Figure 1. Multiscale Implicit Representations. We present a novel implicit
representation framework called MINER that is well suited for very large
visual signals such as images, videos, and 3D volumes. We leverage the self-
similarity of visual signals across scales to iteratively represent models
from coarse to fine scales, resulting in a dramatic decrease in inference and
training time, while requiring fewer parameters and less memory than state-of-
the-art representations. This figure demonstrates fitting of the Lucy 3D mesh
over four scales with scale 3 being the coarsest and 0 being the finest. MINER
achieves high quality results across all scales with high IoU value and
achieves an IoU of 0.999 at the finest scale in less than 30 minutes. In
comparison, the state-of-the-art approach (ACORN) results in an IoU of 0.97 in
that time, while requiring far more parameters.

# MINER: Multiscale Implicit Neural Representations

Vishwanath Saragadam, Jasper Tan, Guha Balakrishnan,
Richard G. Baraniuk, Ashok Veeraraghavan  ECE Department, Rice
UniversityHoustonUSA<EMAIL_ADDRESS>
(2022)

###### Abstract.

We introduce a new neural signal representation designed for the efficient
high-resolution representation of large-scale signals. The key innovation in
our multiscale implicit neural representation (MINER) is an internal
representation via a Laplacian pyramid, which provides a sparse multiscale
representation of the signal that captures orthogonal parts of the signal
across scales. We leverage the advantages of the Laplacian pyramid by
representing small disjoint patches of the pyramid at each scale with a tiny
MLP. This enables the capacity of the network to adaptively increase from
coarse to fine scales, and only represent parts of the signal with strong
signal energy. The parameters of each MLP are optimized from coarse-to-fine
scale which results in faster approximations at coarser scales, thereby
ultimately an extremely fast training process. We apply MINER to a range of
large-scale signal representation tasks, including gigapixel images and very
large point clouds, and demonstrate that it requires fewer than 25% of the
parameters, 33% of the memory footprint, and 10% of the computation time of
competing techniques such as ACORN to reach the same representation error.

neural representations, Laplacian pyramids

††copyright: acmcopyright††journalyear: 2022††doi:
10.1145/1122445.1122456††journal: TOG††journalvolume: 37††journalnumber:
4††article: 111††publicationmonth: 1††submissionid: 301††ccs: Computing
methodologies Neural networks††ccs: Computing methodologies Hierarchical
representations

##  1\. Introduction

Figure 2. MINER trains and predicts Laplacian pyramids. Visual signals are
similar across scales and are compactly represented by Laplacian pyramids.
MINER follows a similar scheme where each scale of the Laplacian pyramid is
represented by multiple, local MLPs with a small number of parameters. The
number of such MLPs increase by a factor of 2 from coarse to fine scale,
thereby representing a fixed spatial size at each scale. This multi-scale
representation naturally lends itself to a sequential, coarse-to-fine scale
training process that is fast and memory efficient.

The bedrock of modern signal processing is a concise signal representation
that can then be utilized for all manner of applications from compression to
inverse estimation. Neural implicit representations offer such a concise way
to represent discrete data as samples of a continuous manifold. This enables
an efficient way to interpolate data between samples and has found numerous
applications in 3D representations (Mildenhall et al., 2020; Peng et al.,
2020; Kuznetsov et al., 2021; Srinivasan et al., 2021), images (Chen et al.,
2021b), video (Chen et al., 2021a), and linear inverse problems (Sun et al.,
2021; Chen et al., 2021b).

Despite their numerous advantages, neural representations suffer from high
computational complexity, large memory footprint while training, and a large
number of parameters to achieve high accuracy. While there have been several
modifications to the network architecture (Sitzmann et al., 2020; Reiser et
al., 2021; Martel et al., 2021) and inference (Yu et al., 2021), the existing
approaches are not practical for very large signals such as multi-gigapixel
images, or 3D point clouds with billions of data points.

We introduce Multiscale Implicit Neural Representations (MINER) that predicts
a Laplacian pyramid at multiple spatial scales. Our key observation is that a
Laplacian pyramid offers a sparse and multiscale decomposition that separates
different frequency content of the signal into different bands (Adelson et
al., 1984; Do and Vetterli, 2003), meaning that signals at different bands are
approximately orthogonal to one another. We leverage this observation by
training MLPs to predict the signal at the coarsest scale and residues at
finer scales, achieving a compact representation. A single global MLP at each
scale is however ill-suited to leverage the sparseness of the Laplacian
transform of visual signals. MINER instead represents local 2D and 3D patches
of images and videos with small MLPs, which is similar in structure to
recently proposed KiloNeRF (Reiser et al., 2021) representation. The multi-
patch decomposition is performed at every scale, with the number of spatial or
volume blocks doubling along each dimension as we go from coarse to fine
scale. MINER offers a fast and flexible multi-resolution inference, as
computing the signal value at lower resolution requires evaluation of fewer
MLPs. An overview of MINER is presented in Fig. 2.

The multiscale, multi-MLP architecture is well-suited for fast and memory-
efficient training procedure. At each spatial scale, the parameters of the
MLPs are trained for the downsampled version of the signal at that scale. We
then _sequentially_ train MLPs from coarsest scale to the finest scale. To
leverage correlations across scales during the training phase, we propose two
innovations. First, we demonstrate that training on a Laplacian pyramid
(composed of difference signals) is significantly faster than training on a
Gaussian pyramid (composed of downsampled signals) as MLPs at finer scales
represent information orthogonal to the representation at coarser scales.
Second, we compare the upsampled signal from the fine block and target signal
at fine block – if the error in representation is smaller than a threshold, we
prune out the blocks before training starts. Since visual signals are sparse
in their Laplace transform, the fitting converges for parent blocks at a
coarser scale, leading to fewer blocks to train at subsequent resolutions.
These two modifications result in a dramatic increase in training speed
without compromising on memory requirements or total number of parameters. As
a result, MINER is $10\times$ or more faster compared to state-of-the-art
implicit representations in terms of training process for a comparable number
of parameters and target accuracy. MINER can represent gigapixel images with
greater than 38dB accuracy in less than three hours, compared to more than a
day with techniques such as ACORN (Martel et al., 2021). For 3D point clouds,
MINER achieves an intersection over union (IoU) of $0.999$ or higher in less
than three minutes, resulting in two orders of magnitude speed up over ACORN.
Figure 1 shows results obtained by MINER at the end of training at each
spatial scale, along with result from ACORN for the same training time –
showcasing how MINER achieves high quality results gradually from low
resolution to high resolution.

