrum that can be
defined for noncommuting $d$-tuples of Hermitian elements of a
$C^{*}$-algebra. The emphasis is on theoretical calculations of examples, in
particular for noncommuting pairs and triple of operators on infinite
dimensional Hilbert space. In particular, we look at the universal pair of
projections in a $C^{*}$-algebra, the usual position and momentum operators,
and triples of tridiagonal operators.

###### Key words and phrases:

Pseudospectrum, noncommutative, linear operator

###### 1991 Mathematics Subject Classification:

47A13, 46L05

##  1\. Introduction

Given $\mathbb{A}=(A_{1},\dots,A_{d})$, a noncommuting $d$-tuple of bounded
linear Hermitian operators on Hilbert space, so
$A_{j}=A_{j}^{*}\in\mathcal{B}(\mathcal{H})$, there are many competing notions
of a joint spectrum. The one that has been involved in recent developments in
many areas of physics [3, 4, 8, 9, 11, 12, 20, 25] is the Clifford spectrum
$\Lambda(\mathbb{A})$. This is a closed, bounded subset of $\mathbb{R}^{d}$.
To define it, we define first the _spectral localizer_

$L_{\boldsymbol{\lambda}}(\mathbb{A})=\sum_{j=1}^{d}(A_{j}-\lambda_{j}I)\otimes\Gamma_{j}.$

Here $(\Gamma_{1},\dots,\Gamma_{d})$ are matrices that satisfy the relations

$\displaystyle\Gamma_{j}^{*}$ $\displaystyle=\Gamma_{j},\quad(j=1,\dots,d)$
$\displaystyle\Gamma_{j}^{2}$ $\displaystyle=1,\quad(j=1,\dots,d)$
$\displaystyle\Gamma_{j}\Gamma_{k}$
$\displaystyle=-\Gamma_{k}\Gamma_{j},\quad(j\neq k)$

that we call a _representation of the Clifford relations_.

It does not matter what representation we use [5, Lemma 1.2], so generally we
take an irreducible representation, which means these matrices are of size
$2^{\left\lfloor d/2\right\rfloor}$. For $d=2$ a standard choice is

(1.1) $\Gamma_{1}=\sigma_{x},\,\Gamma_{2}=\sigma_{y}$

and for $d=3$ standard choice is

(1.2) $\Gamma_{1}=\sigma_{x},\,\Gamma_{2}=\sigma_{y},\,\Gamma_{3}=\sigma_{z}.$

Here we are using the Pauli spin matrices

$\sigma_{x}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\,\sigma_{y}=\begin{bmatrix}0&-i\\\
i&0\end{bmatrix},\,\sigma_{z}=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}.$

The _Clifford spectrum_ is then

(1.3)
$\Lambda(\mathbb{A})=\left\\{\left.\boldsymbol{\lambda}\in\mathbb{R}^{d}\right|L_{\boldsymbol{\lambda}}(\mathbb{A})\text{
is not invertible}\right\\}.$

We will see that an interesting subset of this is the _essential Clifford
spectrum_ , defined as

$\Lambda^{e}(\mathbb{A})=\left\\{\left.\boldsymbol{\lambda}\in\mathbb{R}^{d}\right|L_{\boldsymbol{\lambda}}(\mathbb{A})\text{
is not Fredholm}\right\\}.$

From a mathematical point of view, there is much that is unknown about the
Clifford spectrum. We are not even sure if it can be the empty set [5, §8],
although a lot of evidence says this does not happen. Most of the examples
examined so far have required computer calculations, so most examples have
been finite-dimensional. Here we move on to infinite-dimensional examples. In
most cases, we work inside the Toeplitz algebra, so that we work with easy
$C^{*}$-algebras inside the Calkin algebra to calculate most of the Clifford
spectrum, and then reduce the search for the rest of the Clifford spectrum to
a calculation involving some difference equations. We are then able to find
infinite-dimensional examples where the Clifford spectrum looks very different
from the Clifford spectra that arise from similar examples finite-dimensions.

If $\mathbb{A}=(A_{1},\dots,A_{d})$, where each $A_{j}$ is an Hermitian
element of a unital $C^{*}$-algebra $\mathcal{A}$, the definition of Clifford
spectrum still makes sense. Taking the $\Gamma_{j}$ to be matrices in
$M_{2^{\left\lfloor d/2\right\rfloor}}(\mathbb{C})$, we treat
$L_{\boldsymbol{\lambda}}(\mathbb{A})$ as an element of $M_{2^{\left\lfloor
d/2\right\rfloor}}(\mathcal{A})\cong\mathcal{A}\otimes M_{2^{\left\lfloor
d/2\right\rfloor}}(\mathbb{C})$. We now are treating $I$ as the identity
element of $\mathcal{A}$. As $\Lambda(\mathbb{A})$ is defined in Eq. 1.3, in
term of invertiblity, we find we have spectral permanence. That is, if
$\mathcal{A}$ is a unital $C^{*}$-subalgebra of $\mathcal{B},$ then we get the
same result if we compute $\Lambda(\mathbb{A})$ working in $M_{2^{\left\lfloor
d/2\right\rfloor}}(\mathcal{B})$ as we do if working in $M_{2^{\left\lfloor
d/2\right\rfloor}}(\mathcal{A})$.

##  2\. Symmetries and $*$-homomorphisms

We collect here some basic lemmas on the Clifford spectrum in
$C^{*}$-algebras. We already discussed spectral permanence, so we already know
the behavior of the Clifford spectrum with respect to an embedding of
$C^{*}$-algebras.

######  Theorem 2.1.

Suppose that $\varphi:\mathcal{A}\rightarrow\mathcal{B}$ is a unital
$*$-homomorphism between unital $C^{*}$-algebras. If $A_{1},\dots,A_{d}$ are
Hermitian elements of $\mathcal{A}$ then

(2.1)
$\Lambda(\varphi(A_{1}),\dots,\varphi(A_{d}))\subseteq\Lambda(A_{1},\dots,A_{d}).$

If $\varphi$ is one-to-one, then the inclusion in Equation 2.1 becomes an
equality.

###### Proof.

Since

$\left(\varphi\otimes
I\right)\left(L_{\boldsymbol{\lambda}}\left(A_{1},\dots,A_{d}\right)\right)=L_{\boldsymbol{\lambda}}\left(\varphi(A_{1}),\dots,\varphi(A_{d})\right),$

we know that if $L_{\boldsymbol{\lambda}}\left(A_{1},\dots,A_{d}\right)$ is
invertible then
$L_{\boldsymbol{\lambda}}\left(\varphi(A_{1}),\dots,\varphi(A_{d})\right)$ is
also invertible. ∎

######  Corollary 2.2.

If $A_{1},\dots,A_{d}$ in $\mathcal{B}(\mathcal{H}$) are Hermitian and $U$ is
a unitary operator on $\mathcal{H}$ then

$\Lambda(UA_{1}U^{*},\dots,UA_{d}U^{*})=\Lambda(A_{1},\dots,A_{d})$

and

$\Lambda^{\textup{e}}(UA_{1}U^{*},\dots,UA_{d}U^{*})=\Lambda^{\textup{e}}(A_{1},\dots,A_{d}).$

For many purposes, such as proving that a symmetry in $\mathbb{A}$ leads to a
symmetry in $\Lambda(\mathbb{A})$, we need to know that we have complete
flexibility in selecting the $\Gamma_{j}$.

######  Lemma 2.3.

Suppose $\Gamma_{1},\dots,\Gamma_{d}$ form a representation of the Clifford
relations in $\boldsymbol{M}_{r}(\mathbb{C})$ and
$\Gamma^{\prime}_{1},\dots,\Gamma^{\prime}_{d}$ form a representation of the
Clifford relations in $\boldsymbol{M}_{s}(\mathbb{C})$. If $A_{1},\dots,A_{d}$
are Hermitian elements of unital $C^{*}$-algebra $\mathcal{A}$ then

$\sum_{j=1}^{d}\left(A_{j}-\lambda_{j}\right)\otimes\Gamma_{j}$

is invertible if, and only if,

$\sum_{j=1}^{d}\left(A_{j}-\lambda_{j}\right)\otimes\Gamma^{\prime}_{j}$

is invertible.

The proof of this lemma is essentially the same as the proof of [5, Lemma 1.2]
and is omitted. We need also the following lemmas from [5], also generalized
to the $C^{*}$-algebra setting. Again the proofs are almost identical to the
matrix case and are omitted. We will denote by $O(d)$ the real-valued
orthogonal matrices of size $d.$

######  Lemma 2.4.

Suppose $\left(A_{1},\dots,A_{d}\right)$ is a $d$-tuple of Hermitian elements
of unital $C^{*}$-algebra $\mathcal{A}$ and that $U\in O(d)$. Suppose
$\boldsymbol{\mathbb{\lambda}}\in\mathbb{R}^{d}$. The $d$ elements

$\hat{A}_{j}=\sum_{s}u_{js}A_{s}$

are also Hermitian and

$\boldsymbol{\mathbb{\lambda}}\in\Lambda\left(A_{1},\dots,A_{d}\right)\iff
U\boldsymbol{\mathbb{\lambda}}\in\Lambda(\hat{A}_{1},\dots,\hat{A}_{d}).$

######  Theorem 2.5.

Suppose $\left(A_{1},\dots,A_{d}\right)$ are Hermitian elements in the unital
$C^{*}$-algebra $\mathcal{A}$ and that $U\in O(d)$. Let

$\hat{A}_{j}=\sum_{s}u_{js}A_{s}.$

If there exists a unitary $Q$ in $\mathcal{A}$ such that
$Q\hat{A_{j}}Q^{*}=A_{j}$ for all $j$ then

$\boldsymbol{\mathbb{\lambda}}\in\Lambda\left(A_{1},\dots,A_{d}\right)\iff
U\boldsymbol{\mathbb{\lambda}}\in\Lambda\left(A_{1},\dots,A_{d}\right).$

##  3\. The commuting and essentially commuting cases

We know that a single operator’s spectrum can look very different in the
infinite-dimensional case when compared with the spectrum of a finite-
dimensional counterpart. In the finite-dimensional case, any operator (or
matrix) $T$ will have finite spectrum. In contrast, in infinite dimensions we
can get any nonempty closed and bounded subset of the complex plane. This
phenomenon gives us our first examples where the Clifford spectrum looks
different from how things looked in finite dimensions [10, 28].

