figurec

# GeMSE: a Low-Background Facility for Gamma-Spectrometry at Moderate Rock
Overburden

Diego Ramírez García  Daniel Baur  Jaron Grigat  Beda A. Hofmann  Sebastian
Lindemann  Darryl Masson  Marc Schumann  Moritz von Sivers  Francesco Toschi

###### Abstract

The GeMSE (Germanium Material and meteorite Screening Experiment) facility
operates a low-background HPGe crystal in an underground laboratory with a
moderate rock overburden of 620 m.w.e. in Switzerland. It has been optimized
for continuous remote operation. A multi-layer passive shielding, a muon veto,
and a boil-off nitrogen purge line inside the measurement cavity minimize the
instrument’s background rate, which decreased by 33% to ($164\pm 2$)
counts/day (100 – 2700 keV) after five-years of underground operation. This
agrees with the prediction based on the expected decay of short-lived
isotopes. A fit to the known background components, modeled via a precise
simulation of the detector, shows that the GeMSE background is now muon-
dominated. We also present updates towards a more accurate detection
efficiency calculation for the screened samples: the thickness of the
crystal’s outer dead-layer is precisely determined and the efficiency can now
be easily calculated for any sample geometry. The advantage of this feature is
showcased via the determination of the 40K content in the screening of a
complex-shaped object: a banana.

##  1 Introduction

Low-background gamma-ray spectrometry based on high-purity germanium (HPGe)
detectors is an indispensible tool for rare event search experiments in
astroparticle physics, as it provides the means to select detector
construction materials and components with the lowest possible contamination
of radioactive impurities [1]. Other disciplines can also highly benefit from
this technology. One example is geology and meteorite research, where short-
lived (and thus hard to detect) isotopes are used to date the terrestrial age
of a meteorite and can help to identify samples from the same fall [2, 3].

The GeMSE (Germanium Material and meteorite Screening Experiment) facility is
installed in the Vue-des-Alpes tunnel in Switzerland [4]. It is mainly used
for material screening for the XENON [5, 6, 7] and DARWIN [8] dark matter
projects and has a dedicated meteorite program, mostly centered around a large
collection of meteorites collected in the Oman deserts [9, 10]. The facility
was described in detail in [11] and is thus only briefly summarized here:
GeMSE is using a Canberra p-type coaxial HPGe crystal of 85 mm diameter and 65
mm height ($m\approx 2.0$ kg) installed in an ultra low-background U-style
oxygen-free high thermal conductivity (OFHC) copper cryostat. It is connected
to a liquid nitrogen (LN2) bath via a coldfinger. GeMSE’s massive shell-
structured shield consists of (from inside to outside) 80 mm of oxygen-free
copper (Cu-OFE), 50 mm of low-background lead (210Pb concentration: ($7.2\pm
0.5$) Bq/kg) and 150 mm of standard lead. The large sample cavity is 240
$\times$ 240 mm2 wide and 350 mm high and accessed from the top via a massive
sliding door. The entire shield is enclosed by a sealed glove box which is
constantly purged with gaseous boil-off nitrogen (GN2), guided directly into
the cavity. The depth of 620 m.w.e. (235 m of rock overburden) of the
laboratory in the Vue-des-Alpes tunnel reduces the muon flux to $8.3\times
10^{-2}$ m-2s-1. Two plastic scintillator panels of $1400\times 1050$ mm2 are
installed above and behind the glove box, read out by 3 photomultiplier tubes
(PMTs) each, and serve as a muon veto.

Here we report on several notable updates and new features which significantly
improved the performance of the instrument: like most HPGe screening
facilities, GeMSE operates continuously. Over the last years it has been
automated to an extent that visits to the laboratory are only necessary to
refill the LN2 inventory required to cool the crystal and to purge the shield
(every $\sim$ 6 weeks), as well as for sample changes. Full remote control
further minimizes the workload for the operators. Detector operation is
described in Section 2. The background of the facility improved considerably
over the last years, mainly due to the decay of cosmogenically-activated
isotopes. The new background characterization presented in Section 3 shows
that the facility is now limited by muon-induced backgrounds which could be
further reduced by rather straightforward upgrades to the muon veto. Finally,
Section 4 presents several upgrades to the analysis procedure: most
importantly we are now able to calculate very accurate detection efficiencies
also for complex-shaped samples, which we demonstrate by screening a widely-
known 40K souce – a banana. The article closes with a summary and an outlook
in Section 5.

##  2 Detector Operation and Control

The Vue-des-Alpes laboratory is a small underground space in the Swiss Jura
mountains [4]. It does not provide any service besides electricity and network
access, in particular there are no personnel on-site. This means that
operators have to travel there by car for every action on the GeMSE detector.
In order to simplify operation and to reduce travel to the absolute minimum,
i.e., to refill the $\sim$ 360 l of LN2 storage in the laboratory and for
sample changes, all standard operation procedures were automated and/or can be
controlled remotely via the custom-designed slow control system Doberman
(Detector OBsERving and Monitoring ApplicatioN) [12].

Figure 1: Capture of the slow control monitor, showing a remote-controlled LN2
refill during a stable operation period. From top to bottom and left to right,
the panels show: HPGe pre-amplifier voltage; GN2-purge flow and the flow
controller’s temperature; temperature inside the glovebox and HPGe LN2 dewar
valve state; muon veto trigger rate, with an average value of $\sim$ 650 Hz;
HPGe bias voltage and current; and levelmeter capacitance, which can be
correlated to a liquid level inside the HPGe LN2 dewar. The correlation
between the open valve state (reading = 1) and the levelmeter capacitance
(liquid level) increase is apparent, as well as the temperature decrease
inside the glovebox.

The slow control system was updated to enable operation of the facility for
extended periods with only remote access. Slow control readings are handled
asynchronously by a scheduling system to ensure independent, race-free readout
of parameters from the same readout device. The data storage for the readings
is realized with InfluxDB [13] and a graphical web monitor is based on the
Grafana dashboard [14]. A screen capture of the GeMSE slow control dashboard
is shown in figure 1. The Doberman slow control system monitors all critical
performance parameters, e.g., the level inside the HPGe’s LN2 dewar or the
HPGe leakage current, and automatically sends SMS alarms in case readings fall
out of pre-defined parameter ranges. It also allows for remote actions such as
(un)biasing the HPGe crystal, refilling the LN2 dewar connected to the
crystal, and controlling the GN2 purge flow. Further automation of the LN2
handling can optimize the visit schedule and time of on-site detector
operation: the slow control system can trigger a refill of the LN2 dewar
cooling the HPGe crystal from a storage vessel, using the readings of the
capacitive sensor measuring the LN2 level inside the dewar (see figure 1). The
GeMSE data acquisition system can be also operated remotely but is not
integrated in the slow control readings.

Figure 2: Measured event rate of 10 Hamamatsu R11410-21 PMTs in the energy
region between 50 and 2700 keV. (Left) Each data point corresponds to an hour.
The dashed red line represents the expected rate without GN2 purging, assuming
the first measured data point is entirely attributed to 222Rn contamination.
The non-flat part of the spectrum (1.75 days, gray band) is discarded from the
analysis. (Right) The gray histogram corresponds to the first 1.75 days and is
populated by photopeaks from the 222Rn progeny (e.g., the 1765 keV
$\upgamma$-line from 214Bi). The magenta histogram with significantly fewer
entries corresponds to the remaining $\sim$ 17 days of measurement. Peaks
from, e.g., 60Co (1172 keV, 1332 keV) and 40K (1461 keV) are clearly visible
as they do not stem from the air contamination from the first part of the
acquisition.

A typical sample is measured from a few days, for freshly fallen meteorites,
up to 6 weeks, for ultra-pure material candidates for rare-event search
experiments, depending on its size and radioactive contamination. A 14-bit
digital multichannel analyzer (CAEN DT5781A MCA), with real-time digital pulse
processing [15] records the signals from the HPGe detector. It is operated via
the manufacturer’s software in time-stamped list mode [16] which enables the
recording of pulse height and time of every event at 10-ns resolution. The
latter is especially useful for data selection: periods with higher rate in
the detector cavity, right after a sample exchange, or failures in the GN2
purge or the muon veto, can be removed from data before sample analysis. The
effect of purging on the effective acquisition rate decrease is shown in
figure 2 where 10 low-background Hamamatsu R11410-21 PMTs used for the XENONnT
experiment were screened [17, 7].

The signals of the muon veto PMTs are amplified before being fed into a
discriminator and then a coincidence trigger. A gate generator ensures a fixed
width of the trigger signal which is then fed into the MCA as a veto. A muon
veto occurs when at least one of the scintillator panels detects a hit which
involves two of the three PMTs in coincidence. This signal vetoes the MCA for
10 $\upmu$s, inducing an acquisition dead time of about 0.6%.

##  3 Background Characterization

An integral background rate of ($246\pm 2$) counts/day in the region of
interest 100 – 2700 keV was measured in 2016, right after the detector
installation underground [11]. Cosmic neutron-induced spallation is
responsible for the production of most of the short-lived cosmogenic
radionuclides in the detector and shield materials at the earth’s surface,
while this production mechanism can be already considered negligible at depths
of a few tens of m.w.e. [1, 18, 19, 20, 21]. At a moderate rock over
overburden of 620 m.w.e. these isotopes are thus expected to decay away,
further reducing the already ultra-low initial background rate.

### Background Description

A new combined background run of 109.7 days was acquired in 2020. Figure 3
shows the comparison with the background from 2016 with some $\gamma$-lines of
interest labeled. The new background rate is ($164\pm 2$) counts/day or
($82\pm 1$) counts/(day $\cdot\,$kg) in the 100 – 2700 keV region (($91\pm 1$)
counts/(day $\cdot\,$kg) in 40 – 2700 keV). This number is in excellent
agreement with the predicted rate of $\sim$ 80 counts/(day $\cdot\,$kg) in
[11], calculated by applying the half-life correction of the activated
isotopes to the 2016 background spectrum. This background level places GeMSE
among the most sensitive HPGe detectors available [22], even though it is
operated only at a moderate depth.