MINER combines the compact multiscale representations offered by Laplacian
pyramids along with the advantages of implicit representations. Due to the
multiscale representation, MINER can be used for streaming reconstruction of
images, as with JPEG2000 (Shapiro, 2009), or efficiently sampling for
rendering purposes with octrees (Yu et al., 2021) – making neural
representations ready for extremely large scale visual signals.

##  2\. Prior Work

MINER draws inspiration from classical multiscale techniques and more recent
neural representations. We outline some of the salient works here to set
context.

##### Implicit neural representations.

Implicit neural representations learn a continuous mapping from local
coordinates to the signal value such as intensity for images and videos, and
occupancy value for 3D volumes. The learned models are then used for a myriad
of tasks including image representations (Chen et al., 2021b), multi-view
rendering (Mildenhall et al., 2020), and solving linear inverse problems (Sun
et al., 2021). Recent advances in the choice of coordinate representation
(Tancik et al., 2020) and non-linearity (Sitzmann et al., 2020) have resulted
in training processes that have high fitting accuracy. Salient works related
to implicit representations include the NeRF representations (Mildenhall et
al., 2020) and its many derivatives that seek to learn the 3D geometry from a
set of multi-view images. Despite the interest and success of these implicit
representation, current approaches often require millions of parameters. This
culminates in a large memory footprint and training times, precluding
representation of very high-dimensional signals.

##### Architectural changes for faster learning.

Several interesting modifications have been proposed to increase training or
inference speed. KiloNeRF (Reiser et al., 2021) replaced the large MLP with
multiple small MLPS that fit only a small, disjoint part of the 3D space. This
drastically speeds up the inference time (often by $60\times$), but has no
effect on the training process itself. ACORN (Martel et al., 2021) utilized an
adaptive coordinate decomposition to efficiently fit various signals. By
utilizing a combination of integer programming and interpolation, ACORN
reduced training time for fitting of images and 3D point clouds by one to two
orders of magnitude compared to techniques like SIREN (Sitzmann et al., 2020)
and convolutional occupancy network (Peng et al., 2020). However, ACORN does
not leverage cross-scale similarity of visual signals which often leads to
long times to convergence for very large signals. Moreover, the adaptive
optimized blocks calls for complex integer programming which can be
prohibitively expensive.

##### Multi-scale representations.

Visual signals are similar across scales, and this has been exploited for a
wide variety of applications. In computer vision and image processing, wavelet
transform and Laplacian pyramids are often used to efficiently perform tasks
such as image registration (Thévenaz and Unser, 2000), optical flow
computation (Weber and Malik, 1995), and feature extraction (Lowe, 2004).
Multi-scale representations such as octrees (Meagher, 1982; Chien and
Aggarwal, 1986) and mip-mapping are used to speed up the rendering pipeline.
This has also inspired neural mipmapping techniques (Kuznetsov et al., 2021)
that utilize neural networks to represent texture at each scale. Multi-scale
representations are also utilized for several linear inverse problems such as
multi-scale dictionary learning for denoising (Sulam et al., 2014),
compressive sensing (Park and Wakin, 2009), and sparse approximation (Mairal
et al., 2007). Some recent works have focused on a level-of-detail approach to
neural representations (Takikawa et al., 2021) where the multiple scales are
_jointly_ learned. While efficient in rendering, the technique does not target
extremely large meshes. MINER also results in an LOD representation, but the
underlying approach is significantly different. MINER relies on a block-wise
representation at each scale, which enables more compact representation with
faster training times.

##  3\. MINER

MINER leverages a block decomposition of signal along with a Laplacian
pyramid-like representation. We now details the proposed signal model and the
training process.

###  3.1. Signal model

Let ${\bf x}$ be the coordinate and $\text{I}({\bf x})$ be the target. We will
assume that the coordinates lie in $[-1,1]$. Let $\mathcal{D}_{j}$ be the
domain specific operator that downsamples the signal by $j$ times, and
$\mathcal{U}_{j}$ be the domain-specific operator that upsamples the signal by
$j$ times. We will leverage $J$ implicit representations, $I_{j}({\bf
x})\approx N_{j}$ for $j\in[0,J-1]$, where $N_{j}$ is the MLP at the
$j^{\text{th}}$ level of a Laplacian pyramid, a multiscale representation
which separates the input signal into scales capturing unique spatial
frequency bands. Two desirable properties of such a bandpass pyramid is that
signals across scales are approximately orthogonal to one another (Do and
Vetterli, 2003), and are sparse. We find in our experiments that these
properties help reduce training and inference times compared to a lowpass
pyramid such as the Gaussian pyramid (see Fig. 3).