In the case $d=2$ the Clifford spectrum is a minor variation on the ordinary
spectrum. With the standard $\Gamma$ matrices, as in Eq. 1.1, the spectral
localizer is

$L_{(x,y)}(A_{1},A_{2})=\left[\begin{array}[]{cc}0&A_{1}-iA_{2}-(x-iy)I\\\
A_{1}+iA_{2}-(x+iy)I&0\end{array}\right].$

This tells us immediately that $(x,y)\in\Lambda(A_{1},A_{2})$ exactly when
$x+iy$ is in the ordinary spectrum of $A_{1}+iA_{2}$. For example, we can have
$A_{1}+iA_{2}$ be the bilateral shift and so have an example where

(3.1) $\Lambda(A_{1},A_{2})=\mathbb{T}^{1}.$

In finite dimensions, we have conjectured [5, §8] that one-dimensional
manifolds cannot arise as the Clifford spectrum of three matrices. In infinite
dimensions, we can get the circle as the Clifford spectrum of three operators
with commutative and noncommutative examples.

######  Lemma 3.1.

If $A_{1},\dots,A_{d}$ are pair-wise commuting Hermitian elements of a unital
$C^{*}$-algebra $\mathcal{A}$ then the Clifford spectrum of
$\mathbb{A}=(A_{1},\dots,A_{d})$ equals the standard joint spectrum.

###### Proof.

Because of spectral permanence and Corollary 2.2, we can assume that
$\mathcal{A}=C(X)$ for some compact Hausdorff space and that $A_{j}=f_{j}$ for
some continuous $f_{j}:X\rightarrow\mathbb{R}$. In this case, we can work
pointwise and we find

$L_{\boldsymbol{\lambda}}(\mathbb{A})=g:X\rightarrow M_{2^{\left\lfloor
d/2\right\rfloor}}(\mathbb{C})$

where

$g(x)=\sum_{j=1}^{d}(f_{j}(x)-\lambda_{j})\Gamma_{j}.$

An easy calculation shows that, for scalars $\alpha_{j}$,

$\sigma\left(\sum\alpha_{j}\Gamma_{j}\right)=\left\\{\pm\sqrt{\sum\alpha_{j}^{2}}\right\\}.$

Thus $g$ is singular only when there is a point $x_{0}$ in $X$ such that
$f_{j}(x_{0})=\lambda_{j}$ for all $j$. Thus the Clifford spectrum is just the
joint spectrum. ∎

######  Example 3.2.

If $X$ is a compact nonempty subset of $\mathbb{R}^{d}$ then there is a
commutative example of $d$ Hermitian operators on separable Hilbert space
whose Clifford spectrum equals $X$. Notice that $X$ is metrizable so $C(X)$ is
separable and so can be represented on a separable Hilbert space.

######  Theorem 3.3.

Suppose $A_{1},\dots,A_{d}$ are Hermitian elements of a unital $C^{*}$-algebra
$\mathcal{A}$ and that $1\leq r<d$. If $A_{j}$ commutes with $A_{k}$ whenever
$j\leq r$ and $k>r$ then

$\Lambda(A_{1},\cdots,A_{d})\subseteq\Lambda(A_{1},\cdots,A_{r})\times\Lambda(A_{r+1},\cdots,A_{d}).$

###### Proof.

We always have

$\left(L_{\boldsymbol{\lambda}}(A_{1},\cdots,A_{d})\right)^{2}=\sum(A_{j}-\lambda_{j})^{2}\otimes
I+\sum_{j<k}\left[A_{j},A_{k}\right]\otimes\Gamma_{j}\Gamma_{k}$

but with the given assumptions many commutators vanish. Here we obtain

$\left(L_{\boldsymbol{\lambda}}(A_{1},\cdots,A_{d})\right)^{2}=\sum_{j=1}^{d}(A_{j}-\lambda_{j})^{2}\otimes
I+\sum_{j<k\leq
r}\left[A_{j},A_{k}\right]\otimes\Gamma_{j}\Gamma_{k}+\sum_{r\leq
j<k}\left[A_{j},A_{k}\right]\otimes\Gamma_{j}\Gamma_{k}$

which implies

(3.2)
$\left(L_{\boldsymbol{\lambda}}(A_{1},\cdots,A_{d})\right)^{2}=\left(L_{(\lambda_{1},\dots,\lambda_{r})}(A_{1},\cdots,A_{r})\right)^{2}+\left(L_{(\lambda_{r+1},\dots,\lambda_{d})}(A_{r+1},\cdots,A_{d})\right)^{2}.$

If $(\lambda_{1},\dots,\lambda_{r})\notin\Lambda(A_{1},\cdots,A_{r})$ then
there is a positive $a$ such that

$a\leq\left(L_{(\lambda_{1},\dots,\lambda_{r})}(A_{1},\cdots,A_{r})\right)^{2}.$

We always have

$0\leq\left(L_{(\lambda_{r+1},\dots,\lambda_{d})}(A_{r+1},\cdots,A_{d})\right)^{2}$

and so

$a\leq\left(L_{\boldsymbol{\lambda}}(A_{1},\cdots,A_{d})\right)^{2}.$

Thus

$(\lambda_{1},\dots,\lambda_{r})\notin\Lambda(A_{1},\cdots,A_{r})\implies(\lambda_{1},\dots,\lambda_{d})\notin\Lambda(A_{1},\cdots,A_{d}).$

By symmetry,

$(\lambda_{r+1},\dots,\lambda_{d})\notin\Lambda(A_{r+1},\cdots,A_{d})\implies(\lambda_{1},\dots,\lambda_{d})\notin\Lambda(A_{1},\cdots,A_{d}).$

∎

Notice that the reverse inclusion in Thereom 3.3 is already false in the case
where all the $A_{j}$ commute. We do get equality in a simple special case.

######  Theorem 3.4.

Suppose $A_{1},\dots,A_{d-1}$ are Hermitian elements of a unital
$C^{*}$-algebra $\mathcal{A}$. For any real scalar $\alpha$ we have

$\Lambda(A_{1},\cdots,A_{d-1},\alpha
I)=\Lambda(A_{1},\cdots,A_{d-1})\times\\{\alpha\\}.$

###### Proof.

Equation 3.2 here becomes

$\left(L_{\boldsymbol{\lambda}}(A_{1},\cdots,A_{d-1},\alpha
I)\right)^{2}=\left(L_{(\lambda_{1},\dots,\lambda_{d-1})}(A_{1},\cdots,A_{d-1})\right)^{2}+\left(\alpha-\lambda_{d}\right)^{2}I$

and the result follows. ∎

The following provides more evidence to support the conjecture [5] that when
the Clifford spectrum of a $d$-tuple of $n$-by-$n$ matrices is finite, it must
have cardinality at most $n$.

######  Theorem 3.5.

If $A_{1}$, $A_{2}$ and $A_{3}$ are $n$-by-$n$ Hermitian matrices and $A_{1}$
commutes with both $A_{2}$ and $A_{3}$ then the Clifford spectrum of
$(A_{1},A_{2},A_{3})$ is a set containing at most $n$ points.

###### Proof.

First we look at the special case where the first matrix is scalar. Here
Theorem 3.4 tells us that $\Lambda(\alpha I,A_{2},A_{3})$ equals
$\\{\alpha\\}\times\sigma(A_{2}+iA_{3})$ which can have no more than $n$
points.

In the general case, let $\alpha_{1},\dots,\alpha_{r}$ denote the distinct
eigenvalues of $A_{1}$. We can conjugate all the matrices by a unitary to
ensure that

$A_{1}=\left[\begin{array}[]{cccc}\alpha_{1}I\\\ &\alpha_{2}I\\\ &&\ddots\\\
&&&\alpha_{r}I\end{array}\right].$

The fact that the other matrices commute with $A_{1}$ forces them to be block
diagonal,

$A_{2}=\left[\begin{array}[]{cccc}B_{1}\\\ &B_{2}\\\ &&\ddots\\\
&&&B_{r}\end{array}\right],\quad A_{3}=\left[\begin{array}[]{cccc}C_{1}\\\
&C_{2}\\\ &&\ddots\\\ &&&C_{r}\end{array}\right].$

Therefore

$\Lambda(A_{1},A_{2},A_{3})=\bigcup_{j=1}^{r}\Lambda(\alpha_{j}I,B_{j},C_{j})$

and the result follows. ∎

##  4\. Singular values in infinite dimensions

The spectrum of a matrix is generally less informative when that matrix is not
normal. In essence, this is why we cannot build as wide-ranging functional
calculus in the nonnormal case as in the normal case. Some additional
information can be found in the pseudospectum of a nonnormal matrix. The
pseudospectrum [29] of a square matrix $X$ is based on look formally at the
function

(4.1) $\alpha\mapsto\left\|\left(X-\alpha I\right)^{-1}\right\|^{-1}$

which takes on a gradation of values for $\alpha\in\mathbb{C}$, including $0$
by default when $X-\lambda$ is not invertible. This function can be seen as
the pseudospectrum, but traditionally one looked at the inverse image of
$[0,\epsilon)$ and called that the $\epsilon$-pseudospectrum.