Figure 3: GeMSE background spectrum during the early detector operation (2016
[11], blue) and now (2020, red). Some $\upgamma$-ray lines are labeled, as
well as the e$+$e$-$ pair production line at 511 keV. A rate reduction of
$\sim$ 33% occurred after about four years of underground operation due to the
decay of the short-lived cosmogenic isotopes. By operating GeMSE underground,
the activation rate is strongly suppressed, therefore the background rate
decreases with the half-life of these isotopes (mainly 56-58Co, 54Mn, 65Zn and
68Ge).

Figure 3 reveals a significant reduction of the background from short-lived
cosmogenically-activated isotopes which were present during early detector
operation. The most important differences between the background spectra from
2016 and 2020 are:

  * •

the suppressed continuous $\upbeta$-spectrum of 68Ga ($Q=2921.1$ keV), a
short-lived daughter of 68Ge ($T$1/2 = 270.8 d), which leads to the visible
rate reduction in the central region of the plot (maximum around $\sim$1300
keV, shown in detail in figure 4);

  * •

the activity decrease for the characteristic 65Zn ($T$1/2 = 244 d) peak at
1115.5 keV;

  * •

the activity decrease for the 54Mn ($T$1/2 = 312.2 d) $\upgamma$-line at 834.8
keV;

  * •

the activity decrease for some prominent $\upgamma$-peaks from the short-lived
cobalt isotopes: the 122.1 keV and 136.5 keV $\upgamma$-lines from 57Co
(T1/2=271.7 d), following the electron capture decay into 57Fe, and the
$\upgamma$-line of 58Co ($T$1/2 = 70.9 d) at 810.8 keV;

  * •

the current rate of the 511 keV line from e$+$e$-$ pair production is slightly
larger than in 2016. This peak is characteristic for large energy depositions
in the HPGe crystal, mostly induced by muons, which points to a potentially
deteriorated muon veto tagging efficiency. This is further discussed below
after evaluating the rate for each background contribution to the overall
spectrum.

### Background Model and Analysis

The background is analyzed by matching the measured counting rates with the
spectra from the different predicted background contributors. These are
obtained by means of detailed Monte Carlo simulations using the Geant4 toolkit
[23, 24]. The GeMSE simulations framework is based on Geant4 version 10.7 and
is publicly available [25]. The flux of cosmic-ray muons is simulated using
the Shielding Geant4 physics list, which accounts for spallation processes and
the propagation of secondaries. Entries in the crystal are vetoed for 10
$\upmu$s following events producing an energy deposit $>$1 MeV in one of the
muon veto panels. The measured muon flux of $(8.3\pm 0.2)\times 10^{-2}$ m-2
s-1 is used as a prior for the fit for this component.

Besides the muons, the model includes the combined contributions from
radioactive isotopes in the HPGe crystal as well as in the Cu and Pb
shielding. The background contribution from the residual 222Rn in the sample
cavity ($\sim$ 0.3 mBq/m3) and the crystal’s Cu cryostat ($\sim$ 0.8 mBq/kg)
are negligible and therefore not included in the final fit. However, some
degeneracy for the background from the Cu shielding and the detector cryostat
is expected given their similar spectral shape. The combined rate for the 56Co
and 58Co isotopes in the Ge crystal and Cu shielding was derived as an upper
limit in the 2016 analysis. Since $\sim$ 20 half-lives for these isotopes have
elapsed between the two background measurements, and under the reasonable
assumption of no new cosmogenic activation underground at 620 m.w.e., these
isotopes are also excluded from the new fit.

The simulated spectra are fitted to the new background measurement and
Bayesian inference is used to determine the detected values/upper limits of
the isotopic activities [11]. Figure 4 shows the best fits to the newly
measured background spectrum from 2020, as well as the one from 2016 for
comparison. A variable bin size proportional to the detector resolution
function is used. The p-value of the fit is calculated according to [26] and
is 0.01. This indicates a good match with the background model, considering
the wide fit range and the fact that some underprediction is expected
especially at low energies, where the measurement contains contributions from
the Compton spectra of all subdominant background sources ignored in the
model.

Figure 4: Best fit (solid red line) to the measured GeMSE background spectrum
(black points) in 2016 (top) and 2020 (bottom). The contributions from the Ge
crystal, Cu and Pb shield, and muons are shown as dashed lines. The
deactivation of the Ge crystal (cyan), discussed in the text, is mainly due to
the suppression of the 68Ga $\upbeta$-spectrum. The fit for the 511 keV
e$+$e$-$ peak shows that the contribution attributed to muon-induced events
has slightly increased compared to the background measurement from 2016. The
fit slightly underpredicts the measured data in the $\sim$ 100 – 200 keV
region, whis is expected as the measured Compton continuum in this range also
includes not simulated residual background sources.

The background contributions induced by the Cu shield (orange) and the HPGe
crystal itself (cyan) have significantly decreased as a result of the decay of
cosmogenically-activated isotopes. This effect is most striking for the HPGe
crystal, which dominated the background in 2016 with the 68Ga
$\upbeta$-spectrum. Now the expected features from other isotopes are also
distinguishable, e.g., the 60Co $\upgamma$-rays at 1173 keV and 1332 keV and
their sum peak at 2505 keV111The fin-like asymmetric shape and the relative
height of the sum peak have geometrical reasons: since the 60Co$\to$60Ni∗
decay happens within the HPGe crystal, the $\upbeta^{-}$ emission is recorded
in most of the cases in coincidence with the gamma lines resulting from the
60Ni∗ de-excitation.. The 210Pb activity extracted from the fit agrees with
the value from external measurements of lead used in the GeMSE shield.

The "effective" (i.e., not vetoed) muon flux derived from the fit has
increased by 20% compared to 2016, probably due to a worsened tagging
efficiency of the setup. The fit successfully models this component, although
the model lacks precise knowledge about the features of muon identification in
the laboratory such as the exact rock composition/profile and the validity on
the energy threshold for the simulated muon veto tagging efficiency. The
resulting activity of the considered isotopes is summarized in table 1 and
compared to these derived from the 2016 background spectrum [11].

Table 1: Radioactive contaminants contributing to the GeMSE background
extracted from the fit of the simulated components to the measured spectrum. A
detection is quoted with uncertainties (68% confidence interval) while upper
limits are defined by the 95% confidence interval. The values from 2016 are
taken from [11]. The muon flux quoted here corresponds to an "effective" flux
passing the muon veto.

| Specific isotope activity [$\upmu$Bq/kg]
---|---
Source |  |  |  |  |  |  |  |  |  |
|  65Zn |  68Ge |  60Co |  58Co |  57Co |  56Co |  54Mn |  238U |  232Th |  40K
Ge crystal (2016) | 77${+15\atop-13}$ | 313${+17\atop-23}$ | $<118$ | $<3.5$ | 11${+2\atop-1}$ | $<43$ | $<23$ | – | – | –
Ge crystal (2020) | $<5.7$ | $<15.1$ | $<26.4$ | – | 3.6${+1.3\atop-1.1}$ | – | $<5.4$ | – | – | –
Cu shield (2016) | – | – | 43${+7\atop-5}$ | $<26$ | 374${+48\atop-39}$ | $<7.6$ | $<56$ | 104${+13\atop-15}$ | 42${+9\atop-10}$ | $<196$
Cu shield (2020) | – | – | 35${+3\atop-3}$ | – | 121${+23\atop-24}$ | – | $<11.5$ | 53${+7\atop-7}$ | 30${+5\atop-6}$ | 103${+36\atop-29}$
Pb shield (2016) |  210Pb activity $<8.2$ Bq/kg
Pb shield (2020) |  210Pb activity $=7.4{+0.5\atop-0.5}$ Bq/kg
Muons (2016) | Flux = $1.19{+0.07\atop-0.09}\times 10^{-5}$ cm-2 s-1
Muons (2020) |  Flux = $1.44{+0.04\atop-0.05}\times 10^{-5}$ cm-2 s-1

### Muon Veto Efficiency

The results presented in figure 4 and table 1 show that about 70% of the
current background in the GeMSE detector is caused by muon-induced events,
even after the reduction by the muon veto. This motivates studies to upgrade
the muon veto by enlarging the geometrical coverage of the cavity with
additional plastic scintillator panels.

In order to better understand the current muon veto capabilities of GeMSE, as
well as to confirm the spectral shape of this component used in the background
model, a measurement simultaneously recording interactions in the Ge crystal
and the muon veto was performed. Contrary to the default settings, in which
reading the signals from the HPGe detector is directly vetoed after a muon
veto trigger, in this measurement the (logic) muon veto trigger signal was
read out by using the second channel of the MCA recording the time stamps and
pulse heights of the signals from the HPGe crystal. Both MCA channels feature
the same clock, i.e., the time stamps from both channels can be directly
compared; it was verified that muon veto signals occurring in coincidence with
HPGe signals were not delayed with respect to the latter by their acquisition
electronics.

Figure 5: Measured HPGe signals in coincidence with the muon veto detector
(purple dots). Their spectral shape matches that of the predicted muon-induced
background extracted from the multi-component fit to the background data
(magenta histogram), i.e., remaining after the active muon veto. The
comparison of vetoed (data points) and total (data points plus histogram) muon
events yields a muon veto tagging efficiency ($31\pm 4$)%, shown in the bottom
panel with the 1$\sigma$ uncertainty. The increased efficiency for energies
$<200$ keV can be explained by the accidental pairing of events in this region
(happening at higher rate) with fake muon veto triggers produced by readout
noise in the scintillator panels. The gray histogram shows the fitted
background contribution (magenta histogram) scaled according to this tagging
efficiency.

To emulate the finite veto window of the standard acquisition mode, the muon
veto data were searched for entries within the 10 $\upmu$s interval preceding
every recorded HPGe signal. This allowed selecting the Ge events with a
coincident muon veto signal which are otherwise discarded. The measured
spectrum of muon-induced HPGe events is shown in figure 5 (purple data points)
together with the simulated muon veto spectrum fitted to the GeMSE background
(magenta histogram) from figure 4. Their spectral shapes agree very well. It
must be noted that the data points represent entries recorded by the HPGe
detector which were identified by the muon veto, while the histogram from the
fit corresponds to the remaining, non-tagged muon-induced background. The
ratio of tagged and total muon events can be used to estimate the muon veto
tagging efficiency, shown in the bottom panel. The average efficiency in the
range 100 – 2700 keV (dashed purple line) is ($31\pm 4$)%. The agreement of
the spectral shapes of the simulated and the recorded muon-induced data
validates the Monte Carlo prediction for this background component.