Letting $R_{j}$ denote the MLP modeling the residual signal at scale $j$, our
Laplacian pyramid representation may be written as:

(1) $\displaystyle I_{J-1}({\bf x})$ $\displaystyle=\mathcal{D}_{J-1}(I)({\bf
x})\approx N_{J-1}({\bf x})$ (2) $\displaystyle I_{J-2}({\bf x})$
$\displaystyle=\mathcal{D}_{J-2}(I)({\bf x})\approx R_{J-2}({\bf
x})+\mathcal{U}_{2}(N_{J-1}({\bf x}/2))$ $\displaystyle\vdots$ (3)
$\displaystyle I({\bf x})$ $\displaystyle\approx R_{0}({\bf
x})+\mathcal{U}_{2}(N_{1}({\bf x}/2))$ (4) $\displaystyle\approx N_{0}({\bf
x})+\mathcal{U}_{2}(N_{1}({\bf x}/2))+\mathcal{U}_{2^{J-1}}(N_{J-1}({\bf
x}/2^{J-1})),$

where Equation (1) is the coarsest representation of the signal. At finer
scales (as in Equation (2)), we write the signal to be approximated as a sum
of the upsampled version of the previous scale and a residual term. This leads
to a recursive multi-resolution representation that naturally shares
information across scales.

We make two observations about the proposed multiscale representation:

  * •

Signals at coarser resolutions are smaller, and therefore require smaller
MLPs. These MLPs are faster for inference, which is beneficial for tasks such
as mipmapping and LOD-based rendering.

  * •

The parameters of the MLPs up to scale $j$ only rely on the signal at scales
$q=j,j+1,\cdots,J-1$. This implies the MLPs can be trained _sequentially_ from
coarsest to finest scale. We will see next that this offers a dramatic
reduction in training time without sacrificing quality.

####  3.1.1. Using multiple MLPs per scale.

Equation (4) implies that obtaining a value at a spatial point ${\bf x}$
requires evaluating a total of $J$ MLPs across scales. This is not beneficial
compared to a single scale solution such as SIREN (Sitzmann et al., 2020).
Further, the residual signals at finer scales are often low-amplitude, a
consequence of visual signals being composed of several smooth areas. To
leverage this fact and make inference faster, we split the signal into equal
sized blocks at each scale. We create an MLP for each block that requires
significantly fewer parameters than a single full MLP at that scale. Moreover,
blocks with small residual energy can be represented as a zero signal, and do
not even need to be represented with an MLP.

We now combine the Laplacian representation with the multi-MLP approach stated
above. Let $\widetilde{{\bf x}}$ be a local coordinate at the finest scale in
a block with coordinate $(m,n)$, where $m\in{1,2,\cdots M}$ is the number of
vertical blocks, and $n\in{1,2,\cdots,N}$ is the number of horizontal blocks.
To evaluate the signal at ${\bf x}$ we compute,

$\displaystyle I({\bf x})$ $\displaystyle=I\left(\widetilde{{\bf
x}}+\left[\frac{mH}{M},\frac{nW}{N}\right]\right)=\mathcal{R}^{(m,n)}_{0}\left(\widetilde{{\bf
x}}+\left[\frac{mH}{M},\frac{nW}{N}\right]\right)+\cdots$ (5)
$\displaystyle+\cdots\mathcal{U}_{2}\left(\mathcal{N}^{(\lfloor
m/2\rfloor,\lfloor n/2\rfloor)}_{1}\left(\widetilde{{\bf
x}}+\left[\left\lfloor\frac{mH}{2M}\right\rfloor,\left\lfloor\frac{nW}{2N}\right\rfloor\right]\right)\right),$

where $\lfloor\cdot\rfloor$ is the floor operator, and
$\mathcal{N}^{(m,n)}_{j}$ is the MLP for block at $(m,n)$ and at scale $j$.
With this formulation, we require evaluation of at most $J$ _small_ MLPs
instead of large MLPs, thereby dramatically reducing inference time.

###  3.2. Training MINER

MINER requires estimation of parameters at each scale and each block. We now
present an efficient _sequential_ training procedure that starts at the
coarsest scale and trains up to the finest scale.

##### Training at coarsest scale.

The training process starts by fitting $\text{I}_{J-1}({\bf x})$, the image at
the coarsest scale. We estimate the parameters of each of the MLPs
$\mathcal{N}^{(m,n)}_{J-1}$ by solving the objective function,

(6)
$\displaystyle\min_{\mathcal{N}_{J-1}^{(m,n)}}\left\|I_{J-1}\left(\widetilde{{\bf
x}}+\left(\frac{mH}{2^{J-1}M},\frac{nW}{2^{J-1}N}\right)\right)-\mathcal{N}_{J-1}^{(m,n)}(\widetilde{{\bf
x}})\right\|^{2}.$

Let $\widehat{I}_{J-1}({\bf x}_{J-1})$ be the estimate of the image at this
stage.

##### Pruning at convergence.

As the training proceeds, some MLPs, particularly for simpler blocks, will
converge to a target mean squared error (MSE) earlier than others. We remove
those MLPs that have finished training from the optimization process and
continue with the others.

##### Training at finer scales.

As with the coarsest scale, we continue to fit small MLPs to blocks at each
finer scale. For example, for scale $J-2$, the target signal is given by

(7) $\displaystyle R_{J-2}({\bf x})=\text{I}_{J-2}({\bf
x})-\mathcal{U}_{2}(\widehat{\text{I}}_{J-1})({\bf x}/2).$

We leverage the fact that blocks within each scale occupy disjoint regions and
can optimize each MLP independently of one another:

(8)
$\displaystyle\min_{\mathcal{R}_{J-2}^{(m,n)}}\left\|R_{J-2}\left(\widetilde{{\bf
x}}+\left(\frac{mH}{2^{J-2}M},\frac{nW}{2^{J-2}N}\right)\right)-\mathcal{R}_{J-2}^{(m,n)}(\widetilde{{\bf
x}})\right\|^{2}.$

##### Pruning before optimization.

Nominally, the number of blocks and MLPs double along each dimension at finer
scales. However, some blocks may already be adequately represented by the
corresponding MLP at the coarser scale. In such a case, we do not assign an
MLP to that block and set the estimate of the residue to all zeros. Depending
on the frequency content in the image, this decision dramatically reduces the
number of total MLP parameters, and thereby the overall training and inference
times.