In applied math, the pseudospectrum is only defined for a single, typically
non-normal, square matrix. One can easily translate this into a theory
applying to two Hermitian matrices by considering $X=A_{1}+iA_{2}$. The norm
in Eq. 4.1 can be interpreted in many ways. Here we are only interested in the
operator norm. We will find in more convenient to compute $\|X^{-1}\|^{-1}$ as
the smallest singular value of a matrix. Various papers have looked at the
pseudospectrum of a single nonnormal operator on separable Hilbert space,
including [2, 15, 16] in operator theory and [18] in physics.

We need a replacement for the smallest singular value of a non-square matrix
that works for a bounded linear operator
$T:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}.$ For now, we are content to deal
with the bounded linear operators on separable Hilbert space. In the finite-
dimensional case, one characterization of the smallest singular value is the
minimum value of $\|T\boldsymbol{v}\|$ as $\boldsymbol{v}$ ranges over all
unit vectors. We take this as a definition in the infinite-dimensional case,
except we use infimum,

$s_{\min}(T)=\inf_{\|\boldsymbol{v}\|=1}\|T\boldsymbol{v}\|.$

In the special case of a compact operator, there is a spectral decomposition
[13] and so all singular values are defined. We need only the smallest, and of
course the largest since we are looking at the spectral norm.

######  Lemma 4.1.

If $T:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ is invertible, then

$s_{\min}(T)=\|T^{-1}\|^{-1}.$

###### Proof.

By homogeneity we can compute $s_{\min}(T)$ as

$\displaystyle s_{\min}(T)$ $\displaystyle=\inf_{\boldsymbol{v}\neq
0}\frac{\|T\boldsymbol{v}\|}{\|\boldsymbol{v}\|}.$

If $T\boldsymbol{v}=\boldsymbol{w}$ then

$\left(\frac{\left\|T\boldsymbol{v}\right\|}{\left\|\boldsymbol{v}\right\|}\right)^{-1}=\frac{\left\|T^{-1}\boldsymbol{w}\right\|}{\left\|\boldsymbol{w}\right\|}$

and the result now follows. ∎

######  Lemma 4.2.

Suppose that $T:\mathcal{H}\rightarrow\mathcal{H}$ is a bounded linear
operator. If $T$ is normal, then

$s_{\min}(T)=\min\left\\{\left.|\lambda|\,\right|\lambda\in\sigma(T)\right\\}.$

###### Proof.

Suppose $\lambda$ is an element in the spectrum of $T$ of smallest absolute
value. Since $T$ is normal, $\lambda$ is in the approximate point spectrum so
there is a sequence of unit vectors $\boldsymbol{v}_{n}$ such that

$\|T\boldsymbol{v}_{n}-\lambda\boldsymbol{v}_{n}\|\leq\frac{1}{n}.$

Thus

$\|T\boldsymbol{v}_{n}\|\leq|\lambda|+\frac{1}{n}$

and so $s_{\min}(T)\leq|\lambda|$.

Now we prove the opposite inequality. Since $s_{\min}(T)$ and $\sigma(T)$ are
invariant under conjugation by a unitary, we can use the spectral theorem [14]
to reduce to the case where $\mathcal{H}=L^{2}(X,\mu)$ and there is some
bounded measurable $f:X\rightarrow\mathbb{C}$ so that $T\xi(x)=f(x)\xi(x)$.
Let $[a,b]$ denoted the smallest closed interval that contains the essential
range of $|f|$. Then $a$ is the smallest element in the spectrum of $|f|$,
which by the spectral mapping theorem equals the smallest absolute value of an
element of the spectrum of $f$. Since

$\left\|T\xi\right\|^{2}=\int|f(x)|^{2}|\xi(x)|^{2}\,dx\geq
a^{2}\int|\xi(x)|^{2}\,dx=a^{2}\left\|\xi\right\|^{2}$

we know that $s_{\min}(T)\geq a$. ∎

Lemma 4.2 allows us to extend the definition of $s_{\min}$ to normal elements
of a $C^{*}$-algebra. In particular, we can now extend the definition of the
Clifford pseudospectrum for the $C^{*}$-algebra setting.

######  Definition 4.3.

Suppose $A_{1},\dots,A_{d}$ are Hermitian elements of a unital $C^{*}$-algebra
$\mathcal{A}$. The _Clifford pseudospectrum_ for
$\mathbb{A}=(A_{1},\dots,A_{d})$ is the function
$\boldsymbol{\lambda}\mapsto\mu_{\boldsymbol{\lambda}}^{\textup{C}}(\mathbb{A})$
where

$\mu_{\boldsymbol{\lambda}}^{\textup{C}}(\mathbb{A})=s_{\min}\left(L_{\boldsymbol{\lambda}}(\mathbb{A})\right).$

######  Lemma 4.4.

Suppose $S,T:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ are both bounded
linear operators, then

$\left|s_{\min}(S)-s_{\min}(T)\right|\leq\left\|S-T\right\|.$

###### Proof.

Let $\epsilon>0$ be given. Then there is a unit vector $\boldsymbol{v}$ so
that

$\|T\boldsymbol{v}\|\leq s_{\min}(T)+\epsilon.$

Then

$s_{\min}(S)\leq\|S\boldsymbol{v}\|\leq\|T\boldsymbol{v}\|+\|(S-T)\boldsymbol{v}\|\leq
s_{\min}(T)+\epsilon+\|S-T\|$

proving

$s_{\min}(S)-s_{\min}(T)\leq\|S-T\|+\epsilon.$

As this is true for all $\epsilon>0$, we have shown

$s_{\min}(S)-s_{\min}(T)\leq\|S-T\|.$

We are done, by symmetry. ∎

######  Remark 4.5.

Unlike the case in finite dimensions, $s_{\min}(T)\neq 0$ does not always
imply $T$ is invertible. However, when $T^{*}=T$ it is true that
$s_{\min}(T)\neq 0$ if and only if $T^{-1}$ exists.

######  Lemma 4.6.

If $T:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ and
$S:\mathcal{H}_{2}\rightarrow\mathcal{H}_{3}$ are bounded linear operators
then

$\displaystyle s_{\min}(ST)$ $\displaystyle\geq s_{\min}(S)s_{\min}(T)$

###### Proof.

Recall that

$\displaystyle s_{\min}(T)$ $\displaystyle=\inf_{\boldsymbol{v}\neq
0}\frac{\|T\boldsymbol{v}\|}{\|\boldsymbol{v}\|}.$

If $T$ has no kernel then, for every nonzero $\boldsymbol{v}$, we have

$\frac{\|ST\boldsymbol{v}\|}{\|\boldsymbol{v}\|}=\frac{\|ST\boldsymbol{v}\|}{\|T\boldsymbol{v}\|}\frac{\|T\boldsymbol{v}\|}{\|\boldsymbol{v}\|}\geq
s_{\min}(S)s_{\min}(T)$

so

$\displaystyle s_{\min}(ST)$ $\displaystyle\geq s_{\min}(S)s_{\min}(T).$

If $T$ has a kernel then $ST$ has a kernel so both $s_{\min}(ST)$ and
$s_{\min}(T)$ are zero and we are done for trivial reasons. ∎

######  Remark.

One might think that $T$ and $T^{*}$ and

$\left[\begin{array}[]{cc}0&T^{*}\\\ T&0\end{array}\right]$

all have the same $s_{\min}$. After all, this is true in the finite-
dimensional case. However, if we let $T$ be the forward shift, then
$s_{\min}(T)=1$ since
$\left\|T\boldsymbol{v}\right\|=\left\|\boldsymbol{v}\right\|$ for all
$\boldsymbol{v}$ (that is, $T$ is an isometry). In contrast, there is a unit
vector $\boldsymbol{v}$ so that that the backwards shift sends
$\boldsymbol{v}$ so zero, so that $\left\|T^{*}\boldsymbol{v}\right\|=0$. Thus
also

$\left[\begin{array}[]{cc}0&T^{*}\\\
T&0\end{array}\right]\left[\begin{array}[]{c}0\\\
\boldsymbol{v}\end{array}\right]=\left[\begin{array}[]{c}0\\\
0\end{array}\right]$

so we can conclude

$s_{\min}\left[\begin{array}[]{cc}0&T^{*}\\\
T&0\end{array}\right]=s_{\min}(T^{*})=0.$

Finally, notice that $\sqrt{T^{*}T}=I$ while $\sqrt{TT^{*}}$ has a kernel, so

$s_{\min}\left(|T^{*}|\right)\neq s_{\min}\left(|T|\right)$

in infinite dimensions.

For two Hilbert spaces $\mathcal{H}_{i},\ i=1,2,$ let
$\mathcal{L}(\mathcal{H}_{1},\mathcal{H}_{2})$ denote the set of all bounded
linear operators from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}.$ Recall that for
$T\in\mathcal{L}(\mathcal{H}_{1},\mathcal{H}_{2}),$
$|T|\in\mathcal{L}(\mathcal{H}_{1})$ denotes the modulus of $T$, that is the
unique positive semi-definite operator so that $|T|^{2}=T^{*}T$ and
$\ker(T)=\ker(|T|).$

######  Lemma 4.7.

If $T:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ is a bounded linear operator
then

$s_{\min}(T)=s_{\min}(|T|)=\sqrt{s_{\min}(T^{*}T)}.$

###### Proof.

Since $\|T\boldsymbol{v}\|=\|\,|T|\boldsymbol{v}\,\|$ for all
$\boldsymbol{v}\in\mathcal{H}_{1},$ the consequence that
$s_{\min}(T)=s_{\min}(|T|)$ is straightforward. The equality of
$s_{\min}(|T|)$ with $\sqrt{s_{\min}(T^{*}T)}$ follows from Lemma 4.2 and the
spectral mapping theorem.