##  4 Improvements in Sample Analysis

The analysis of the sample spectra measured with GeMSE generally follows the
procedure described in [11], but features several upgrades to the simulations
framework used to estimate the detection efficiency as well as the MCA
software and firmware. The utilized software framework is available at [27] as
a Docker container [28] and includes the Bayesian Analysis Toolkit (BAT) [29]
required to evaluate activities for ultra low-background counting rates.

In order to obtain the activity for a given radioisotope, corrections need to
be applied for the branching ratio of the observed $\upgamma$-lines as well as
for the detection efficiency of the HPGe crystal. This efficiency depends on
the energy and the geometrical placement of the sample inside the measurement
cavity with respect to the HPGe crystal and shield. It is therefore necessary
to accurately model the combined detector-sample geometry in the GeMSE
simulations, also to ensure that self-shielding effects are properly accounted
for.

### Estimating the Dead Layer Thickness

A 1 mm-thick OFHC copper cryostat prevents low-energy X-rays from penetrating
the Ge crystal. In addition to the cryostat shielding, special care must be
taken to properly model the dead layer of the crystal itself. This dead layer
refers to the n+-contact consisting of Li atoms diffused into the Ge; it
behaves as an insensitive region which only absorbs radiation. Its thickness
is smallest just after the production of the HPGe crystal and gradually
increases with time as the Li atoms diffuse into the crystal, a process
greatly inhibited by cryogenic conditions [30, 31, 32]. An increase of the
inactive dead layer reduces the remaining sensitive volume and leads to a
decrease in the detector efficiency. The p+-contact forming the p-n junction
of the detector, implanted in the inner region of the crystal, has a thickness
of $\sim$ 0.3 $\upmu$m and thus does not significantly affect the efficiency
[33].

A dead layer thickness of ($0.67\pm 0.01$) mm was inferred in 2016 by using a
certified multi-gamma calibration source [11], which was then implemented in
the simulated model. An increase in the dead layer thickness was expected due
to the few thermal cycles that the detector underwent in the past years,
including a longer three-month shutdown during the start of the COVID-19
pandemic in Europe. A new determination of the dead layer was conducted using
the peak-ratio approach [34, 35]: a spectrum from a calibrated 133Ba point-
source located at a precisely known distance from the top of the detector
cryostat (138 mm) was recorded, and the geometry of this calibration source
and its holder was implemented in the Geant4 detector model. The ratio of the
signal strengths of the 81 keV and 356 keV full-absorption peaks was measured
and compared to the ratio obtained from simulations for different thicknesses
of the passive layer.

Figure 6: The dead layer thickness is obtained using the ratio of the signal
strengths of the 81 keV and 356 keV full absorption peaks of a calibrated
133Ba point-source placed 138 mm above the cryostat endcap. Every data point
corresponds to one simulation ($4\times 10^{6}$ primaries) for a given dead
layer. The exponential fit to this data and the 1$\sigma$ uncertainty are
represented by the dashed line and the gray band, respectively. The
experimentally measured ratio and its uncertainties are given by the red solid
and dashed/dotted lines, respectively. The obtained dead layer thickness is
($0.94\pm 0.06$) mm.

Figure 6 shows the simulated signal strength ratios, which can be well
described by an exponential function (dashed black line), as well as the
measured ratio (red solid line) with uncertainties. By comparing the
experimental ratio and the simulated line, and repeating the process for the
measured uncertainty and the 1$\sigma$ band of the exponential fit, the
detector’s effective dead layer thickness of ($0.94\pm 0.06$) mm is obtained.
This corresponds to an increase of $\sim$ 0.37 mm over the five years of
operation.

This value is reasonably small compared to dead layer increases reported for
similar detectors and time [31, 35, 36], although a direct comparison is not
possible given the different thermal cycles and crystal sizes. The newly
measured thickness has been implemented in the Geant4 GeMSE model. Simulations
of the commonly evaluated radioisotopes in low-background measurements
indicate that the efficiency loss in the 100 – 2700 keV range is not strongly
affected by this increase, but becomes relevant for lines around the lower
energy threshold with branching ratios below the percent level.

### Modeling Complex Geometries

A common problem for the proper determination of the detection efficiency, and
thus the sample activity, is the accurate implementation of the shape and
placement of screened samples in the simulation. This is equally relevant for
rare-event searches, for which counting rates are extremely low, and meteorite
research, where the usual samples have complex, non-uniform shapes and
therefore their $\upgamma$-ray self-absorption is not uniform either. The
efficiencies for all meteorites screened with GeMSE so far [9, 10] were
approximated by assuming the shape of an oblate ellipsoid.

To address this problem, the option to import 3D CAD models into Geant4 was
added to the GeMSE simulation. The CADMesh interface [37, 38] utilizes
dedicated libraries to read the CAD file and directly converts it to a
Geant4-readable format. The 3D CAD files can be generated using precision
optical 3D scanners or even smartphone applications. The CAD models of the
samples are imported as triangular facet surface meshes in binary format and
returned as a G4TessellatedSolid object which can be handled in the Geant4
detector construction class.

As a complex-shaped example to showcase this feature, a banana was screened
with GeMSE. Bananas are rich in potassium (0.36% per mass [39]), which
includes 40K ($T$1/2 = 1.25 $\times$ $10^{9}$ y) with a natural isotopic
abundance of 0.0117%. The 40K decay via electron capture, occurring in 11% of
the cases, leads to an 40Ar excited state which emits a 1461 keV
$\upgamma$-ray when de-exciting to the ground state. Given the high potassium
content of a banana and the low-background regime of GeMSE, a measurement time
of a few hours is already sufficient to identify this peak. However, its
particularly complex shape is hard to implement with the default volume
generators provided by Geant4.

Figure 7: Geant4 rendering of the screened banana inside the GeMSE sample
cavity. The sample was 3D-scanned and imported into the detector model. (Left)
The contours of the three-layer passive shielding: Cu-OFE (orange), low-
activity Pb (cyan) and standard Pb (dark blue). The cold finger and the
crystal’s OFHC Cu cryostat (orange), the PTFE holder (white) and the banana
are shown as solid bodies. (Right) A close-up visualization of the precise
implementation of the PTFE holder and the banana with respect to the HPGe
endcap.

A high-resolution 3D scanner was used to generate the CAD model of the banana
before placing it inside the detector cavity. For this purpose a light-weight
PTFE sample holder was designed and also imported into the detector model. The
simulated geometry is shown in figure 7. 108 primary 40K isotope decays were
generated within the simulated banana volume and the energy deposits in the
HPGe crystal were recorded. The resulting detection efficiency of the Ge
crystal for the full-absorption 1461 keV $\upgamma$-rays is ($0.947\pm
0.009$)%, almost 10% higher than the value obtained for a simplified cylinder
placed on top of the PTFE sample holder, with similiar radius and length and
equal volume and density as the banana..

The spectrum acquired during a five-day run is shown in figure 8 along with
the simulated spectrum corrected for the detector energy resolution function.
The measured and simulated width (FWHM) and the area of the 1461 keV peak
match, although the measured spectrum features a tail towards lower energies
and thus does not have the same height as the simulated $\upgamma$-peak.
Events which survived the pile-up rejection of the data acquisition system or
for which the charges were not entirely collected can be responsible for this.
The impact of these artifacts is generally negligible as their contribution to
the observed 40K peak is below the 0.1% level. The Compton edge at $\sim$ 1250
keV and the backscatter peak ($\sim$ 250 keV) agree very well in data and
simulation. The GeMSE background spectrum (red histogram from figure 3),
several orders of magnitude lower than the banana spectrum, is also shown.

Figure 8: Measured banana spectrum (brown) and GeMSE background (red, taken
from figure 3). The simulated spectrum is also shown (blue) after a
convolution with the detector resolution function. The backscatter peak at
$\sim$ 250 keV is visible, as well as the Compton edge at $\sim$ 1250 keV. The
area under the Gaussian full absorption peak at 1460.8 keV matches that of the
simulation.

The 40K activity was calculated following the standard analysis approach
described in [11], where the activity of the low-mass, low-background PTFE
sample holder used for this measurement was neglected. The obtained activity
is 42.6${+0.9\atop-0.8}$ Bq, which agrees with the usually quoted 40K activity
for a standard banana. The mass proportion of potassium in banana pulp and
skin are $\sim$ 0.36% [39] and $\sim$ 0.93% [40], respectively, and the banana
skin contributes on average 36.6% to the total mass [41]. Given the total
banana mass of $m=265.15$ g, this corresponds to 1.50 g of natK in the fruit.
The measured specific activity of 1 g of natK is thus 42.6 Bq / 1.50 g = (28.4
$\pm$ 0.6) Bq/g, in excellent agreement with the literature values of 27.3 –
31.3 Bq/g [42]. This exercise demonstrates the potential improvement of
screening measurements by simulating the precise three-dimensional shape of
the screened sample. This is especially relevant for the radioassay of
meteorite samples which in most cases must not be manipulated and often come
in complex shapes.

##  5 Summary and Outlook

The GeMSE low-background screening facility was installed in the Vue-des-Alpes
underground laboratory with a moderate rock overburden in 2015 and has been
operative since, with very limited downtime periods. This article presented
the current status of the experiment focusing on three notable aspects:
autonomous remote operation, the detector’s ultra-low background rate which is
now muon dominated, and improvements on the sample analysis. Thanks to
upgrades to the monitoring sensors, such as the newly developed LN2 levelmeter
for the HPGe dewar, as well as to the custom-made slow control system Doberman
[12], the facility installed in a rather remote location in a non-staffed
laboratory is now automated and remotely controlled to a large extent. The
autonomous operation time of the detector without the need for people on-site
is now $\gtrsim$ 6 weeks, larger than the typical sample measuring time.

The most recent background run acquired over 109.7 days in 2020 resulted in a
background rate of ($164\pm 2$) counts/day or ($82\pm 1$) counts/(day
$\cdot\,$kg) in the range 100 – 2700 keV. The significant decrease of 33%
compared to the rate presented in [11] after five years of underground
operation agrees with the projected activity drop for the cosmogenically-
activated isotopes. This has been verified and quantified by matching the
experimental data with the simulated background contributions from a precise
Geant4 model of the detector. GeMSE’s background rate is now dominated by
muon-induced events.