Figure 3. Laplacian pyramid enables faster convergence. The plot above shows
training error across time for a $2048\times 2048$ image of Pluto. MINER when
combined with a Laplacian pyramid offers a significantly faster convergence as
the MLPs at finer scale capture orthogonal information compared to the coarser
ones. This also results in very small jumps in training accuracy that is
strongly prevalent when MINER is trained with a Gaussian pyramid, or ACORN.

##  4\. Experimental Results

##### Baselines.

For fitting to images and 3D volumes, we compared MINER against SIREN
(Sitzmann et al., 2020), KiloNeRF (Reiser et al., 2021), and ACORN (Martel et
al., 2021). We also compared MINER Against convolutional occuppancy networks
(Peng et al., 2020) for 3D volumes. We used code from the respective authors
and optimized the training parameters to ensure a fair comparison.

##### Training details.

We implemented MINER with the PyTorch (Paszke et al., 2019) framework.
Multiple MLPs were trained efficiently using the block matrix multiplication
function (`torch.bmm`) and hence we required no complex coding outside stock
pytorch implementations. All our models were trained on a system unit equipped
with Intel Xeon 8260 running at 2.4Ghz, 128GB RAM, and NVIDIA GeForce RTX 2080
Ti with 12GB memory.

##### Fitting images.

We split up RGB images into $32\times 32\times 3$ patches at all spatial
scales. For each patch and at each scale, we trained a single MLP with four
layers with sinusoidal activation function. We fixed the number of features to
be $20$ for each layer. We used the ADAM (Kingma and Ba, 2015) optimizer with
a learning rate of $5\times 10^{-4}$ and an exponential decay with
$\gamma=0.999$. At each scale, we trained either for 500 epochs, or until the
change in loss function was greater than $2\times 10^{-7}$. We used an
$\ell_{2}$ loss function at all scales with no additional prior.

Figure 3 shows training error for a 4 megapixel (MP) image of Pluto across
epochs for MINER by representing a Laplacian pyramid, a Gaussian pyramid, and
ACORN. MINER converges rapidly to an error of $10^{-4}$ compared to other
approaches. Moreover, the periodic and abrupt increase in error are more
prevalent in Gaussian representation, and ACORN, which further hamper their
performance, but not with Laplacian pyramid due to near-orthogonality of
signal across scales. Results on other high resolution images (including 64
megapixel image of Pluto) along with analysis on effect of parameters such as
number of scales and patch size is included in section 2.

Figure 4. MINER adaptively selects window sizes. MINER adaptively selects the
appropriate scale for each local area resulting in patch sizes that are chosen
according to texture variations within the window. The figure above shows a
macro photograph of a grasshopper fit by MINER. Large parts of the image such
as background have very smooth texture implying that they can be fit
accurately at a coarser scale – which translates to large spatial size for low
frequency areas. In contrast, area around the antennae are made of high
spatial frequencies, which required fitting at finer scales. Figure 5. MINER
converges $10\times$ faster than state-of-the-art approaches. The plot shows
PSNR as a function of time for various approaches for the image shown in Fig.
4. MINER achieves higher accuracy at all times, and converges significantly
faster than competing approaches. Moreover, the drop in accuracy when changing
from coarse to fine scale is less severe for MINER compared to when ACORN re-
estimates coordinate decomposition.

Figure 6. MINER requires shorter training time and memory footprint. The plot
shows the time taken to achieve 36 dB and the GPU memory utilization to fit a
16MP image (Pluto) with ACORN and MINER for varying number of parameters.
MINER is an order of magnitude faster than ACORN and requires less than one
third of the GPU memory as ACORN – implying MINER is well-suited to train very
large models. Figure 7. Fitting gigapixel images. The figure shows the results
on fitting a gigapixel image ($20,480\times 56,420$) with MINER and ACORN.
MINER required 188 million parameters and converged to 38.6dB in 2.6 hours. In
contrast, ACORN required 167 million parameters and converged to 32.6dB in 48
hours.

Figure 4 shows an example of fitting a 2 megapixel image with active blocks at
all scales for a macro photograph. Notice how the blocks are concentrated
around the high frequency areas (such as the antennae of the grasshopper) as
the scale increase from coarse to fine. MINER took a less than 10s to converge
to 40dB fitting accuracy. In contrast, KiloNeRF took 6 minutes to converge and
ACORN took 7 minutes to converge to 40dB with approximately equal number of
parameters. Figure 5 shows a plot of PSNR as a function of time for various
approaches. We also note that ACORN curve shows significant drop in accuracy
as a result of re-computation of coordinate blocks. In contrast, the drop in
accuracy for the MINER curve that arises due to scale change is significantly
smaller than ACORN.

Figure 6 compares MINER, KiloNeRF, and ACORN in terms of time taken to achieve
36dB, and GPU memory for fitting 16MP image of Pluto. For all three cases, we
varied the number of hidden features and fixed the number of layers to vary
the number of parameters. MINER consistently achieves 36dB faster than
competing methods and requires significantly smaller memory footprint, making
it highly scalable for large-sized problems.

MINER scales up graciously for extremely large signals. We trained ACORN on a
gigapixel image shown in Fig. 7 over 7 scales. We set the number of features
per each block to be 9, and used a patch size of $32\times 32$. MINER
converged to a PSNR of 38.6dB in 2.6h and required a total of 188 million
parameters. In contrast, ACORN converged to the same accuracy in 36h and
required a total of 167 million parameters. Note that although MINER requires
more parameters than ACORN, the training time is shorter. Since MINER
leverages numerous small MLPs, the backpropagation step rquires significantly
fewer computations.