∎

With these lemmas, we can now extend some of the result on Clifford spectrum
and pseudospectrum that have been previously worked out in the case of finite
matrices [7, 21]. Theorem 4.8 is a generalization of Lemma 2.3 of [26]. That
result handles one general operator, or equivalently two Hermitian operators.

######  Theorem 4.8.

Suppose $A_{1},\dots,A_{d}$ are Hermitian elements of a unital $C^{*}$-algebra
$\mathcal{A}$. The Clifford pseudospectrum of $\mathbb{A}$ is Lipschitz,
specifically with

(4.2)
$\left|\mu_{\boldsymbol{\lambda}}^{C}(A_{1},\dots,A_{d})-\mu_{\boldsymbol{\nu}}^{C}(A_{1},\dots,A_{d})\right|\leq\|\boldsymbol{\lambda}-\boldsymbol{\nu}\|$

where the norm is taken to be the Euclidean norm on $\mathbb{R}^{n}$. Moreover

(4.3)
$\left|\mu_{\boldsymbol{\lambda}}^{C}(\mathbb{A})-\|\boldsymbol{\lambda}\|\right|\leq\left\|L_{\boldsymbol{0}}(\mathbb{A})\right\|.$

###### Proof.

These statements all follow from the fact that $s_{\min}(A)$ is Lipshitz in
the operator norm (Lemma 4.4). This result is a generalization of results in
Section 7 of [21], and is proven by related methods. The essential
calculations are

$\sigma\left(L_{\boldsymbol{0}}\left(\lambda_{1}I,\cdots,\lambda_{d}I\right)\right)=\left\\{\left\|\boldsymbol{\lambda}\right\|,-\left\|\boldsymbol{\lambda}\right\|\right\\}$

and its corollary

$\left\|L_{\boldsymbol{0}}\left(\lambda_{1}I,\cdots,\lambda_{d}I\right)\right\|=\left\|\boldsymbol{\lambda}\right\|.$

For (4.2) note that

$L_{\boldsymbol{\lambda}}\left(A_{1},\cdots,A_{d}\right)-L_{\boldsymbol{\nu}}\left(A_{1},\cdots,A_{d}\right)=L_{\boldsymbol{0}}\left((\nu_{1}-\lambda_{1})I,\cdots,(\nu_{d}-\lambda_{d})I\right)$

which implies that

$\left\|L_{\boldsymbol{\lambda}}\left(A_{1},\cdots,A_{d}\right)-L_{\boldsymbol{\nu}}\left(A_{1},\cdots,A_{d}\right)\right\|=\|\boldsymbol{\lambda}-\boldsymbol{\nu}\|.$

For (4.3), we know that

$s_{\min}\left(L_{\boldsymbol{0}}\left(\lambda_{1}I,\cdots,\lambda_{d}I\right)\right)=\|\boldsymbol{\lambda}\|$

and

$\left\|L_{\boldsymbol{0}}\left(\lambda_{1}I,\cdots,\lambda_{d}I\right)+L_{\boldsymbol{\lambda}}\left(A_{1},\cdots,A_{d}\right)\right\|=\left\|L_{\boldsymbol{0}}\left(A_{1},\cdots,A_{d}\right)\right\|.$

∎

######  Corollary 4.9.

The Clifford spectrum for a $d$-tuple of Hermitian elements of a
$C^{*}$-algebra is always compact.

###### Proof.

Since Lipshcitz implies continuous we get continuity from Equation 4.2.
Equation 4.3 tells us that then
$\left\|\boldsymbol{\lambda}\|>\|L_{\boldsymbol{0}}(\mathbb{A})\right\|$ we
cannot have $\mu_{\boldsymbol{\lambda}}^{C}(\mathbb{A})=0$. Therefore the
Clifford spectrum of $(A_{1},\dots,A_{d})$ is a closed subset of the ball at
the origin of radius $\left\|L_{\boldsymbol{0}}(A_{1},\dots,A_{d})\right\|$. ∎

##  5\. Other forms of multivariable pseudospectrum

###  5.1. The quadratic pseudospectrum and more

There are many ways to define a pseudospectrum for
$\mathbb{A}=(A_{1},\dots,A_{d})$ that are Hermitian. If these are operators on
Hilbert space $\mathcal{H}$, Mumford [24] suggests associating to
$\boldsymbol{\lambda}$ the number

(5.1)
$\min\left\\{\left.\max\left\|A_{j}\boldsymbol{v}-\lambda_{j}\boldsymbol{v}\right\|\,\right|\,\|\boldsymbol{v}\|=1\text{
and }\textup{E}_{\boldsymbol{v}}[A_{j}]=\lambda_{j}\text{ for all }j\right\\}$

where $\textup{E}_{\boldsymbol{v}}[A_{j}]=\left\langle
A_{j}\boldsymbol{v},\boldsymbol{v}\right\rangle$ is the expectation for the
observable $A_{j}$ when the system is in state $\boldsymbol{v}$. This is
expected [21, §1] to be a very difficult, but important, minimization problem
related to joint measurement.

Fortunately, there is a slight modification that can be made to Equation 5.1
that turns it into something very computable, as the minimization problem
leads to a number equal to the smallest spectral value of a Hermitian matrix.
The quadratic pseudospectrum [7] is defined as

(5.2)
$\mu_{\boldsymbol{\lambda}}^{\textup{Q}}(A_{1},\dots,A_{d})=\min\left\\{\left.\sqrt{\sum_{j}\left\|A_{j}\boldsymbol{v}-\lambda_{j}\boldsymbol{v}\right\|^{2}}\,\right|\,\|\boldsymbol{v}\|=1\right\\}.$

The practical way to compute this is to use the known equality [7]

(5.3)
$\mu_{\boldsymbol{\lambda}}^{\textup{Q}}(A_{1},\dots,A_{d})=\sqrt{s_{\min}\left(Q_{\boldsymbol{\lambda}}\left(\mathbb{A}\right)\right)}$

where

(5.4)
$Q_{\boldsymbol{\lambda}}\left(\mathbb{A}\right)=\sum_{j}\left(A_{j}-\lambda_{j}\right)^{2}.$

This can be defined also in the case where the $A_{j}$ are Hermitian elements
in a unital $C^{*}$-algebra.

We can also select a bump function $g:\mathbb{R}\rightarrow\mathbb{R}$ with
$0\leq g\leq 1$ and $g(0)=1$, some manner of a windowing function. Lin (in
[19]) looks at

$1-\left\|W(\mathbb{A})\right\|$

where

$W(\mathbb{A})=g\left(A_{1}-\lambda_{1}\right)g\left(A_{2}-\lambda_{2}\right)\cdots
g\left(A_{d}-\lambda_{d}\right).$

Notice there needs to be some choice of the order in the product. This is
related to the windowed LDOS [22] that looks at the trace of
$\left(W(\mathbb{A})\right)^{*}W(\mathbb{A})$. It can be difficult to compute
$g\left(A_{j}-\lambda_{1}\right)$, but it should be should be possible to
numerically compute this in settings where some of the matrices are diagonal.
For theoretical calculations, this can be a practical form of pseudospectrum
[19].

###  5.2. Relation between the Clifford and quadratic pseudospectrum

It is expected that all variations on multivariable operator pseudospectrum
will be related somehow. Indeed, they should all be equal in the commutative
case, and be close when the $A_{j}$ have small commutators. This is known in
one case, in that the Clifford and quadratic are of bounded distance apart.
That is, we know [7] that

(5.5)
$\left|\biggl{(}\mu_{\boldsymbol{\lambda}}^{C}\left(\mathbb{A}\right)\biggr{)}^{2}-\biggl{(}\mu_{\boldsymbol{\lambda}}^{Q}\left(\mathbb{A}\right)\biggr{)}^{2}\right|\leq\sum_{j<k}\|[A_{j},A_{k}]\|.$

It is becoming clear that one cannot make do with a single form of
pseudospectrum when studying physical systems. The Clifford pseudospectrum is
related to $K$-theory and is finding applications in photonics [4, 6],
acoustics [9], aperiodic structures [12], nonlinear systems [30], and even
non-Hermitian systems [3, 8, 11, 20, 25]. On the other hand, showing that
points in the Clifford spectrum correspond to states approximately localized
in energy and position seems to require using something like Equation 5.5.

###  5.3. Symmetry in the quadratic pseudospectrum

There are symmetry properties in the Clifford spectrum that must follow from
symmetry properties in $\mathbb{A}$ [5]. Here we prove the same implication,
but for the quadratic case.