The spectrum of such muon-induced events in the HPGe crystal has been measured
by using the muon veto to tag these events rather than discarding them. This
allows extracting a muon veto efficiency of about 31%, in good agreement with
the Geant4 simulations of the detector, and confirms the shape of the muon-
induced sprectrum used to fit the data.

The ongoing deactivation of the remaining short-lived isotopes in the Ge
crystal and the shielding will only mildy reduce the background rate down to
78 counts/(day $\cdot\,$kg) in 2025, hence the next step to further decrease
the background is to target the dominating muon background: an additional top
scintillator muon veto panel with larger geometrical coverage would achieve
efficiencies of 50 – 60%, reducing the background rate down to 55 – 40
counts/(day $\cdot\,$kg).

The GeMSE detector model was improved to more accurately determine the
detection efficiency for samples. The thickness of the detector’s dead layer
was investigated using the peak-ratio approach with a certified 133Ba
calibration source and yielded a value of ($0.94\pm 0.06$) mm, 0.37 mm thicker
than the measured value in 2016 [11], probably caused by several thermal
cycles over the past years. Furthermore, 3D geometries of complex-shaped
objects can now be imported into the simulations model, thus mitigating the
inaccuracies in the detection efficiency estimations for the examined samples.
To exemplify this feature, a banana was screened with GeMSE and the detection
efficiency was derived from the accurately implemented 3D model. This method
will be highly advantageus for the future meteorite screening program.

## Acknowledgments

This work was supported by the European Research Council (ERC) grant number
724320 (ULTIMATE) and the German Ministry for Education and Research (BMBF)
grant number 05A17VF1. The facility was initially funded by the Swiss National
Science Foundation (SNF) interdisciplinary grant number 152941. We gratefully
acknowledge the continuous and remarkable support from the Albert Einstein
Center for Fundamental Physics at the University of Bern, especially from
Prof. Michele Weber, Roger Hänni and his team, and Jean-Michel Villeumier. We
thank Dr. Rugard Dressler (PSI Villigen) for providing the calibration sources
used to characterize the detector. We also thank the Bachelor students from
the University of Freiburg who helped to improve GeMSE over the last years:
Thomas Kok, Anne K. Becker, Lukas B. Grunwald and Baris Kiyim.

## References

  * [1] G. Heusser, _Low-radioactivity background techniques_ , _Ann. Rev. Nucl. Part. Sci._ 45 (1995) 543.
  * [2] I. Leya and J. Masarik, _Cosmogenic nuclides in stony meteorites revisited_ , _Meteoritics & Planetary Science_ 44 (2009) 1061.
  * [3] P. Weber, B. A. Hofmann, T. Tolba et al., _A gamma-ray spectroscopy survey of Omani meteorites_ , _Meteoritics & Planetary Science_ 52 (2017) 1017 [1602.00915].
  * [4] Y. Gonin, J. Busto and J.-L. Vuilleumier, _The “La Vue-des-Alpes” underground laboratory_ , _Review of Scientific Instruments_ 74 (2003) 4663.
  * [5] XENON collaboration, E. Aprile et al., _Material radioassay and selection for the XENON1T dark matter experiment_ , _Eur. Phys. J. C_ 77 (2017) 890 [1705.01828].
  * [6] XENON collaboration, E. Aprile et al., _Projected WIMP sensitivity of the XENONnT dark matter experiment_ , _JCAP_ 11 (2020) 031 [2007.08796].
  * [7] XENON collaboration, E. Aprile et al., _Material radiopurity control in the XENONnT experiment_ , 2112.05629.
  * [8] DARWIN collaboration, J. Aalbers et al., _DARWIN: towards the ultimate dark matter detector_ , _JCAP_ 11 (2016) 017 [1606.07001].
  * [9] Å. V. Rosén, B. A. Hofmann, M. von Sivers et al., _Radionuclide activities in recent chondrite falls determined by gamma-ray spectrometry: Implications for terrestrial age estimates_ , _Meteoritics & Planetary Science_ 55 (2020) 149.
  * [10] Å. V. Rosén, B. A. Hofmann, F. Preusser et al., _Meteorite terrestrial ages in Oman based on gamma spectrometry and sediment dating, focusing on the Ramlat Fasad dense collection area_ , _Meteoritics & Planetary Science_ 56 (2021) 2017.
  * [11] M. von Sivers, B. Hofmann, Å. V. Rosén et al., _The GeMSE facility for low-background $\gamma$-ray spectrometry_, _JINST_ 11 (2016) P12017 [1606.03983].
  * [12] P. Zappa, L. Bütikofer, D. Coderre et al., _A versatile and light-weight slow control system for small-scale applications_ , _JINST_ 11 (2016) T09003 [1607.08189].
  * [13] “InfluxDB Time Series Platform.” https://www.influxdata.com/products/influxdb/.
  * [14] “Grafana Labs - Grafana.” https://grafana.com/grafana/.
  * [15] “DT5781 - Dual/Quad Digital Multi Channel Analyzer.” https://www.caen.it/products/dt5781.
  * [16] “MC2 Analyzer - Graphical software tool for digitizers running DPP-PHA firmware.” https://www.caen.it/products/mc2analyzer.
  * [17] XENON collaboration, E. Aprile et al., _Lowering the radioactivity of the photomultiplier tubes for the XENON1T dark matter experiment_ , _Eur. Phys. J. C_ 75 (2015) 546 [1503.07698].
  * [18] S. Cebrián, _Cosmogenic activation of materials_ , _Int. J. Mod. Phys. A_ 32 (2017) 1743006 [1708.07449].
  * [19] S. Cebrián, _Cosmogenic activation in double beta decay experiments_ , _Universe_ 6 (2020) 162 [2010.02381].
  * [20] C. Zhang, D.-M. Mei, V. Kudryavtsev et al., _Cosmogenic activation of materials used in rare event search experiments_ , _Astropart. Phys._ 84 (2016) 62 [1603.00098].
  * [21] L. Baudis, A. Kish, F. Piastra et al., _Cosmogenic activation of xenon and copper_ , _Eur. Phys. J. C_ 75 (2015) 485 [1507.03792].
  * [22] M. Laubenstein and I. Lawson, _Low Background Radiation Detection Techniques and Mitigation of Radioactive Backgrounds_ , _Front. in Phys._ 8 (2020) 506.
  * [23] Geant4 collaboration, S. Agostinelli et al., _Geant4: A Simulation toolkit_ , _Nucl. Instrum. Meth. A_ 506 (2003) 250.
  * [24] Geant4 collaboration, J. Allison et al., _Recent developments in Geant4_ , _Nucl. Instrum. Meth. A_ 835 (2016) 186.
  * [25] “GeMSE_MC - GeMSE Geant4 code to simulate detector efficiencies.” https://github.com/AG-Schumann/GeMSE_MC.
  * [26] F. Beaujean, A. Caldwell, D. Kollár et al., _$p$ -values for model evaluation_, _Phys. Rev. D_ 83 (2011) 012004.
  * [27] “GeMSE_environment - Software environment for GeMSE analysis and simulations.” https://github.com/AG-Schumann/GeMSE_environment.
  * [28] D. Merkel, _Docker: lightweight linux containers for consistent development and deployment_ , _Linux journal_ 2014 (2014) 2\.
  * [29] A. Caldwell, D. Kollár and K. Kröninger, _BAT – The Bayesian analysis toolkit_ , _Comput. Phys. Commun._ 180 (2009) 2197 [0808.2552].
  * [30] J.-M. Laborie, G. Le Petit, D. Abt et al., _Monte Carlo calculation of the efficiency calibration curve and coincidence-summing corrections in low-level gamma-ray spectrometry using well-type HPGe detectors_ , _Appl. Radiat. Isot._ 53 (2000) 57.
  * [31] N. Quang Huy, _The influence of dead layer thickness increase on efficiency decrease for a coaxial HPGe p-type detector_ , _Nucl. Instrum. Meth. A_ 621 (2010) 390.
  * [32] N. Huy, B. Do and V. Xuân, _Study on the increase of inactive germanium layer in a high-purity germanium detector after a long time operation applying MCNP code_ , _Nucl. Instrum. Meth. A_ 573 (2007) 384.
  * [33] T. S. Kyle, G. Geurkov, E. Roth et al., _Improved Efficiency at Low Energies with P-Type High Purity Germanium Detectors_ , in _IEEE NSS/MIC Conference Proceedings, Seattle, WA_ , p. 8, 2014.
  * [34] H. Jiang, Q. Yue, Y.-L. Li et al., _Measurement of the dead layer thickness in a p-type point contact germanium detector_ , _Chinese Physics C_ 40 (2016) 096001 [1601.04305].
  * [35] L. Truong, V. Ba, N. T. Truong Huu et al., _Determination of the dead-layer thickness for both p- and n-type HPGe detectors using the two-line method_ , _J. Radioanal. Nucl. Chem._ 315 (2018) 95.
  * [36] A. Azbouche, M. Belamri and T. Théophile, _Study of the germanium dead layer influence on HP(Ge) detector efficiency by Monte Carlo simulation_ , _Radiat. Detect. Technol. Methods_ 2 (2018) 45.
  * [37] C. M. Poole, I. Cornelius, J. V. Trapp et al., _A CAD Interface for GEANT4_ , _Australas. Phys. Eng. Sci. Med._ 35 (2012) 329 [1105.0963].
  * [38] C. Poole, I. Cornelius, J. Trapp et al., _Fast tessellated solid navigation in GEANT4_ , _IEEE Transactions on Nuclear Science_ 59 (2012) 1695.
  * [39] J. S. Sidhu and T. A. Zafar, _Bioactive compounds in banana fruits and their health benefits_ , _Food Quality and Safety_ 2 (2018) 183.
  * [40] J. Archibald, _Nutrient Composition of Banana Skins_ , _Journal of Dairy Science_ 32 (1949) 969.
  * [41] M. Soltani Firouz, R. Alimardani and M. Omid, _Some physical properties of full-ripe banana fruit (cavendish variety)_ , _Intl. J. Agr. Sci. Res. Technol._ 1 (2011) 1.
  * [42] S. B. Samat, S. Green and A. H. Beddoe, _The activity of one gram of potassium_ , _Physics in Medicine and Biology_ 42 (1997) 407.