##### Fitting 3D point clouds.

Figure 8. Active blocks reduce with increasing scale. The figure shows MINER
results at the end of training at each scale (left column) and the active
blocks at each scale (right column). As the iterations progress, only the
blocks on the surface of the object remain, which leads to a dramatic
reduction in non-zero blocks, and hence the total number of parameters. Figure
9. Comparisons against state-of-the-art for 3D volume fitting. The figure
compares 3D occupancy fitting for a fixed duration with MINER, ACORN,
Convolutional occupancy (Peng et al., 2020), and screening poisson
reconstruction (Kazhdan and Hoppe, 2013). The number of parameters of MINER
was chosen automatically according to model complexity. MINER achieves high
accuracy in a very short duration for arbitrarily complex shapes, which is not
possible with prior works, even though some models such as the engine (second
row) require significantly larger number of parameters.

Inpsired by Convolutional occupancy networks (Peng et al., 2020), we utilized
signed density function where the value was 1 inside the mesh and 0 outside.
We sampled a total of one billion points, resulting in a $1024\times
1024\times 1024$ occupancy volume. We then optimized MINER over four scales
for a maximum of 2000 iterations at each scale. We experimented with logistic
loss and MSE and found the MSE resulting in signficantly faster convergence.
The learning rate was set to $10^{-3}$. We set the number of features to 22
and the number of hidden layers to 2 for MLP for each block at all scales. We
then constructed mesh from the resultant occupancy volumes using marching
cubes (Lorensen and Cline, 1987). We compared our results against ACORN, and
convolutional occupancy networks for accuracy and timing comparison, and
screened poisson reconstruction (SPSR) (Kazhdan and Hoppe, 2013) for comparing
quality of reconstruction. For SPSR, which takes in point clouds, we sampled 5
million points from the object’s surface as input.

Figure 1 shows the active blocks and reconstructed mesh at each scale. MINER
converges in less than 25 minutes to an Intersection over Union (IoU) of
0.999. In the same time, ACORN achieved an IoU of 0.97 with worse results than
MINER. ACORN took greater than 7 hours to converge to an IoU of 0.999, clearly
demonstrating the advantages of MINER for 3D volumes. We also note that MINER
required less than a third as the number of parameters as ACORN – this is a
direct consequence of using block-based representation – most blocks outside
the mesh and inside the mesh converge rapidly within the first few scales,
requiring far fewer representations than a single scale representation. An
example of active blocks at each scale is shown in Fig. 8. As iterations
progress, the number of active blocks after pruning reduce, which in turn
results in more compact representation, and fewer parameters.

Figure 9 visualizes the meshes fit with various reconstruction approaches for
a fixed duration. The time for exach experiment was chosen to be when MINER
Achieved an IoU of 0.999. MINER has superior reconstruction quality compared
to ACORN and convolutional occupancy networks (Peng et al., 2020), and
comparable performance to SPSR. We do note that the quality of reconstruction
specifically for the engine model is superior to SPSR. Since the engine model
has a large number of sharp edges, SPSR tends to oversmooth the result. Since
MINER combined with marching cubes relied only on local information for
reconstruction, the resultant mesh was more accurate.

##  5\. Conclusions

We have proposed a novel multi-scale neural representation that trains faster,
requires same or fewer parameters, and has lower memory footprint than state-
of-the-art approaches. We demonstrated that the advantages of a Laplacian
pyramid naturally extend to MINER. We showed that leveraging self-similarity
across scales is beneficial in reducing training time drastically while not
affecting the training accuracy. MINER naturally lends itself to rendering
where level-of-detail is of importance including representation and mipmapping
for texture mapping. Future directions will focus on combining MINER with
multi-view synthesis to achieve extremely fast fitting and visualization
pipelines that will open avenues for training highly complex geometric shapes.

##  6\. Acknowledgments

This work was supported by NSF grants CCF-1911094, IIS-1838177, and
IIS-1730574; ONR grants N00014-18-12571, N00014-20-1-2534, and MURI
N00014-20-1-2787; AFOSR grant FA9550-22-1-0060; and a Vannevar Bush Faculty
Fellowship, ONR grant N00014-18-1-2047.

##  7\. Appendix 1: Algorithm Details

Algorithm 1 shows the overall MINER flow with initialization, pruning, and
parameter update. The function `DOWNSAMPLE` implements a domain-specific
downsampling operator. For images it is,

(9) $\displaystyle\mathcal{D}_{j}(I)=I_{j}({\bf
x})=\int_{0}^{2^{j}/H}\int_{0}^{2^{j}/W}\text{I}(2^{j}{\bf x}+d{\bf x})dxdy.$

For three-dimensional signals such as videos ($I(x,y,t)$) and 3D occupancy
volumes (I(x, y, z)) we have,

(10) $\displaystyle\mathcal{D}_{j}(I)=I_{j}({\bf
x})=\int_{0}^{2^{j}/H}\int_{0}^{2^{j}/W}\int_{0}^{2^{j}/T}\text{I}(2^{j}{\bf
x}+d{\bf x})dxdydz.$

##  8\. Appendix 2: Experimental Results

##### Baselines.

We compare MINER against three competing baselines for 2D images, and three
for 3D volumes.

  1. (1)

SIREN (Sitzmann et al., 2020) fits a single large MLP at a single scale and
utilizes a sinusoidal activation function for accelerated training. We varied
the number of hidden units for each experiment to ensure that the number of
parameters matched that of MINER.

  2. (2)

KiloNeRF (Reiser et al., 2021) fits multiple small MLPs at a single scale
instead of a single large MLP. The number of hidden units for each MLP was
chosen to be the same as that for MINER.

  3. (3)

ACORN (Martel et al., 2021) fits a single large MLP at a single scale with
adaptive coordinate decomposition.