######  Lemma 5.1.

Suppose $\left(A_{1},\dots,A_{d}\right)$ is a $d$-tuple of Hermitian elements
in the unital $C^{*}$-algebra $\mathcal{A}$ and that $U\in O(d)$. Suppose
$\boldsymbol{\mathbb{\lambda}}\in\mathbb{R}^{d}$. The $d$ element

$\hat{A}_{j}=\sum_{s}u_{js}A_{s}$

are also Hermitian, and

$\mu_{U\boldsymbol{\lambda}}^{Q}\left(\hat{A}_{1},\dots,\hat{A}_{d}\right)=\mu_{\boldsymbol{\lambda}}^{Q}\left(A_{1},\dots,A_{d}\right).$

###### Proof.

The elements $\hat{A}_{j}$ can be shown to be Hermitian, by the same proof [5,
§2] as for the matrix case. We find

$\displaystyle\sum_{j}\hat{A}_{j}^{2}$
$\displaystyle=\sum_{r}\sum_{s}\sum_{j}u_{jr}u_{js}A_{r}A_{s}$
$\displaystyle=\sum_{r}A_{r}^{2}$ $\displaystyle=\sum_{j}A_{j}^{2}.$

Setting

$\tilde{\lambda}_{j}=\sum_{s}u_{js}\lambda_{s}.$

we next compute

$\displaystyle\sum_{j}\tilde{\lambda}_{j}\hat{A}_{j}$
$\displaystyle=\sum_{r}\sum_{s}\sum_{j}u_{js}u_{jr}\lambda_{s}A_{r}$
$\displaystyle=\sum_{r}\lambda_{r}A_{r}$
$\displaystyle=\sum_{j}\lambda_{j}A_{j}$

Finally

$\displaystyle\sum_{j}\tilde{\lambda}_{j}^{2}$
$\displaystyle=\sum_{r}\sum_{s}\sum_{j}u_{rj}u_{sj}\lambda_{r}\lambda_{s}$
$\displaystyle=\sum_{r}\lambda_{r}^{2}.$

Thus

$\displaystyle
Q_{\widetilde{\boldsymbol{\lambda}}}\left(\hat{A}_{1},\dots,\hat{A}_{d}\right)$
$\displaystyle=\sum_{j}\hat{A}_{j}^{2}-2\sum_{j}\tilde{\lambda}_{j}\hat{A}_{j}+\sum_{j}\tilde{\lambda}_{j}^{2}I$
$\displaystyle=\sum_{j}A_{j}^{2}-2\sum_{j}\lambda_{j}A_{j}+\sum_{j}\lambda_{j}I$
$\displaystyle=Q_{\boldsymbol{\lambda}}\left(A_{1},\dots,A_{d}\right).$

∎

If $Q$ is a unitary and we let $\mathbb{B}=(QA_{1}Q^{*},\dots,QA_{1}Q^{*})$
then $L_{\boldsymbol{\lambda}}(\mathbb{B})$ is unitarily equivalent to
$L_{\boldsymbol{\lambda}}(\mathbb{A})$ and
$Q_{\boldsymbol{\lambda}}(\mathbb{B})$ is unitarily equivalent to
$Q_{\boldsymbol{\lambda}}(\mathbb{A})$. Thus we can use Lemma 2.4 and Lemma
5.1 to obtain the following theorem.

######  Theorem 5.2.

Suppose $\left(A_{1},\dots,A_{d}\right)$ are Hermitian elements in the unital
$C^{*}$-algebra $\mathcal{A}$ and that $U\in O(d)$. Let

$\hat{A}_{j}=\sum_{s}u_{js}A_{s}.$

If there exists a unitary $Q$ in $\mathcal{A}$ such that
$Q\hat{A_{j}}Q^{*}=A_{j}$ for all $j$ then

$\mu^{\textup{C}}_{U\boldsymbol{\lambda}}\left(A_{1},\dots,A_{d}\right)=\mu^{\textup{C}}_{\boldsymbol{\lambda}}\left(A_{1},\dots,A_{d}\right)$

and

$\mu^{\textup{Q}}_{U\boldsymbol{\lambda}}\left(A_{1},\dots,A_{d}\right)=\mu^{\textup{Q}}_{\boldsymbol{\lambda}}\left(A_{1},\dots,A_{d}\right).$

##  6\. The $C^{*}$-algebra generated by two projections

The universal unital $C^{*}$-algebra for the relations that define two
orthogonal projections [27] is

(6.1) $C^{*}(\mathbb{Z}_{2}*\mathbb{Z}_{2})=\left\\{f\in
C([-1,1],\boldsymbol{M}_{2})\,\mid f(-1)\text{ and }f(1)\text{ are
diagonal}\right\\}.$

We will now compute both forms of pseusospectrum for this universal pair of
projections. The formulas derived will come out simpler if we instead think in
terms of a universal pair of unitary operators of order two. We might see
these as two incompatible dichotomous observables.

The generators of $C^{*}(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ are $U$ and $V$ where

(6.2) $U(z)=\left[\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right],\
V(z)=\left[\begin{array}[]{cc}z&\sqrt{1-z^{2}}\\\
\sqrt{1-z^{2}}&-z\end{array}\right].$

First we work on the representation corresponding to evaluation at a fixed
$z$. Here the Hilbert space is only two-dimensional and the calculations are
not too extensive. They are a bit tedious so we utilized a computer algebra
package here.

Figure 6.1. Plotted together the Clifford and quadratic pseudospectrum for
$(U,V_{z})$, as defined in (6.3,) for various values of of $z$. From the top-
left to the bottom-right, the values used are $z=1$, $z=0.6$, $z=0$, $z=-0.5$,
$z=-1$.

######  Lemma 6.1.

Suppose $-1\leq z\leq 1$. If

(6.3) $U=\left[\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right],\
V_{z}=\left[\begin{array}[]{cc}z&\sqrt{1-z^{2}}\\\
\sqrt{1-z^{2}}&-z\end{array}\right]$

then

$\mu_{(x,y)}^{\textup{Q}}(U,V_{z})=\sqrt{x^{2}+y^{2}+2-2\sqrt{x^{2}+2zxy+y^{2}}}$

and

$\mu_{(x,y)}^{\textup{C}}(U,V_{z})=\sqrt{x^{2}+y^{2}+2-2\sqrt{x^{2}+2xyz+y^{2}+1-z^{2}}}.$

###### Proof.

We compute the smallest eigenvalue of

$Q_{(x,y)}(U,V_{z})=\left(U-xI\right)^{2}+\left(V_{z}-yI\right)^{2},$

finding that this equals

$Q_{(x,y)}=\left[\begin{array}[]{cc}x^{2}+y^{2}-2x-2zy+2&-2y\sqrt{1-z^{2}}\\\
-2y\sqrt{1-z^{2}}&x^{2}+y^{2}+2x+2zy+2\end{array}\right].$

This has eigenvalues

$x^{2}+y^{2}+2\pm 2\sqrt{x^{2}+2zxy+y^{2}}.$

The smaller of the eigenvalues is with the minus sign. To get to
$\mu_{(x,y)}^{Q}(P,Q_{z})$ we apply square root.

For the Clifford pseudospectrum, we need the eigenvalues of

$\left[\begin{array}[]{cc}0&\left(U-xI\right)-i\left(V_{z}-yI\right)\\\
\left(U-xI\right)+i\left(V_{z}-yI\right)&0\end{array}\right]$

which are determined by the singular values of

$\left(U-xI\right)+i\left(V_{z}-yI\right).$

These are

$\sqrt{x^{2}+y^{2}+2\pm 2\sqrt{x^{2}+2xyz+y^{2}+1-z^{2}}}.$

∎

Figure 6.2. The Clifford and quadratic pseudospectrum for the universal pair
of order-two unitary matrices $(U,V)$ are plotted together here.

######  Theorem 6.2.

For the universal pair of unitary operators of order two, so $U$ and $V$ as in
(6.2) in the universal $C^{*}$-algebra (6.1), the quadratic pseudospectrum is

$\mu_{(x,y)}^{\textup{Q}}(U,V)=\textup{dist}\left((x,y),\left\\{(-1,-1),(-1,1),(1,-1),(1,1)\right\\}\right)$

and the Clifford pseudospectrum is

(6.4)
$\mu_{(x,y)}^{\textup{C}}(U,V)=\sqrt{x^{2}+y^{2}+2-2\sqrt{x^{2}y^{2}+x^{2}+y^{2}+1}}$

when $-1\leq xy\leq 1$, and otherwise

$\mu_{(x,y)}^{\textup{C}}(U,V)=\mu_{(x,y)}^{\textup{Q}}(U,V).$

In particular, the quadratic spectrum is the set of the four points $(\pm
1,\pm 1)$ and the Clifford spetrum is a cross, the union of the line segment
from $(-1,-1)$ to $(1,1)$ and the line segment from $(-1,1)$ to $(1,-1)$.

###### Proof.

First the quadratic case. For each $(x,y)$ we need to minimize

$\mu_{(x,y)}^{\textup{Q}}(U,V_{z})=\sqrt{x^{2}+y^{2}+2-2\sqrt{x^{2}+2zxy+y^{2}}}$

over the range $-1\leq z\leq 1$. This is constant when $x=0$ or $y=0$ and in
the remaining cases, is maximized when $z=-1$ or when $z=1$. Since

$x^{2}\pm 2xy+y^{2}=(x\pm y)^{2}$

we find that

$\mu_{(x,y)}^{\textup{Q}}(U,V_{1})=\sqrt{x^{2}+y^{2}+2-2\sqrt{x^{2}+2xy+y^{2}}}=\begin{cases}\sqrt{\left(x+1\right)^{2}+\left(y+1\right)^{2}}&\text{if
}x+y\leq 0\\\ \sqrt{\left(x-1\right)^{2}+\left(y-1\right)^{2}}&\text{if
}x+y\geq 0\end{cases}.$

This means $\mu_{(x,y)}^{\textup{Q}}(U,V_{1})$ is the distance of $(x,y)$ to
the set $\\{(-1,-1),(1,1)\\}$. Similarly $\mu_{(x,y)}^{\textup{Q}}(U,V_{-1})$
is the distance of $(x,y)$ to the set $\\{(1,-1),(-1,1)\\}$.