# Maximal estimates for fractional Schrödinger equations in scaling critical
magnetic fields

Haoran Wang  Department of Mathematics, Beijing Institute of Technology,
Beijing 100081, China<EMAIL_ADDRESS>and  Jiye Yuan  Department of
Mathematics, Beijing Key Laboratory on Mathematical Characterization,
Analysis, and Applications of Complex Information, Beijing Institute of
Technology, Beijing 100081, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we combine the argument of [12] and [27] to prove the maximal
estimates for fractional Schrödinger equations
$(i\partial_{t}+\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})u=0$ in the purely
magnetic fields which includes the Aharonov-Bohm fields. The proof is based on
the cluster spectral measure estimates. In particular $\alpha=1$, the maximal
estimate for wave equation is sharp up to the endpoint.

Key Words: Fractional Schrödinger equation; Maximal estimate; Aharonov-Bohm
potential; Spectral measure.
AMS Classification: 42B25, 35Q40, 35Q41.

##  1\. Introduction and main results

In this paper, we consider the Cauchy problem of fractional Schrödinger
equation

$\displaystyle\begin{cases}&(i\partial_{t}+\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})u=0,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{2}\setminus\\{0\\},\\\
&u(0,x)=f(x),\end{cases}$ (1.1)

where $\alpha$ is a positive real number and the operator
$\mathcal{L}_{\mathbf{A}}$ is the magnetic Schrödinger operator given by

$\mathcal{L}_{\mathbf{A}}=\Big{(}i\nabla+\frac{\mathbf{A}(\hat{x})}{|x|}\Big{)}^{2},\quad
x\in\mathbb{R}^{2}\setminus\\{0\\},\quad\hat{x}=\frac{x}{|x|}\in\mathbb{S}^{1},$

where $\mathbf{A}\in W^{1,\infty}(\mathbb{S}^{1},\mathbb{R}^{2})$ satisfies
the transversality condition

${\mathbf{A}}(\hat{x})\cdot\hat{x}=0,\qquad\text{for all }x\in\mathbb{R}^{2}.$
(1.2)

In particular, a typical example of ${\bf A}$ is known as the Aharonov-Bohm
potential

${\bf
A}(\hat{x})=\sigma\Big{(}-\frac{x_{2}}{|x|},\frac{x_{1}}{|x|}\Big{)},\quad\sigma\in\mathbb{R},$
(1.3)

which was initially used to study one of the most interesting and intriguing
effect of quantum physics by Aharonov and Bohm in [1]. This Aharonov-Bohm
effect occurs when electrons propagate in a domain with a zero magnetic field
but with a nonzero vector potential, see [23] and the references therein. In
another physic effect, the fermionic charges can be non-integer multiples of
the Higgs charges in another typical cosmic-string scenarios observed by
Alford and Wilczek [2]. As the flux is quantized with respect to the Higgs
charge, this will lead to a non-trivial Aharonov-Bohm scattering of these
fermions. From the mathematical point, the operator $\mathcal{L}_{\mathbf{A}}$
can be extended as a self-adjoint operator (see [11]) and the fractional
operator $\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}$ can be defined from the
spectral perspective, see [14] for the fractional operator. The solution of
(1.1) can be written as

$u(t,x)=e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f(x),$ (1.4)

which includes the usual wave and Schrödinger equation as two special cases.
The dispersive and Strichartz estimates associated with the magnetic
Schrödinger operator $\mathcal{L}_{\mathbf{A}}$ have been extensively studied
in [11, 13, 16] and we also refer to [12, 15] for the resolvent estimates.

In this paper, we consider the the minimal regularity of initial data for
which the above solution (1.4) pointwisely converges to the initial data $f\in
H_{\bf A}^{s}(\mathbb{R}^{2})$,

$\lim_{t\to 0}e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f(x)=f(x),\quad
a.\,e.$ (1.5)

that is, the pointwise convergence problem related to the operator
$\mathcal{L}_{\mathbf{A}}$. The problem for free Schrödinger
$(i\partial_{t}-\Delta)u=0$ with initial data $f\in H^{s}$ was first proposed
by Carleson in [7] (see also [17] ). In this classical case, the solution can
be formally expressed by using Fourier transform as

$u(t,x)=e^{it\Delta}f(x)=\int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}e^{-2\pi
it|\xi|^{2}}\hat{f}(\xi)d\xi.$ (1.6)

The standard routine for the pointwise convergence of the free Schrödinger
operator (1.6) is to find the minimal regularity of $f$ for which the
corresponding maximal estimate holds as follows

$\left\|\sup_{0<t<1}|e^{it\Delta}f(x)|\right\|_{L^{q}(B^{n}(0,1))}\leq
C_{n,q,s}\|f\|_{H^{s}(\mathbb{R}^{n})}$ (1.7)

where $B^{n}(0,1)$ is the unit ball in $\mathbb{R}^{n}$.

In one dimension, Carleson proved that (1.7) holds for $s\geq\frac{1}{4}$. In
1981, Dahlberg and Kenig [8] proved that the result obtained by Carleson is
sharp. In higher dimension for $n\geq 2$, Sjölin [26] and Vega [28]
independently showed that (1.7) holds for $s>\frac{1}{2}$. Later, this range
was improved by Lee [19] to $s>\frac{3}{8}$ for $n=2$, and by Bourgain [5] to
$s>\frac{1}{2}-\frac{1}{4n}$ for $n\geq 3$. Bourgain also proved that
$s\geq\frac{1}{2}-\frac{1}{n}$ is necessary for (1.7) to hold. Du, Guth and Li
[9] obtained $s>\frac{1}{3}$ in dimension $n=2$. Recently, Du and Zhang [10]
proved that (1.7) holds for $s>\frac{n}{2(n+1)}$ for $n\geq 3$. Luc$\grave{a}$
and Rogers obtained the necessary condition $s\geq\frac{n}{2(n+2)}$ and later
Bougain obtained the best necessary condition $s\geq\frac{n}{2(n+1)}$ up to
now.

It is nature to ask the same problem for more general dispersive equations
associated with Schrödinger operator with potentials. However, due to the lack
of Fourier transform, there are much fewer results compared with the free
Schrödinger case. The picture in this direction is far to be complete since
many powerful tools (e.g. decoupling, polynomial decomposition) cannot be
applied in this setting. It worths to mention that one of the most interesting
potentials, the so-called inverse-square potential, which prevents us from
using Fourier transform to give the explicit expression of the solution and
hence the related techniques of Fourier analysis fail to work. However, by
replacing the Fourier transform by the Hankel transform, the authors of [21]
established the maximal estimates for Schrödinger equation with inverse-square
potential, in which they write the solution in terms of spherical harmonics.
For more details, we refer the interested readers to [21] and the references
therein. As for other types of maximal estimates, one is referred to [3, 4]
for quadratic Weyl sum and [18] for average over space curve. Even though,
almost all of the results are far to be sharp in the setting of equations with
potentials.

To the best of our knowledge, there are few papers on the maximal estimates
related to the magnetic operator $\mathcal{L}_{\mathbf{A}}$ available, for
which we start a new program to study the maximal estimates associated with
the operator $\mathcal{L}_{\mathbf{A}}$. To be specific, we try to establish
the local- and global-in-time maximal estimates associated with the operator
$\mathcal{L}_{\mathbf{A}}$. The proof is based on the cut-off spectral measure
estimates. To state the main theorems, we define the distorted Sobolev space
as follows

$\displaystyle
H^{s}_{\mathbf{A}}(\mathbb{R}^{2}):=(I+\mathcal{L}_{\mathbf{A}})^{-\frac{s}{2}}L^{2}(\mathbb{R}^{2}).$
(1.8)

In particular, we shall write

$\|f\|_{H^{s}_{\mathbf{A}}(\mathbb{R}^{2})}:=\|(I+\mathcal{L}_{\mathbf{A}})^{\frac{s}{2}}f\|_{L^{2}(\mathbb{R}^{2})},$
(1.9)

in what follows. Now we are in the position to state our main results.

######  Theorem 1.1 (Local estimate).

Let $u(t,x)=e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f(x),\alpha>0$ be
the solution to (1.1) with $f\in H^{s}_{\mathbf{A}}(\mathbb{R}^{2})$, then for

$\bullet$ either $2\leq q\leq 6$ and
$s>\frac{\alpha}{4}(\frac{6}{q}-1)+(1-\frac{2}{q})$,

$\bullet$ or $6\leq q<\infty$ and $s>1-\frac{2}{q}$,

there exists a constant $C$ such that

$\begin{split}\big{\|}\sup_{|t|\leq
1}|e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f|\big{\|}_{L^{q}(\mathbb{R}^{2})}\leq
C\|f\|_{H_{\mathbf{A}}^{s}(\mathbb{R}^{2})}.\end{split}$ (1.10)

As a direct consequence of Theorem 1.1 with $q=2$ for
$0<\alpha\leq\frac{4}{3}$ and $q=6$ for $\alpha>\frac{4}{3}$, we have the
following pointwise convergence result.

######  Corollary 1.2.

Let $s>\frac{\alpha}{2}$ when $0<\alpha\leq\frac{4}{3}$ or $s>\frac{2}{3}$
when $\alpha>\frac{4}{3}$, then there holds the pointwise convergence (1.5)
for $f\in H^{s}_{\mathbf{A}}(\mathbb{R}^{2})$.

######  Remark 1.3.

In particular $\alpha=1$, i.e., the half-wave operator, from Rogers and
Villarroya in [25, Theorem 1], the result is sharp up to endpoint
$s=\frac{1}{2}$. But the result is far to be sharp in the most interesting and
challenging Schrödinger case, that is, $\alpha=2$. We will continue
investigating the problem in future work.