  4. (4)

Convolutional occupancy network (Peng et al., 2020) utilizes convolutions to
capture local correlations. We used this only for 3D volume comparisons.

We used code from the respective authors and optimized the training parameters
to ensure a fair comparison.

Figure 10. Image fit over time. The figure compares fitting of the 16
megapixel pluto image at various times during the training process. A distinct
advantage of MINER is that the signal is similar to the final output (albeit
downsampled) from the starting itself which enables an easy visual debug of
the fitting process.

##### Fitting images.

Figure 11. Fitting gigapixel images. The figure shows the results on fitting a
gigapixel image ($20,480\times 56,420$) with MINER and ACORN. MINER required
188 million parameters and converged to 38.6dB in 2.6 hours. In contrast,
ACORN required 167 million parameters and converged to 32.6dB in 48 hours.
Algorithm 1 MINER algorithm.

$I({\bf x})$, number of scales $J$, block size $b$, number of features per
layer $N_{\text{feat}}$, number of layers $N_{\text{layers}}$

for $j=J-1,J-2,\ldots,0$ do $\triangleright$ Loop over spatial scales

if $j$ is $J-1$ then

$R_{j}({\bf x}_{j})\leftarrow\text{DOWNSAMPLE}(I({\bf x}),1/2^{J-1})$

end if

$Q_{j}\leftarrow\frac{HW}{2^{j}b^{2}}$ $\triangleright$ Number of blocks

$A_{j}\leftarrow\\{1,2,\ldots,Q_{j}\\}$ $\triangleright$ Active set

for $q=1,2,\ldots,Q_{j}$ do $\triangleright$ Loop over blocks

if $||R_{j}({\bf x}_{j}^{q})||\leq\tau_{j}$ then

$A_{j}\leftarrow A_{j}\backslash q$ $\triangleright$ Remove converged blocks

else

$\mathcal{N}_{j}^{q}\leftarrow\text{MLP}(N_{\text{feat}},N_{\text{layers}})$
$\triangleright$ MLPs for each block

if $j<J-2$ then

$k_{q}\leftarrow\text{PARENT}(q)$

$\theta_{j}^{q}\leftarrow\theta_{j}^{k_{q}}/2$ $\triangleright$ Weight sharing
for images

$\theta_{j}^{q}\leftarrow\theta_{j}^{k_{q}}/\sqrt{8}$ $\triangleright$ Weight
sharing for 3D volumes

end if

end if

end for

for $i=1,2,\ldots,N_{\text{iter}}$ do

for $q$ in $A_{j}$ do

$\widehat{I}({\bf x}_{j}^{q})\leftarrow\mathcal{N}_{j}^{q}({\bf x}_{j}^{q})$
$\triangleright$ Compute MLP output

$\epsilon_{j}^{q}\leftarrow||\widehat{I}({\bf x}_{j}^{q})-I({\bf
x}_{j}^{q})||^{2}$ $\triangleright$ Compute MSE loss

Backpropagate $\epsilon_{j}^{q}$ to update $\theta_{j}^{q}$

if $\epsilon_{j}^{q}\leq\tau_{j}$ then

$A_{j}\leftarrow A_{j}\backslash q$ $\triangleright$ Prune converged blocks

end if

end for

end for

end for

Figure 10 shows fitting result for 64 megapixel Pluto image across training
iterations. The times correspond to the instances when MINER converged at a
given scale. MINER maintains high quality reconstruction at all instances due
to the multiscale training scheme and rapidly converges to a PSNR of 40dB
within 50 seconds. In contrast, ACORN achieves qualitatively good results
after 10s and achieves a PSNR of $30.8$ dB after 50s, and KiloNeRF achieves a
qualitatively good result only after 50s. SIREN Results are not shown in the
plot as the first result was produced after 4 minutes. Figure 11 shows the
gigapixel image from the main paper fit over 7 scales along with more insets.

##### Fitting 3D point clouds.

Figure 12. Comparisons against state-of-the-art for 3D volume fitting. The
figure compares 3D occupancy fitting for a fixed duration with MINER, ACORN,
Convolutional occupancy (Peng et al., 2020), and screening poisson
reconstruction (Kazhdan and Hoppe, 2013). The number of parameters of MINER
was chosen automatically according to model complexity. MINER achieves high
accuracy in a very short duration for arbitrarily complex shapes, which is not
possible with prior works, even though some models such as the engine (second
row) require significantly larger number of parameters.

Figure 12 visualizes the meshes fit with various reconstruction approaches for
a fixed duration. MINER has superior reconstruction quality compared to ACORN
and convolutional occupancy networks (Peng et al., 2020), and comparable
performance to SPSR. We do note that the quality of reconstruction
specifically for the engine model is superior to SPSR. Since the engine model
has a large number of sharp edges, SPSR tends to oversmooth the result. Since
MINER combined with marching cubes relied only on local information for
reconstruction, the resultant mesh was more accurate.

###  8.1. Analysis of parameter space

The training time of MINER is affected by number of scales, size of each
patch, stopping criteria when switching to a finer scale, and parameters of
each MLP including number of layers, number of features per layer and the type
of non-linearity. We now provide a thorough analysis of the parameters.

##### Performance with varying patch size.

The optimal patch size is highly dependent on the signal itself. To understand
the empirical relationship, we fit three types of images with low, medium, and
high texture content using MINER to achieve 40dB accuracy. In each case, we
varied the patch size from 8 pixels to 64 pixels. We proportionally increased
the number of features per layer for each patch to keep the total number of
parameters approximately the same. Figure 13 shows the plot of time taken to
achieve 40 dB as a function of patch size for the three images. We notice that
the optimal patch size for least training time increases with reducing texture
content – which can be used as a guideline when choosing the appropriate patch
size.