Now we look to the Clifford case. For each $(x,y)$ we need to minimize

(6.5)
$\mu_{(x,y)}^{\textup{C}}(U,V_{z})=\sqrt{x^{2}+y^{2}+2-2\sqrt{x^{2}+2xyz+y^{2}+1-z^{2}}}.$

This will happen at the maximum of

$g(z)=x^{2}+2xyz+y^{2}+1-z^{2}.$

This has derivative

$g^{\prime}(z)=2xy-2z$

so the maximum value of $g$ seems like it will be at $z=z_{0}$ where
$z_{0}=xy$ but we need to ensure this is with $-1\leq z\leq 1$. If $(x,y)$ is
in the region specified by $-1\leq xy\leq 1$, a region that includes
$[-1,1]\times[-1,1]$, then the value of $\mu_{(x,y)}^{\textup{C}}(U,V_{z})$ is
found at $z_{0}$. Substituting $z_{0}$ for $z$ in Eq. 6.5 yields Eq. 6.4.

∎

##  7\. A hemisphere as Clifford spectrum

Example 3.2 tells us we can find essentially any finite-dimensional compact
space as the Clifford spectrum of some set of commuting Hermitian operators.
In the finite-dimensional case we need to introduce noncommutativity to get
Clifford spectrum in dimension greater that zero. Many papers have examined
the spaces that one can get from finite matrices, including [1, 5, 10, 28].
With three matrices it is possible to have as Clifford spectrum a $2$-manifold
or a surface with cusps. With four matrices one can get $2$-manifolds and
$3$-manifold, for example. The simplest example leading to a closed surface is
the Pauli matrices themselves [17], where

$\Lambda(\sigma_{x},\sigma_{y},\sigma_{z})=S^{2}.$

We find some odd behavior when we modify this example to be infinite-
dimensional.

Consider

(7.1) $A_{1}=\frac{1}{2}\left[\begin{array}[]{ccccc}0&1\\\ 1&0&1\\\ &1&0&1\\\
&&1&\ddots&\ddots\\\ &&&\ddots&\ddots\end{array}\right],\
A_{2}=\frac{1}{2}\left[\begin{array}[]{ccccc}0&-i\\\ i&0&-i\\\ &i&0&-i\\\
&&i&\ddots&\ddots\\\ &&&\ddots&\ddots\end{array}\right],\
A_{3}=\left[\begin{array}[]{ccccc}b\\\ &0\\\ &&0\\\ &&&\ddots\\\
&&&&\ddots\end{array}\right].$

If we set $b=1/2$ we have operators that are somewhat like
$\tfrac{1}{2}\sigma_{x},\tfrac{1}{2}\sigma_{y},\tfrac{1}{2}\sigma_{z}$. We
will find that for this class of examples, we get a hemisphere and not a
sphere. This is in some way a manifestation that the $K$-theory of
$\mathcal{B}(\mathcal{H})$ is trivial so there can be no local index
stabilizing the Clifford spectrum.

Notice that $A_{1}$, $A_{2}$, $A_{3}$ are compact perturbations of

$K_{1}=\frac{1}{2}(S^{*}+S),K_{2}=\frac{i}{2}(S^{*}-S),K_{3}=0$

where $S$ is the usual shift operator. Thus the essential Clifford spectrum in
this example will not depend on $b$.

######  Lemma 7.1.

For any $b\geq 0$, if $H_{1}$, $H_{2}$, $H_{3}$ are defined according to
Equation 7.1, then

$\Lambda^{\textup{e}}(H_{1},H_{2},H_{3})=\mathbb{T}\times\\{0\\}.$

###### Proof.

Since the essential spectrum of the unilateral shift is the unit circle we
know that

$\Lambda^{\textup{e}}(H_{1},H_{2})=\mathbb{T}.$

Now Theorem 3.4 implies

$\Lambda^{\textup{e}}(H_{1},H_{2},H_{3})=\Lambda^{\textup{e}}(H_{1},H_{2})\times\\{0\\}$

so we are done. ∎

######  Example 7.2.

For any $0\leq b\leq 2.25$, if $H_{1}$, $H_{2}$, $H_{3}$ are defined according
to Equation 7.1, then

$\Lambda(H_{1},H_{2},H_{3}).$

equals the set of points $(x,y,z)$ that satisfy the equation

$b^{4}z+b^{3}r^{2}-3b^{3}z^{2}-b^{3}-b^{2}r^{2}z+3b^{2}z^{3}+2b^{2}z+br^{4}-br^{2}-bz^{4}-bz^{2}+r^{2}z=0.$

where $r^{2}=x^{2}+y^{2}$ and subject to the constraints $0\leq z\leq b$ and
$x^{2}+y^{2}\leq 1$.

We will verify this with a rigorous proof in the special case $b=1$. When
$b=0$ this follows easily from Theorem 3.4. For the general case, we use an
experimental mathematics approach, where we use computer algebra to derive an
equality and an inequality that together determine part of the Clifford
spectrum and then estimate the joint solution numerically.

For any $b\geq 0$ it is easy to see that $e^{i\theta}S$ is unitarily
equivalent to $S$. This implies that

$\left(\cos(\theta)H_{1}+\sin(\theta)H_{2},\
\cos(\theta)H_{1}-\sin(\theta)H_{2},\ H_{3}\right)$

is unitarily equivalent to $H_{1},H_{2},H_{3}$. We can now apply Theorem 2.5
to conclude that $\Lambda(H_{1},H_{2},H_{3})$ has rotational symmetry in the
$x$-$y$-plane. Thus, we can restrict our attention to the localizer with
$\boldsymbol{\lambda}=(x,0,z)$ for $x\geq 0$.

The localizer naturally lives in $\mathbf{M}_{2}(\mathcal{B}(\mathcal{H}))$
but we can identify that with $\mathcal{B}(\mathcal{H})$ via the usual
shuffling of basis elements. Making this identification we can compute
$L_{(-x,0,z)}$ as follows. Let $E_{ij}$ denote the usual two-by-two matrix
units, so that $\sigma_{y}=-iE_{12}+iE_{21}$ etc. Since

$\displaystyle A_{1}\otimes\sigma_{x}+A_{2}\otimes\sigma_{y}$
$\displaystyle=A_{1}\otimes(E_{12}+E_{21})+iA_{2}\otimes(-E_{12}+E_{21})$
$\displaystyle=(A_{1}-iA_{2})\otimes E_{12}+(A_{1}+iA_{2})\otimes E_{21}$
$\displaystyle=S^{*}\otimes E_{12}+S\otimes E_{21}$

we find

$A_{1}\otimes\sigma_{x}+A_{2}\otimes\sigma_{y}=\left[\begin{array}[]{ccccccc}0&0&&&&&\\\
0&0&1&&&&\\\ &1&0&0&&&\\\ &&0&0&1&&\\\ &&&1&0&0&\\\ &&&&0&0&\ddots\\\
&&&&&\ddots&\ddots\end{array}\right].$

Therefore

$L_{(x,0,z)}=\left(A_{1}-xI\right)\otimes\sigma_{x}+A_{2}\otimes\sigma_{y}+\left(A_{3}-zI\right)\otimes\sigma_{z}$

works out as

$L_{(x,0,z)}=\left[\begin{array}[]{ccccccc}b-z&-x&&&&&\\\ -x&z-b&1&&&&\\\
&1&-z&-x&&&\\\ &&-x&z&1&&\\\ &&&1&-z&-x&\\\ &&&&-x&z&\ddots\\\
&&&&&\ddots&\ddots\end{array}\right].$

We we already worked out the essential Clifford spectrum so we already know
that $0$ is in the essential spectrum of the spectral localizer. What is left
is find and discrete spectrum. Thus, we search for values of $x$ and $z$ that
lead to null vectors for $L_{(-x,0,z)}$.

Figure 7.1. The Clifford spectrum for the three operators in Equation 7.1.
$b=1.00$ (left); $b=2.00$ (center); $b=2.05$ (right);

The calculations in the general case are complicated so now it is time to
restrict to the case of $b=1$.

We need to discuss several special cases. First we deal with $x=0$. This makes
$L_{(x,0,z)}$ block diagonal, with blocks

$\left[1-z\right],\ \left[\begin{array}[]{cc}z-1&1\\\ 1&-z\end{array}\right],\
\left[\begin{array}[]{cc}z&1\\\ 1&-z\end{array}\right]$

that have determinants $1-z$ $-z^{2}+z-1$ and $-z^{2}-1$, so only $z=1$ leads
to $L_{(x,0,z)}$ being singular.