######  Remark 1.4.

In the free case ${\bf A}\equiv 0$, in which the Fourier transform is
available, Sjölin and Walther independently used the stationary phase to
obtain the corresponding maximal estimate for the operator
$e^{it(-\Delta)^{\frac{\alpha}{2}}}$. In one dimension Sjölin [26] obtained
the pointwise convergence holds for $s\geq\frac{1}{4}$ for $\alpha>1$ which is
sharp, and Walther [29] proved that the pointwise convergence holds for
$s>\frac{\alpha}{4}$ for $0<\alpha<1$, which is almost sharp up to the
endpoint. In higher dimension, for $\alpha>1$ Sjölin [26] obtained the result
of $s>\frac{1}{2}$ in $\mathbb{R}^{n}$ if $n\geq 3$ and $s\geq\frac{1}{2}$ if
$n=2$, and the pointwise convergence for the free Schrödinger equation for
$0<\alpha<1$ in $\mathbb{R}^{n}$ with $n\geq 2$ is left to be solved.

######  Theorem 1.5 (Global estimate).

Let $u(t,x)=e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f(x),\alpha>0$ be
the solution to (1.1) with $f\in\dot{H}^{s}_{\mathbf{A}}(\mathbb{R}^{2})$,
then for $6\leq q<\infty$ and $s>1-\frac{2}{q}$, we have

$\begin{split}\big{\|}\sup_{t\in\mathbb{R}}|e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f|\big{\|}_{L^{q}(\mathbb{R}^{2})}\leq
C\|f\|_{\dot{H}_{\mathbf{A}}^{s}(\mathbb{R}^{2})}.\end{split}$ (1.11)

######  Remark 1.6.

These results extend the local maximal estimates in Theorem 1.1 to global ones
when $q\geq 6$. In the case of $\alpha=1$, which corresponds to the wave
equation with a magnetic field, the global maximal estimate is sharp up to the
endpoint regularity, we refer to [25, Theorem 2]. In the Schrödinger case
$\alpha=2$, from [24, Theorem 8], the regularity assumption $s>1-\frac{2}{q}$
is necessary for (1.11) up to the endpoint $s=1-\frac{2}{q}$.

We close this section by briefly stating the organization of the whole paper
as follows. In Sec.2, we provide the main tools needed in the proof of our
main results, i.e., the spectral measure estimates. In Sec.3 and 4, we prove
Theorem 1.1 and Theorem 1.5 respectively.

##  2\. preliminaries

In this section, we provide the key spectral cluster estimates, which play a
critical role in our proof. For $0<\epsilon\leq 1$, we define the spectral
projector associated with the magnetic Schrödinger operator
$\mathcal{L}_{\mathbf{A}}$ on the frequencies $[k,k+\epsilon]$ by

$\chi_{k,\epsilon}(\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})=\int_{0}^{\infty}\chi_{[k,k+\epsilon]}(\lambda^{\alpha})\,dE_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda),\qquad\alpha>0,$
(2.1)

where $\chi_{E}(s)$ is the characteristic function on the set $E$ and
$dE_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda)$ is the spectral measure given
by

$dE_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda)=\frac{d}{d\lambda}E_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda)d\lambda=\frac{\lambda}{i\pi}\big{(}R(\lambda+i0)-R(\lambda-i0)\big{)}d\lambda.$
(2.2)

Here $R(\lambda\pm i0)$ is the (outcoming/ingoing) resolvent

$R(\lambda\pm i0)=(\mathcal{L}_{\mathbf{A}}-(\lambda^{2}\pm
i0))^{-1}=\lim_{\epsilon\searrow
0}\big{(}\mathcal{L}_{\mathbf{A}}-(\lambda^{2}\pm i\epsilon)\big{)}^{-1}.$

Very recently, Fanelli-Zhang-Zheng [12] used the spectral measure constructed
in [16] to prove the following resolvent estimates.

######  Lemma 2.1.

[12, Theorem 3.1] Let $1\leq p<\frac{4}{3}$ and $4<q\leq\infty$ satisfy
$\frac{2}{3}\leq\frac{1}{p}-\frac{1}{q}<1$. Then there exists a constant
$C=C(p,q)>0$ such that

$\|(\mathcal{L}_{\mathbf{A}}-(\lambda^{2}\pm
i0))^{-1}f\|_{L^{q}(\mathbb{R}^{2})}\leq
C\lambda^{2(\frac{1}{p}-\frac{1}{q})-2}\|f\|_{L^{p}(\mathbb{R}^{2})},\quad$
(2.3)

where $(\mathcal{L}_{\mathbf{A}}-(\lambda^{2}\pm i0))^{-1}:=\lim_{\epsilon\to
0^{+}}(\mathcal{L}_{\mathbf{A}}-(\lambda^{2}\pm i\epsilon))^{-1}$.

In particular, if $6\leq q<\infty$, then (2.3) holds for $q=p^{\prime}$.

Therefore, from (2.2) and (2.3), we have the following spectral measure
estimate

$\|dE_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda)\|_{L^{p}(\mathbb{R}^{2})\to
L^{q}(\mathbb{R}^{2})}\leq C\lambda^{2(\frac{1}{p}-\frac{1}{q})-1}.$ (2.4)

Integrating on a frequency band of width $\epsilon$, we obtain

######  Lemma 2.2.

For the operator
$\chi_{k,\epsilon}(\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})$ defined by
(2.1), we have

$\|\chi_{k,\epsilon}(\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})\|_{L^{p}(\mathbb{R}^{2})\to
L^{q}(\mathbb{R}^{2})}\leq
C\big{(}(k+\epsilon)^{\frac{1}{\alpha}}-k^{\frac{1}{\alpha}}\big{)}(k+\epsilon)^{(2(\frac{1}{p}-\frac{1}{q})-1)/\alpha},$
(2.5)

where $1\leq p<\frac{4}{3}$ and $4<q\leq\infty$ satisfy
$\frac{2}{3}\leq\frac{1}{p}-\frac{1}{q}<1$.

In particular, for $6\leq q=p^{\prime}<+\infty$, then

$\|\chi_{k,\epsilon}(\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})\|_{L^{p}(\mathbb{R}^{2})\to
L^{p^{\prime}}(\mathbb{R}^{2})}\leq
C\big{(}(k+\epsilon)^{\frac{1}{\alpha}}-k^{\frac{1}{\alpha}}\big{)}(k+\epsilon)^{(2(\frac{1}{p}-\frac{1}{p^{\prime}})-1)/\alpha}.$
(2.6)

To prove the local result in Theorem (1.1) for the range $2\leq q\leq 6$, here
we prove the following lemma.

######  Lemma 2.3.

Let $U(t)=e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}$, then there
exists a constant $C$ such that

$\|D_{t}^{s}U(t)\|_{\dot{H}_{\bf A}^{\alpha s}\to L^{2}}\leq C,$

where $0\leq s\leq 1$.

###### Proof.

By the interpolation, we only need to consider $s=0$ and $s=1$. By the
spectral theory, we have

$T^{s}(t):=D_{t}^{s}U(t)=\int_{0}^{\infty}e^{it\lambda^{\alpha}}\lambda^{\alpha
s}\,dE_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda).$

For $\varphi\in C_{c}^{\infty}([1,2])$, we define

$T^{s}_{k}(t)=\int_{0}^{\infty}e^{it\lambda^{\alpha}}\varphi(2^{-k}\lambda)\lambda^{\alpha
s}\,dE_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda),$ (2.7)

then $T^{s}(t)=\sum_{k\in\mathbb{Z}}T^{s}_{k}(t)$. By orthogonality, we have

$\|T^{s}(t)f\|^{2}_{L^{2}(\mathbb{R}^{2})}=\sum_{k\in\mathbb{Z}}\|T^{s}_{k}(t)f\|^{2}_{L^{2}(\mathbb{R}^{2})}=\sum_{k\in\mathbb{Z}}\|T^{s}_{k}(t)f_{k}\|^{2}_{L^{2}(\mathbb{R}^{2})},$

where $f_{k}=\tilde{\varphi}(2^{-k}\sqrt{\mathcal{L}_{\mathbf{A}}})f$ with
$\tilde{\varphi}\in C_{c}^{\infty}([\frac{1}{2},4])$ such that
$\tilde{\varphi}=1$ on supp $\varphi$. It thus suffices to prove

$\|T^{s}_{k}(t)\|_{L^{2}\to L^{2}}\leq C2^{\alpha sk},$

which is equivalent to

$\|\big{(}T^{s}_{k}(t)\big{)}^{*}T^{s}_{k}(t)\|_{L^{2}\to L^{2}}\leq
C^{2}2^{2\alpha sk}.$

To this end, we set $\psi=\varphi^{2}\in C_{c}^{\infty}([1,2])$, then it is
further enough to show

$\begin{split}\Big{|}\int^{\infty}_{0}\psi(2^{-k}\lambda)\lambda^{2\alpha
s}dE_{\sqrt{\mathcal{L}_{\bf
A}}}(\lambda;x,y)\Big{|}\lesssim\frac{2^{2k}2^{2\alpha
sk}}{(1+2^{k}|x-y|)^{K}},\quad k\in\mathbb{Z},\quad\forall K\geq
0,\end{split}$ (2.8)

where $C$ is a constant independent of $k\in\mathbb{Z}$. On the one hand, by
scaling,

$\begin{split}\Big{|}\int^{\infty}_{0}\psi(2^{-k}\lambda)\lambda^{2\alpha
s}&dE_{\sqrt{\mathcal{L}_{\bf A}}}(\lambda;x,y)\Big{|}\\\ &=2^{2k}2^{2\alpha
sk}\Big{|}\int^{\infty}_{0}\psi(\lambda)\lambda^{2\alpha
s}dE_{\sqrt{\mathcal{L}_{\bf A}}}(\lambda;2^{k}x,2^{k}y)\Big{|}.\end{split}$
(2.9)

On the other, by [16, Proposition 1.1] and integration by parts K times, we
have

$\begin{split}\Big{|}\int^{\infty}_{0}\psi(\lambda)\lambda^{2\alpha
s}dE_{\sqrt{\mathcal{L}_{\bf
A}}}(\lambda;x,y)\Big{|}\lesssim\frac{1}{(1+|x-y|)^{K}}+\int_{0}^{\infty}\frac{1}{(1+|d_{s}|)^{K}}|B_{\alpha}(s,\theta_{1},\theta_{2})|ds,\end{split}$

where for the definition of $d_{s}$ and $B_{\alpha}(s,\theta_{1},\theta_{2})$
we can refer to [16]. From [16, (1.25)], one has
$|d_{s}|^{2}\geq(r_{1}+r_{2})^{2}\geq|x-y|^{2}$. Hence, it follows from [16,
(3.3)] that

$\begin{split}\Big{|}\int^{\infty}_{0}\psi(\lambda)\lambda^{2\alpha
s}dE_{\sqrt{\mathcal{L}_{\bf
A}}}(\lambda;x,y)\Big{|}\lesssim\frac{1}{(1+|x-y|)^{K}}.\end{split}$

This together with (2.9) yields (2.8).