##### Performance with number of scales.

Figure 14 shows the time taken to achieve 40dB on the pluto image with varying
number of spatial scales. We note that increasing the number of scales is
beneficial up to four scales, but the performance deteriorates after that.

Figure 13. Effect of patch size. The plot shows the time taken to achieve 40
dB for three different images. We notice that the optimal patch size is highly
dependent on the image. For images with high texture content (image 1), a
smaller patch size of 8 pixels is optimal. For medium texture with large flat
areas (image 2), a medium patch size of 32 pixels is optimal. For images such
as macro photography (image 3) which has strong low frequency content, larger
patch size of 64 is optimal. Figure 14. Effect of number of scales. The plot
shows the time taken to fit a 16MP image (Pluto) with MINER with varying
number of spatial scales. Performance increases up to 4 scales but then
degrades

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# ON A CLASS OF ANHARMONIC OSCILLATORS II. GENERAL CASE

Marianna Chatzakou  Marianna Chatzakou: Department of Mathematics: Analysis,
Logic and Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS>,  Julio Delgado  Julio Delgado: Departamento de
Matemáticas Universidad del Valle, Colombia E-mail address
<EMAIL_ADDRESS>and  Michael Ruzhansky  Michael
Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete Mathematics
Ghent University, Belgium and School of Mathematical Sciences Queen Mary
University of London United Kingdom E-mail address<EMAIL_ADDRESS>
###### Abstract.

In this work we study a class of anharmonic oscillators on ${\mathbb{R}}^{n}$
corresponding to Hamiltonians of the form $A(D)+V(x)$, where $A(\xi)$ and
$V(x)$ are $C^{\infty}$ functions enjoying some regularity conditions. Our
class includes fractional relativistic Schrödinger operators and anharmonic
oscillators with fractional potentials. By associating a Hörmander metric we
obtain spectral properties in terms of Schatten-von Neumann classes for their
negative powers and derive from them estimates on the rate of growth for the
eigenvalues of the operators $A(D)+V(x)$. This extends the analysis in the
first part [8], where the case of polynomial $A$ and $V$ has been analysed.

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and
Partial Differential Equations and by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021). Julio Delgado
is also supported by Vic. Inv Universidad del Valle. Grant No. CI-71281.
Michael Ruzhansky is also supported by EPSRC grants EP/R003025/2 and
EP/V005529.
Keywords: Fractional relativistic Schrödinger operators; anharmonic
oscillators; energy levels; Weyl-Hörmander calculus; microlocal analysis;
growth of eigenvalues

###### Contents

  1. 1 Introduction
  2. 2 Weyl-Hörmander calculus and Schatten-von Neumann classes
  3. 3 The class of operators and the metric
  4. 4 An intrinsic definition of the classes $S(\lambda_{g}^{m},g)$
  5. 5 Schatten-von Neumann classes and spectral properties

##  1\. Introduction

In this manuscript we obtain spectral properties and in particular estimates
for the rate of growth of eigenvalues for a class of operators on
${\mathbb{R}}^{n}$ of the form $A(D)+V(x)$, where $A(\xi)$ and $V(x)$ are
appropriate smooth functions. Some important examples in this class include
fractional relativistic Schrödinger operators, the special cases of
relativistic Schrödinger operators $\sqrt{I-\Delta}+V(x)$, and anharmonic
oscillators with fractional power potentials $(-\Delta)^{\ell}+\langle
x\rangle^{2\kappa}$, where $\ell$ is a positive integer and $\kappa>0$. Our
class also allows lower order terms with respect to each symbol $A(\xi)$ and
$V(x)$; notably real-valued potentials with a finite range of negative values.
Hamiltonians in relativistic quantum mechanics, and quantum field theory are
not, in general, partial differential operators (as in nonrelativistic quantum
mechanics), but pseudo-differential operators. Herein, we frame our class
within the setting of the Weyl-Hörmander calculus by introducing a suitable
Hörmander metric and obtaining a substantial extension of the class considered
in [8].

The mother case of the Hamiltonian $-\Delta+V(x)$ $(\ell=1)$, corresponding to
the Schrödinger equation is one of the milestone objects of study in
mathematical physics, specially since the the study of energy levels for the
Schrödinger equation is reduced to the eigenvalue problem associated the
Hamiltonian. One of the most basic examples of anharmonic oscillators of
polynomial form $-\Delta+|x|^{2k}$, is the one dimensional quartic oscillator
($n=1,\,k=2$) which is an important model in quantum physics and has been
intensively studied in the last 50 years. However, the exact solution for the
eigenvalue problem of this model is unknown (cf. [27], [2]). It is clear that
the situation is even more subtle for non-polynomial Hamiltonians. This kind
of fact is a further motivation for the investigation of different
approximative and qualitative properties for the operators $A(D)+V(x)$.

The case of polynomial Hamiltonians has been widely studied. A class of
anharmonic oscillators arises in the form

$-\frac{d^{2\ell}}{dx^{2\ell}}+x^{2k}+p(x),$

where $p(x)$ is a polynomial of order $2k-1$ on $\mathbb{R}$ and with $k,\ell$
integers $\geq 1$. The spectral asymptotics of such operators have been
analysed by B. Helffer and D. Robert [17, 18, 16]. The authors have recently
studied anharmonic oscillators on ${\mathbb{R}}^{n}$ in [8], where a prototype
operator is of the form