Now assume $x\geq 0$. Let us assume that there exists a nonzero vector
$\boldsymbol{a}=(a_{1},a_{2},\dots)\in l^{2}(\mathbb{N})$ so that
$L_{(x,0,z)}\boldsymbol{a}=0$. We can rescale $\boldsymbol{a}$ so that
$a_{1}=1$. This means

$\displaystyle(1-z)-xa_{2}$ $\displaystyle=0,$
$\displaystyle-x+(z-1)a_{2}+a_{3}$ $\displaystyle=0,$

and for $n\geq 2$,

$\displaystyle a_{2n-2}-za_{2n-1}-xa_{2n}$ $\displaystyle=0$ $\displaystyle-
xa_{2n-1}+za_{2n}+a_{2n+1}$ $\displaystyle=0.$

Thus, for $n\geq 2$,

$a_{2n}=-\frac{z}{x}a_{2n-1}+\frac{1}{x}a_{2n-2}$

and

$\displaystyle a_{2n+1}$ $\displaystyle=-za_{2n}+xa_{2n-1}$
$\displaystyle=-z\left(-\frac{z}{x}a_{2n-1}+\frac{1}{x}a_{2n-2}\right)+xa_{2n-1}$
$\displaystyle=\left(\frac{z^{2}}{x}+x\right)a_{2n-1}-\frac{z}{x}a_{2n-2}.$

If we set

$M=\left[\begin{array}[]{cc}\frac{1}{x}&-\frac{z}{x}\\\
-\frac{z}{x}&\frac{x^{2}+z^{2}}{x}\end{array}\right]$

the relation on $\boldsymbol{a}$ is

$\left[\begin{array}[]{c}a_{2n}\\\
a_{2n+1}\end{array}\right]=M\left[\begin{array}[]{c}a_{2n-2}\\\
a_{2n-1}\end{array}\right].$

We also know

$a_{2}=\frac{1-z}{x}$

and

$\displaystyle a_{3}$ $\displaystyle=x+(1-z)a_{2}$
$\displaystyle=x+(1-z)\frac{1-z}{x}$

so have an initial condition

$\boldsymbol{v}_{0}=\left[\begin{array}[]{c}a_{2}\\\
a_{3}\end{array}\right]=\left[\begin{array}[]{c}\frac{1-z}{x}\\\
x+\frac{\left(1-z\right)^{2}}{x}\end{array}\right].$

Now let’s pause to take care of the case of $z=1$ and $x>0$. In that
situation, we have

$M=\left[\begin{array}[]{cc}\frac{1}{x}&-\frac{1}{x}\\\
-\frac{1}{x}&\frac{x^{2}+1}{x}\end{array}\right]$

so that

$\boldsymbol{v}_{0}=\left[\begin{array}[]{c}0\\\ x\end{array}\right]$

is not going to be an eigenvalue. Thus, the only Clifford spectrum on the line
$z=1$ is the point $(x,0,z)=(0,0,1)$ that we found above.

We need to know about the eigenvalues of the real symmetric matrix $M$. a see
that

$\textup{Tr}(M)=\frac{x^{2}+z^{2}+1}{x}$

and $\det(M)=1$ we find that the eigenvalues of the real symmetric matrix $M$
are

$\frac{1}{2}\left(\textup{Tr}(M)\pm\sqrt{\left(\textup{Tr}(M)\right)^{2}-4}\right).$

We get a double real eigenvalue when

$\frac{x^{2}+z^{2}+1}{x}=2$

but this leads to $x=1$ and $z=0$ which we have excluded. We get two complex
eigenvalues when

$\frac{x^{2}+z^{2}+1}{x}<2$

which is equivalent to

$\left(x-1\right)^{2}+z^{2}<0$

and that cannot happen. What we need look at then is where $M$ has two
positive eigenvalues, one outside of $(0,1)$ and one inside. The only way to
have $\boldsymbol{a}$ square-summable is if the initial vector lands in the
eigenspace for the eigenvalue closer to $0$.

Let us find out when $M\boldsymbol{v}_{0}$ is parallel to
$\boldsymbol{v}_{0}$. Since

$M\boldsymbol{v}_{0}=\frac{1}{x^{2}}\left[\begin{array}[]{c}\left(1-z\right)-z\left(x^{2}+z^{2}-2z+1\right)\\\
-z\left(1-z\right)+\left(x^{2}+z^{2}\right)\left(x^{2}+z^{2}-2z+1\right)\end{array}\right]$

we need to solve

$\frac{x^{-2}\left(\left(1-z\right)-z\left(x^{2}+z^{2}-2z+1\right)\right)}{x^{-1}\left(1-z\right)}-\frac{x^{-2}\left(-z\left(1-z\right)+\left(x^{2}+z^{2}\right)\left(x^{2}+z^{2}-2z+1\right)\right)}{x^{-1}\left(x^{2}+z^{2}-2z+1\right)}=0.$

For $x\neq 0$ and $z\neq 1$ this is equivalent to

$-\frac{x^{4}-z^{4}+3z^{3}-4z^{2}+3z-1}{x^{2}+z^{2}-2z+1}=0$

so we get the curve $e(x,z)=0$ where

$e(x,z)=x^{4}-z^{4}+3z^{3}-4z^{2}+3z-1.$

Figure 7.2. Plots of $e(x,z)=0$ (dotted curves) and $f(x,z)\geq 0$ (light blue
area, switched to $\leq$ above $z=b$) for $b=1.00$, $b=2.00$, $b=2.05$.

What about the associated eigenvalue $\alpha$? Is it between $0$ and $1$?
First, we deal with the case where $z<1$. Here the first coordinate of the
initial vector $\boldsymbol{v}_{0}$ will be positive so $\alpha$ will be
positive when

$\frac{1-z}{x}>\frac{-z\left(1-z\right)+\left(x^{2}+z^{2}\right)\left(x^{2}+z^{2}-2z+1\right)}{x^{2}}.$

As $x^{2}$ is positive this is equivalent to

$x-xz>-x^{2}z-z^{3}+2z^{2}-2z+1$

so we need to know when

$f(x,z)=x^{2}z-xz+x+z^{3}-2z^{2}+2z-1$

is positive. For $z>1$ we need to know when $f(x,z)$ is negative.

We find

$\frac{\partial f}{\partial
z}=x^{2}-x+3z^{2}-4z+2=(x-\tfrac{1}{2})^{2}+3(z-\tfrac{2}{3})^{2}+\tfrac{5}{12}$

is always positive. The plots of the zero-locus of $e$ and $f$ are shown in
Figure 7.2. These seem to cross at $(x,z)=(0,1)$ and $(x,z)=(1,0)$ and the
curve for $f=0$ is below the curve for $e=0$ between those two crossings. If
we believe the computer-generated plots then we can conclude that the discrete
Clifford spectrum is on the curve $e(x,z)=0$ restricted to the region $0\leq
x,z\leq 1$.

We can verify the above claim rigorously, with some uninteresting calculus. We
put the rest of the proof in Appendix A.

When $b\neq 1$ the polynomials $e(x,z)$ and $f(x,z)$ are trickier. As such, we
resorted to using symbolic computer algebra for various values of $b$ and
numerically plot the curve $e(x,z)=0$ and regions associated to $f(x,z)$.
While we do not have a fully rigorous proof, we feel this is sufficient to
understand this example.

##  8\. Momentum and Position – an unbounded example

We hope to push the idea of joint pseudospectrum into the realm of unbounded
operators. Here we consider a classic example, and do not attempt a general
theory. The example is a classic interpretation of position and momentum.

We will work on the Hilbert space $L^{2}(\mathbb{R})$, and in particular on
the subspace $\mathcal{S}$ of Schwartz functions, where we conisder only $f$
for which $x^{m}D^{n}f(x)$ is always bounded. Here we use

$D(f)(x)=\frac{d}{dx}(f(x))$

to denote the standard differential operator. We consider two unbounded
operators, $P,Q:\mathcal{S}\rightarrow\mathcal{S}$ where $P=-iD$ and
$Q(f)(x)=xf(x).$ These are both symmetric (but probably not self-adjoint, so
$Q\subseteq Q^{*}$ is all that is claimed, due to the small domain).

We can consider

$Q_{\boldsymbol{\lambda}}(P,Q)=\left(P-\lambda_{1}I\right)^{2}+\left(Q-\lambda_{2}I\right)^{2}:\mathcal{S}\rightarrow\mathcal{S}.$

and define

$\mu_{\boldsymbol{\lambda}}^{\textup{Q}}(P,Q)=\sqrt{\min_{\|f\|=1}\|Q_{\boldsymbol{\lambda}}(P,Q)(f)\|}.$

The Clifford pseudospectrum is also easy to define, as

$\mu_{\boldsymbol{\lambda}}^{\textup{C}}(P,Q)=\min_{\|f\|=1}\|L_{\boldsymbol{\lambda}}(P,Q)(f)\|$

where

$L_{\boldsymbol{\lambda}}(P,Q)=\left[\begin{array}[]{cc}0&\left(P-\lambda_{1}I\right)-i\left(Q-\lambda_{2}I\right)\\\
\left(P-\lambda_{1}I\right)+i\left(Q-\lambda_{2}I\right)&0\end{array}\right]$

is taken to be an operator on ${S}\oplus\mathcal{S}$. In [10], this example of
Clifford pseudospectrum was computed, with

$\mu_{\boldsymbol{\lambda}}^{\textup{C}}(P,Q)\equiv 0$

being the result.

We will need some alternate characterizations of the quadratic pseudospectrum
of $P$ and $Q$, as proven in [7] for the matrix case. We can consider

$M_{\boldsymbol{\lambda}}(P,Q)=\left[\begin{array}[]{c}P-\lambda_{1}I\\\
Q-\lambda_{2}I\end{array}\right]:\mathcal{S}\rightarrow\mathcal{S}\oplus\mathcal{S}.$

and can check that

$\displaystyle\mu_{\boldsymbol{\lambda}}^{Q}(P,Q)$
$\displaystyle=\sqrt{\min_{\|f\|=1}\left\|\left(M_{\boldsymbol{\lambda}}(P,Q)(f)\right)^{*}\left(M_{\boldsymbol{\lambda}}(P,Q)(f)\right)\right\|}$
$\displaystyle=\min_{\|f\|=1}\|M_{\boldsymbol{\lambda}}(P,Q)(f)\|$
$\displaystyle=\min_{\|f\|=1}\sqrt{\left\|Pf-\lambda_{1}f\right\|^{2}+\left\|Qf-\lambda_{2}f\right\|^{2}}.$

Since

$\left\|A\boldsymbol{v}-\lambda\boldsymbol{v}\right\|^{2}=\Delta_{\boldsymbol{v}}^{2}A+\left(\textup{E}_{\boldsymbol{v}}(A)-\lambda\right)^{2}$

for any Hermitian operator, we can find lower bounds on the quadratic
pseudospectrum by utilizing results on lower bounds on the sum of uncertainty.
For example, there are lower bounds on
$\Delta_{\boldsymbol{v}}^{2}A+\Delta_{\boldsymbol{v}}^{2}B$ in [23]. For
simplicity, we give a simplification of a proof from [23] and directly prove
the following.