∎

##  3\. The proof of Theorem 1.1

In this section, we shall prove the local maximal estimate associated with the
magnetic operator $\mathcal{L}_{\mathbf{A}}$. We need to prove the following
local-in-time estimate

$\begin{split}\big{\|}e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{q}(\mathbb{R}^{2};L^{\infty}([-1,1]))}\lesssim\|f\|_{H_{\mathbf{A}}^{s}(\mathbb{R}^{2})}.\end{split}$
(3.1)

To this end, we first observe that there exists a Schwarz function
$\rho\in\mathcal{S}(\mathbb{R})$ with Fourier support
$\text{supp}\,\hat{\rho}\subset(-2,2)$ such that

$\big{\|}e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{q}(\mathbb{R}^{2};L^{\infty}([-1,1]))}\lesssim\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}.$
(3.2)

Then to prove (3.1) , it is sufficient to show

$\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\|(I+\mathcal{L}_{\mathbf{A}})^{\frac{s}{2}}f\|_{L^{2}(\mathbb{R}^{2})}.$
(3.3)

In the case that $2\leq q\leq 6$, by interpolation, it suffices to prove

$\begin{split}\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{2}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\|f\|_{H_{\mathbf{A}}^{\sigma\alpha}(\mathbb{R}^{2})},\quad\sigma>\frac{1}{2}\end{split}$
(3.4)

and

$\begin{split}\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{6}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\|f\|_{H_{\mathbf{A}}^{\sigma}(\mathbb{R}^{2})},\quad\sigma>\frac{2}{3}.\end{split}$
(3.5)

The inequality (3.4) is a consequence of Lemma 2.3 and the Sobolev embedding
$\dot{H}_{t}^{\frac{1}{2}-\epsilon}(\mathbb{R})\cap\dot{H}_{t}^{\frac{1}{2}+\epsilon}(\mathbb{R})\hookrightarrow
L^{\infty}_{t}(\mathbb{R})$ with $0<\epsilon\ll 1$. Indeed, for
$\sigma>\frac{1}{2}$, by Lemma 2.3 we have

$\displaystyle\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{2}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}$
$\displaystyle=\big{\|}e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}(\rho(t)f)\big{\|}_{L^{2}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}$
$\displaystyle\lesssim\big{\|}D_{t}^{\sigma}e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}(\rho(t)f)\big{\|}_{L^{2}(\mathbb{R}^{2};L^{2}(\mathbb{R}))}$
$\displaystyle\lesssim\big{\|}\rho(t)\big{\|}_{L^{2}(\mathbb{R})}\|f\|_{H_{\mathbf{A}}^{\sigma\alpha}(\mathbb{R}^{2})}$
$\displaystyle\lesssim\|f\|_{H_{\mathbf{A}}^{\sigma\alpha}(\mathbb{R}^{2})}.$

The inequality (3.5) is the endpoint of the following case $q\geq 6$.
Therefore, from now on, we focus on the case that $6\leq q<+\infty$.

Recall (2.1) with $\epsilon=1$

$\chi_{k}^{\alpha}:=\chi_{k,1}(\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})=\int_{0}^{\infty}\chi_{[k,k+1]}(\lambda^{\alpha})\,dE_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda),$
(3.6)

then we have the decomposition $f=\sum_{k=0}^{\infty}\chi_{k}^{\alpha}f$. By
(2.6), for $6\leq q<\infty$, we have

$\|\chi_{k}^{\alpha}f\|_{L^{q}(\mathbb{R}^{2})}\lesssim\big{(}(k+1)^{\frac{1}{\alpha}}-k^{\frac{1}{\alpha}}\big{)}^{\frac{1}{2}}(k+1)^{(\frac{1}{2}-\frac{2}{q})/\alpha}\|f\|_{L^{2}(\mathbb{R}^{2})},\quad\forall
k\geq 0.$ (3.7)

Let $0<\delta\ll 1$ and

$F(t,x):=|D_{t}|^{1/2+\delta}(\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f(x)),$
(3.8)

then by Sobolev embedding, we obtain

$\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\|F(t,x)\|_{L^{q}(\mathbb{R}^{2};L^{2}(\mathbb{R}))}.$
(3.9)

Set

$F_{k}(t,x):=|D_{t}|^{1/2+\delta}(\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}\chi_{k}^{\alpha}f(x)),$
(3.10)

then

$F(t,x)=\sum_{k=0}^{\infty}F_{k}(t,x).$ (3.11)

Taking Fourier transform in variable $t$, we get

$\hat{F}_{k}(\tau,x)=|\tau|^{1/2+\delta}\int_{0}^{\infty}\hat{\rho}(\tau-\lambda^{\alpha})\chi_{[k,k+1]}(\lambda^{\alpha})dE(\lambda)f(x).$
(3.12)

By Plancherel’s identity and the compact support property of $\hat{\rho}$, we
have

$\int_{\mathbb{R}}F_{k}(t,x)\overline{F_{\ell}(t,x)}dt=(2\pi)^{-1}\int_{\mathbb{R}}\hat{F}_{k}(\tau,x)\overline{\hat{F}_{\ell}(\tau,x)}d\tau=0,\quad|k-\ell|>100.$
(3.13)

Therefore we get

$\|F(t,x)\|_{L_{t}^{2}(\mathbb{R})}^{2}\lesssim\sum_{k=0}^{\infty}\|F_{k}(t,x)\|_{L_{t}^{2}(\mathbb{R})}^{2}=(2\pi)^{-1}\sum_{k=0}^{\infty}\|\hat{F}_{k}(\tau,x)\|_{L_{\tau}^{2}(\mathbb{R})}^{2}.$
(3.14)

This together with (3.9) gives

$\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\big{\|}\big{(}\sum_{k=0}^{\infty}\|\hat{F}_{k}(\tau,x)\|_{L_{\tau}^{2}(\mathbb{R})}^{2}\big{)}^{1/2}\big{\|}_{L^{q}(\mathbb{R}^{2})}.$
(3.15)

Notice that $q\geq 6$ and (3.12), we obtain

$\displaystyle\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}^{2}_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\sum_{k=0}^{\infty}\int_{\mathbb{R}}\|\hat{F}_{k}(\tau,x)\|^{2}_{L^{q}(\mathbb{R}^{2})}d\tau$
(3.16)
$\displaystyle=\sum_{k=0}^{\infty}\int_{\mathbb{R}}|\tau|^{1+2\delta}\Big{\|}\int_{0}^{\infty}\hat{\rho}(\tau-\lambda^{\alpha})\chi_{[k,k+1]}(\lambda^{\alpha})dE(\lambda)f(x)\Big{\|}^{2}_{L^{q}(\mathbb{R}^{2})}d\tau.$

By (3.7), we thus get

$\displaystyle\big{\|}\rho(t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\big{\|}^{2}_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\int_{k-3}^{k+3}|\tau|^{1+2\delta}\big{(}(k+1)^{\frac{1}{\alpha}}-k^{\frac{1}{\alpha}}\big{)}(k+1)^{(1-\frac{4}{q})/\alpha}\|\chi_{k}^{\alpha}f\|^{2}_{L^{2}(\mathbb{R}^{2})}d\tau$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\big{(}(k+1)^{\frac{1}{\alpha}}-k^{\frac{1}{\alpha}}\big{)}(k+1)^{\frac{1}{\alpha}(1-\frac{4}{q})+1+2\delta}\|\chi_{k}^{\alpha}f\|^{2}_{L^{2}(\mathbb{R}^{2})}$
$\displaystyle=\sum_{k=0}^{\infty}\|(1+k)^{\frac{1}{\alpha}(1-\frac{2}{q})+\delta}\chi_{k}^{\alpha}f\|^{2}_{L^{2}(\mathbb{R}^{2})}\approx\|(I+\sqrt{\mathcal{L}_{\mathbf{A}}})^{1-\frac{2}{q}+\delta\alpha}f\|^{2}_{L^{2}(\mathbb{R}^{2})}.$

##  4\. Proof of theorem 1.5

In this section, we extend the local estimates in Theorem 1.1 to global
estimates. In fact, we shall prove the following estimate:

$\|u\|_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\|\mathcal{L}_{\mathbf{A}}^{s/2}f\|_{L^{2}(\mathbb{R}^{2})},\quad
6\leq q\leq\infty.$ (4.1)

To prove (4.1) , we only need to show that, for $\epsilon\in(0,1)$, there
exists a constant independent of $\epsilon$ such that

$\|e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\|_{L^{q}(\mathbb{R}^{2};L^{\infty}((-1/\epsilon,1/\epsilon)))}\lesssim\|(\sqrt{\mathcal{L}_{\mathbf{A}}}+\epsilon
I)^{s}f\|_{L^{2}(\mathbb{R}^{2})}.$ (4.2)

Similar to the proof of Theorem 1.1, it is further sufficient to show that

$\|\rho(\epsilon
t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\|_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\|(\mathcal{L}_{\mathbf{A}}+\epsilon
I)^{s/2}f\|_{L^{2}(\mathbb{R}^{2})},$ (4.3)

where $\rho\in\mathcal{S}(\mathbb{R})$ is some fixed Schwarz function with
supp$\hat{\rho}\subset(-2,2)$.

Let

$\chi_{k\epsilon}^{\alpha}:=\chi_{k\epsilon,\epsilon}(\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})=\int_{0}^{\infty}\chi_{[k\epsilon,(k+1)\epsilon]}(\lambda^{\alpha})\,dE_{\sqrt{\mathcal{L}_{\mathbf{A}}}}(\lambda),\quad\epsilon\in(0,1),$
(4.4)

then we have decomposition $f=\sum_{k=0}^{\infty}\chi_{k\epsilon}^{\alpha}f$
for a fixed value of $\epsilon$.

By (2.6) and $TT^{*}$ argument, we have for $6\leq q<\infty$

$\|\chi_{k\epsilon}^{\alpha}f\|^{2}_{L^{q}(\mathbb{R}^{2})}\lesssim(((k+1)\epsilon)^{\frac{1}{\alpha}}-(k\epsilon)^{\frac{1}{\alpha}})((k+1)\epsilon)^{(1-\frac{4}{q})/\alpha}\|f\|^{2}_{L^{2}(\mathbb{R}^{2})},\quad\forall
k\geq 0.$ (4.5)

Sobolev embedding
$\dot{H}_{t}^{\frac{1}{2}+\epsilon}(\mathbb{R})\hookrightarrow
L^{\infty}_{t}(\mathbb{R})$ will give us

$\|\rho(\epsilon
t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\|_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}\lesssim\||D_{t}|^{1/2+\delta}(\rho(\epsilon
t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f)\|_{L^{q}(\mathbb{R}^{2};L^{2}(\mathbb{R}))}.$
(4.6)

Let

$F^{\epsilon}(t,x):=|D_{t}|^{1/2+\delta}(\rho(\epsilon
t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f)(x)$ (4.7)

and take the Fourier transform in time variable, we deduce that

$\widehat{F^{\epsilon}}(\tau,x)=|\tau|^{1/2+\delta}\epsilon^{-1}(\hat{\rho}(\epsilon^{-1}(\tau-\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}))f)(x):=\sum_{k=0}^{\infty}\widehat{F^{\epsilon}_{k}}(\tau,x),$
(4.8)

where

$\widehat{F^{\epsilon}_{k}}(\tau,x)=|\tau|^{1/2+\delta}\epsilon^{-1}(\hat{\rho}(\epsilon^{-1}(\tau-\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}))\circ\chi_{k\epsilon}^{\alpha}f)(x).$
(4.9)

Consequently, we have the following orthogonality

$\int_{\mathbb{R}}F^{\epsilon}_{k}(t,x)\overline{F^{\epsilon}_{\ell}(t,x)}dt=(2\pi)^{-1}\int_{\mathbb{R}}\widehat{F^{\epsilon}_{k}}(\tau,x)\overline{\widehat{F^{\epsilon}_{\ell}}(\tau,x)}d\tau=0,\quad|k-\ell|>100.$
(4.10)

Notice the fact that $f=\sum_{k=0}^{\infty}\chi^{\epsilon}_{k}f$, we obtain

$\displaystyle\|F^{\epsilon}(\cdot,x)\|_{L^{2}(\mathbb{R})}^{2}=\|\sum_{k=0}^{\infty}F^{\epsilon}_{k}(\cdot,x)\|_{L^{2}(\mathbb{R})}^{2}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\|F^{\epsilon}_{k}(\cdot,x)\|_{L^{2}(\mathbb{R})}^{2}$
$\displaystyle=\sum_{k=0}^{\infty}\|\widehat{F^{\epsilon}_{k}}(\cdot,x)\|_{L^{2}(\mathbb{R})}^{2}.$
(4.11)

Collecting the above facts, we deduce that

$\displaystyle\|\rho(\epsilon
t)e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\|^{2}_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\int_{\mathbb{R}}\|\widehat{F^{\epsilon}_{k}}(\tau,\cdot)\|_{L^{q}(\mathbb{R}^{2})}^{2}d\tau$
(4.12)
$\displaystyle=\epsilon^{-2}\sum_{k=0}^{\infty}\int_{(k-3)\epsilon}^{(k+3)\epsilon}|\tau|^{1+\delta}\|\hat{\rho}(\epsilon^{-1}(\tau-\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}))\chi_{k\epsilon}^{\alpha}f\|_{L^{q}(\mathbb{R}^{2})}^{2}d\tau$
$\displaystyle\lesssim\epsilon^{-2}\sum_{k=0}^{\infty}\epsilon((k+1)\epsilon)^{1+\delta}\|\chi_{k\epsilon}^{\alpha}f\|_{L^{q}(\mathbb{R}^{2})}^{2}.$

Using (4.5) and letting $s:=1-\frac{2}{q}+\alpha\delta$, we further have

$\displaystyle\|\rho(\epsilon t)$ $\displaystyle
e^{it\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}}}f\|_{L^{q}(\mathbb{R}^{2};L^{\infty}(\mathbb{R}))}^{2}$
(4.13)
$\displaystyle\lesssim\epsilon^{-2}\sum_{k=0}^{\infty}\epsilon((k+1)\epsilon)^{1+\delta}(((k+1)\epsilon)^{\frac{1}{\alpha}}-(k\epsilon)^{\frac{1}{\alpha}})((k+1)\epsilon)^{(1-\frac{4}{q})/\alpha}\|\chi_{k\epsilon}^{\alpha}f\|_{L^{2}(\mathbb{R}^{2})}^{2}$
$\displaystyle=\epsilon^{-2}\sum_{k=0}^{\infty}\epsilon^{2}((k+1)\epsilon)^{\frac{2}{\alpha}(1-\frac{2}{q})+\delta}\|\chi_{k\epsilon}^{\alpha}f\|_{L^{2}(\mathbb{R}^{2})}^{2}$
$\displaystyle=\sum_{k=0}^{\infty}\|((k+1)\epsilon)^{s}\chi_{k\epsilon}^{\alpha}f\|_{L^{2}(\mathbb{R}^{2})}^{2}\thickapprox\|(\sqrt{\mathcal{L}_{\mathbf{A}}}+\epsilon
I)^{s}f\|_{L^{2}(\mathbb{R}^{2})}^{2}.$

Acknowledgements The authors would like to thank Junyong Zhang for the
encouragement and constructive suggestions. They would also like to thank the
anonymous referees for their helpful comments and suggestions.

## References

  * [1] Y. Aharonov, and D. Bohm, Significance of electromagnetic potentials in quantum theory, Phys. Rev. Lett., 1959, 115: 485-491.
  * [2] M. Alford and F. Wilczek, Aharonov-Bohm interaction of cosmic strings with matter, Phys. Rev. Lett., 1989, 62:1071.
  * [3] R. Baker, $L^{p}$ maximal estimates for quadratic Weyl sums, arXiv:2103.05555v1.
  * [4] A. Barron, An $L^{4}$ maximal estimates for quadratic Weyl sums, arXiv:2011.09885v1.
  * [5] J. Bourgain,  On the Schrödinger maximal function in higher dimension,  Proc. Steklov Inst. Math., 2013, 280:46-60.
  * [6] J. Bourgain, A note on the Schrödinger maximal function, J. Anal. Math., 2016, 130:393-396.
  * [7] L. Carleson, Some analytic problems related to statistical mechanics, in Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, MD, 1979), Lecture Notes in Math. 779, pp. 5-45, Springer, Berlin-Heidelberg, 1980.
  * [8] B. E. J. Dahlberg, C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic Analysis(Minneapolis, Minn., 1981)Vol. 908 of Lecture Notes in Math. Springer, Berlin, 1981, 205-209.
  * [9] X. Du, L. Guth and X. Li, A sharp Schrödinger maximal estimate in $\mathbb{R}^{2}$, Annals of Math., 2017, 186:607-640.
  * [10] X. Du and R. Zhang, Sharp $L^{2}$ estimate of Schrödinger maximal function in higher dimensions, Annals of Math., 2019, 189:837-861.
  * [11] L. Fanelli, V. Felli, M. A. Fontelos and A. Primo, Time decay of scaling critical electromagnetic Schrödinger flows, Comm. Math. Phys.,2013, 324:1033-1067.
  * [12] L. Fanelli, J. Zhang and J. Zheng, Uniform resolvent estimates for Schrödinger operators in critical magnetic fields, arXiv:2109.11327v2.
  * [13] L. Fanelli, J. Zhang and J. Zheng, Dispersive estimates for 2D-wave equations with critical potentials, arXiv:2003.10356v3.
  * [14] R. L. Frank, E. H. Lieb, R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, Journ. Amer. Math. Soc., 2008, 21(4):925-950.
  * [15] X. Gao, J. Wang, J. Zhang and J. Zheng, Uniform resolvent estimates for Schrödinger operators in Aharonov-Bohm magnetic fields, J. Differ. Equ., 2021, 292:70-89.
  * [16] X. Gao, Z. Yin, J. Zhang and J. Zheng, Decay and Strichartz estimates in critical electromagnetic fields, arXiv:2003.03086.
  * [17] C. E. Kenig, A. Ruiz, A strong type (2, 2) estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 1983, 280:239-246.
  * [18] H. Ko, S. Lee and S. Oh, Maximal estimates for average over space curve, arXiv:2102.07175v1.
  * [19] S. Lee,  On pointwise convergence of the solutions to the Schrödinger equation in $\mathbb{R}^{2}$,  Int. Math. Res. Not., 2006, 1-21.
  * [20] R. Luc$\grave{a}$ and K. M. Rogers,  An improved necessary condition for the Schrödinger maxmal estimate,  Classical Analysis and ODEs, 2015.
  * [21] C. Miao, J. Zhang and J. Zheng, Maximal estimates for Schrödinger equations with inverse-square potential, Pacific J. Math., 2015, 273(1):1-20.
  * [22] H. Mizutani, J. Zhang and J. Zheng, Uniform resolvent estimates for Schrödinger operator with an inverse-square potential, Jour. Func. Anal., 2020, 278:108350.
  * [23] M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect, Lect. Notes Phys.,1989, 340.
  * [24] K. M. Rogers and P. Villarroya, Global estimates for the Schrödinger maximal operator, Annales Academia Scientiarum Fennica Math., 2007, 32:425-435.
  * [25] K. M. Rogers and P. Villarroya, Sharp estimates for maximal operators associated to the wave equation, Ark. Mat., 2008, 46:143-151.
  * [26] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math.J., 1987, 55:699-715.
  * [27] Y. Sire, C. D. Sogge, C. Wang and J. Zhang, Reversed Strichartz estimates for wave on non-trapping asymptotically hyperbolic manifolds and applications, arXiv:2108.11596.
  * [28] L. Vega,  Schrödinger equations: pointwise convergence to the initial data,  Proc. Amer. Math. Soc., 1988, 102:874-878.
  * [29] B. G. Walther,  Maximal estimates for oscillatory integrals with concave phase,  Contemp. Math., 1995, 189:485-495.