######  Lemma 8.1.

If $\boldsymbol{v}$ is a unit vector and $A$ and $B$ are Hermitian operators
then

$\left\|A\boldsymbol{v}\right\|^{2}+\left\|B\boldsymbol{v}\right\|^{2}\geq\left|\left\langle
i\left[A,B\right]\boldsymbol{v},\boldsymbol{v}\right\rangle\right|.$

###### Proof.

We first calculate

$\left\|\left(A\mp
iB\right)\boldsymbol{v}\right\|^{2}=\left\|A\boldsymbol{v}\right\|^{2}+\left\|B\boldsymbol{v}\right\|^{2}\mp\left\langle
i\left[A,B\right]\boldsymbol{v},\boldsymbol{v}\right\rangle$

so we have

$\left\|A\boldsymbol{v}\right\|^{2}+\left\|B\boldsymbol{v}\right\|^{2}=\left\|\left(A\mp
iB\right)\boldsymbol{v}\right\|^{2}\pm\left\langle
i\left[A,B\right]\boldsymbol{v},\boldsymbol{v}\right\rangle.$

An easy consequence is then

$\left\|A\boldsymbol{v}\right\|^{2}+\left\|B\boldsymbol{v}\right\|^{2}\geq\pm\left\langle
i\left[A,B\right]\boldsymbol{v},\boldsymbol{v}\right\rangle.$

∎

######  Theorem 8.2.

For $P$ and $Q$ as above,

$\mu_{\boldsymbol{\lambda}}^{\textup{Q}}(P,Q)=1$

for all $\boldsymbol{\lambda}$.

###### Proof.

We start by establishing that $\mu_{\boldsymbol{\lambda}}^{\textup{Q}}(P,Q)$
is constant. The shift operator $f(x)\mapsto f(x-\lambda_{2})$ gives a unitary
$U$ that takes $\mathcal{S}$ onto $\mathcal{S}$. Since

$\displaystyle U^{*}PU(f(x))$
$\displaystyle=-iU^{*}(f^{\prime}(x-\lambda_{2}))$
$\displaystyle=-if^{\prime}(x)$ $\displaystyle=P(f(x))$

and

$\displaystyle U^{*}QU(f(x))$ $\displaystyle=U^{*}(xf(x-\lambda_{2}))$
$\displaystyle=(x+\lambda_{2})f(x)$

we find $U^{*}PU=P$ and $U^{*}QU=Q+\lambda_{2}$. Therefore
$Q_{(\lambda_{1},\lambda_{2})}(P,Q)$ is unitarily equivalent to
$Q_{(\lambda_{1},0)}(P,Q)$ and so

$\mu_{(\lambda_{1},\lambda_{2})}^{\textup{Q}}(P,Q)=\mu_{(\lambda_{1},0)}^{\textup{Q}}(P,Q).$

Likewise, we can use $f(x)\mapsto e^{2\pi i\lambda_{1}x}f(x)$ so set up a
unitary that shows

$\mu_{(\lambda_{1},\lambda_{2})}^{\textup{Q}}(P,Q)=\mu_{(0,\lambda_{2})}^{\textup{Q}}(P,Q).$

Taken together, these facts prove that the quadratic pseudospectrum is
constant.

Since $\left[P,Q\right](f)=-if$, Lemma 8.1 says

$\left\|Pf\right\|^{2}+\left\|Qf\right\|^{2}\geq 1.$

Therefore $\mu_{(\boldsymbol{0})}^{\textup{Q}}(P,Q)\geq 1$ and so

$\mu_{\boldsymbol{\lambda}}^{\textup{Q}}(P,Q)\geq 1$

Now consider $g(x)=e^{-\frac{1}{2}x^{2}}$. Clearly

$Q^{2}(g)(x)=x^{2}e^{-\frac{1}{2}x^{2}}=x^{2}g(x).$

We also find

$\displaystyle P^{2}(g)(x)$
$\displaystyle=P\left(ixe^{-\frac{1}{2}x^{2}}\right)$
$\displaystyle=-i\frac{d}{dx}\left(ixe^{-\frac{1}{2}x^{2}}\right)$
$\displaystyle=\frac{d}{dx}\left(xe^{-\frac{1}{2}x^{2}}\right)$
$\displaystyle=e^{-\frac{1}{2}x^{2}}-x^{2}e^{-\frac{1}{2}x^{2}}$
$\displaystyle=g(x)-x^{2}g(x)$

and so find

$\left(P^{2}+Q^{2}\right)g=g.$

Having found an eigenvector $Q_{(0,0)}(P,Q)=P^{2}+Q^{2}$ we have proven

$\mu_{\boldsymbol{\lambda}}^{\textup{Q}}(P,Q)\leq 1.$

∎

## Acknowledgments

A.C. and V.L. acknowledge support from the Laboratory Directed Research and
Development program at Sandia National Laboratories. T.A.L. acknowledges
support from the National Science Foundation, Grant No. DMS-2110398. This work
was performed in part at the Center for Integrated Nanotechnologies, an Office
of Science User Facility operated for the U.S. Department of Energy (DOE)
Office of Science. Sandia National Laboratories is a multimission laboratory
managed and operated by National Technology & Engineering Solutions of Sandia,
LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S.
DOE’s National Nuclear Security Administration under Contract No. DE-
NA-0003525. The views expressed in the article do not necessarily represent
the views of the U.S. DOE or the United States Government.

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##  Appendix A Calculations for Example 7.2

Here if the rest of the proof for the case $b=1$. We can solve for $x$ in
terms of $z$ on the $e=0$ curve, so

$x=\left(z^{4}-3z^{3}+4z^{2}-3z+1\right)^{1/4}.$

which simplifies to

$x=\left|z-1\right|^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/4}.$

This substitution into $f(x,z)$ where,

$f(x,z)=x^{2}z-x(z-1)+\left(z-1\right)\left(z^{2}-z+1\right),$

leads to the one-variable function

$f_{e}(z)=z\left|z-1\right|\left(z^{2}-z+1\right)^{1/2}-(z-1)\left|z-1\right|^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/4}+\left(z-1\right)\left(z^{2}-z+1\right)$

For $z\leq 1$ this becomes

$f_{e}(z)=z\left(1-z\right)\left(z^{2}-z+1\right)^{1/2}+(1-z)\left(1-z\right)^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/4}-\left(1-z\right)\left(z^{2}-z+1\right).$

For $0\leq z\leq 1$ we have $z^{2}-z+1\leq 1$ so

$\displaystyle f_{e}(z)$ $\displaystyle\geq
z\left(1-z\right)\left(z^{2}-z+1\right)^{1/2}+(1-z)\left(1-z\right)^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/2}-\left(1-z\right)\left(z^{2}-z+1\right)^{\frac{1}{2}}$
$\displaystyle=\left(z(1-z)+(1-z)(1-z)^{\frac{1}{2}}-\left(1-z\right)\right)\left(z^{2}-z+1\right)^{\frac{1}{2}}$
$\displaystyle=(1-z)\left(z+(1-z)^{\frac{1}{2}}-(1-z)\right)\left(z^{2}-z+1\right)^{\frac{1}{2}}$
$\displaystyle=(1-z)\left(2z-1+(1-z)^{\frac{1}{2}}\right)\left(z^{2}-z+1\right)^{\frac{1}{2}}$
$\displaystyle\geq 0.$

For $z\leq 0$ we have $z^{2}-z+1\geq 1$ so

$\displaystyle f_{e}(z)$ $\displaystyle\leq
z\left(1-z\right)\left(z^{2}-z+1\right)^{1/4}+(1-z)\left(1-z\right)^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/4}-\left(1-z\right)\left(z^{2}-z+1\right)$
$\displaystyle=(1-z)\left(z\left(z^{2}-z+1\right)^{1/4}+\left(1-z\right)^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/4}-\left(z^{2}-z+1\right)\right)$
$\displaystyle=(1-z)\left(z^{2}-z+1\right)^{1/4}\left(z+\left(1-z\right)^{\frac{1}{2}}-\left(z^{2}-z+1\right)^{\frac{3}{4}}\right)$

One can see this is at most zero by looking at its derivative.

For $z\geq 1$ the formula for $f_{e}$ becomes

$f_{e}(z)=z\left(z-1\right)\left(z^{2}-z+1\right)^{1/2}-(z-1)\left(z-1\right)^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/4}+\left(z-1\right)\left(z^{2}-z+1\right).$

Here $z^{2}-z+1\geq 1$ so

$\displaystyle f_{e}(z)$
$\displaystyle=\left(z-1\right)\left(z\left(z^{2}-z+1\right)^{1/2}-\left(z-1\right)^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/4}+\left(z^{2}-z+1\right)\right)$
$\displaystyle\geq\left(z-1\right)\left(z\left(z^{2}-z+1\right)^{1/2}-\left(z-1\right)^{\frac{1}{2}}\left(z^{2}-z+1\right)^{1/2}+\left(z^{2}-z+1\right)^{\frac{1}{2}}\right)$
$\displaystyle=\left(z-1\right)\left(z^{2}-z+1\right)^{1/2}\left(z+1-\left(z-1\right)^{\frac{1}{2}}\right)$
$\displaystyle\geq 0.$