$(-\Delta)^{\ell}+|x|^{2k},$ (1.1)

where $k,\ell$ are integers $\geq 1$. By considering $A(\xi)=|\xi|^{2\ell}$
and $V(x)=|x|^{2k}$, these symbols will be absorbed by our new class
considered in this paper. Spectral properties for the type (1.1) in the case
$k=\ell$ can be found in [16] and [35]. The case of a fractional Laplacian and
a quartic potential has been considered by S. Durugo and J. Lörinczi in [13].
Bochner–Riesz means and spectral analysis for the one-dimensional anharmonic
oscillator $-\frac{d^{2\ell}}{dx^{2\ell}}+|x|$ has been recently studied in
[9]. Physical models related to the operator $\sqrt{-\Delta+m^{2}}$ have been
intensely studied in the last 30 years and there exists a huge literature on
the spectral properties of relativistic Hamiltonians, most of it strongly
influenced by the works of Lieb on the stability of relativistic matter (cf.
[25], [24], [23]). On the other hand, the study of general fractional
relativistic Schrödinger operators $(I-\Delta)^{\gamma}+V(x)$, $\gamma>0,$ has
also attracted the interest in the last decades. The operators
$(I-\Delta)^{\gamma}$ are also related to the so-called relativistic
$\gamma$-stable process. See for instance [1], [14] and the references therein
for recent investigation on the fractional relativistic Schrödinger operators.

Herein we study the operator $A(D)+V(x)$ within the setting of Hörmander’s
$S(m,g)$ classes by introducing a Hörmander metric $g^{(A,V)}$ in Section 3.
In Section 4 we give an intrinsic formulation of such classes:

For $m\in\mathbb{R}$, appropriate functions $A(\xi),V(x)$ on
${\mathbb{R}}^{n}$ and suitable values of $\gamma$ and $\kappa$, the class
$\Sigma_{A,V}^{m}$ consists of all
$C^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ functions enjoying the
property that for all multi-indexes $\alpha,\beta$ there exists a constant
$C_{\alpha\beta}$ such that

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leq
C_{\alpha\beta}(q+V(x)+A(\xi))^{m-\frac{|\beta|}{2\kappa}-\frac{|\alpha|}{2\gamma}}\,,$
(1.2)

holds all $x,\xi\in\mathbb{R}^{n}$.

By looking at the negative powers of our operators $A(D)+V(x)$ and studying
the corresponding Schatten-von Neumann properties within the setting of
Hörmander $S(m,g)$ classes, we will deduce the rate of decay of the
eigenvalues for those negative powers. Thus, from the inverses of these
negative powers we obtain estimates for the rate of growth of eigenvalues of
the operators $A(D)+V(x)$. The investigation of such properties within these
classes started with Hörmander [20]. Other works on Schatten-von Neumann
classes within the Weyl-Hörmander calculus can be found in [38], [39]. See
also [10], [11], [12] for several symbolic and kernel criteria on different
types of domains. For the spectral theory of non-commutative versions of the
harmonic oscillator we refer the reader to the works of Parmeggiani et al
[28], [29], [30], [31], [32] and [33].

The main results of this work give the order of the corresponding Schatten-von
Neumann class for the negative powers of the operators $A(D)+V(x)$. The
special case of the trace class is also distinguished. That is the contents of
Theorem 5.5 and Corollary 5.7. From those we derive estimates for the rate of
growth of our operators $A(D)+V(x)$ in (5.8). In Theorem 5.9 we give a sharp
condition to guarantee the membership of the negative powers of our main
examples to the Schatten-von Neumann classes. At the end of the section we
also provide examples for the special case of the fractional relativistic
Schrödinger operators with fractional potentials.

##  2\. Weyl-Hörmander calculus and Schatten-von Neumann classes

In this section we briefly review some basic elements of the Weyl-Hörmander
calculus and the Schatten-von Neumann classes. For a comprehensive study on
the Weyl-Hörmander calculus, we refer the interested reader to [21], [22],
[4].

Let us briefly recall the main concepts and ideas that will be involved in
this work. Before doing so, let us note that, in the sequel, we shall use
capital latin letters, e.g. $X,Y,T$ to denote elements in the phase space.
These do, in turn, correspond to pairs of the form
$(x,\xi),(y,\eta),(t,\tau)$, where Latin and Greek letters have been used for
elements in the configuration space $\mathbb{R}^{n}$, and its dual
$(\mathbb{R}^{n})^{*}$, respectively.

For $a=a(x,\xi)\in S^{\prime}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$
($x\in\mathbb{R}^{n}$ and $\xi\in\mathbb{R}^{n}$) and $t\in\mathbb{R}$, we
define the t-quantization of the symbol $a$ as being the operator
$a_{t}(x,D):S(\mathbb{R}^{n})\rightarrow S(\mathbb{R}^{n})$ given by the
formula

$a_{t}(x,D)u(x)=(2\pi)^{-n}\int_{{\mathbb{R}}^{n}}\int_{{\mathbb{R}}^{n}}e^{i(x-y)\xi}a(tx+(1-t)y,\xi)u(y)dyd\xi.$

The operator $a_{1}(x,D)$, or for simplicity $a(x,D)$, is known as the Kohn-
Nirenberg quantization, while the operator $a_{\frac{1}{2}}(x,D)$, denoted
also by $a^{w}(x,D)$, is called the Weyl quantization. In the particular cases
of operators that we consider in this work, the corresponding symbols that
appear stay invariant (up to lower under terms) under different quantizations.

To define Weyl-Hörmander classes of symbols the following notions shall be
recalled; if $g_{X}(\cdot)$ is a positive definite quadratic form on the phase
space, then $g_{\cdot}(\cdot)$ is a Hörmander’s metric if the following three
conditions are satisfied:

  1. (1)

Continuity or slowness\- There exist a constant $C>0$ such that

$g_{X}(X-Y)\leq C^{-1}\implies\left(\frac{g_{X}(T)}{g_{Y}(T)}\right)^{\pm
1}\leq 1\,,\quad\forall T\,.$

  2. (2)

Uncertainty principle\- We say that $g$ satisfies the uncertainty principle
if

$\lambda_{g}(X)=\inf_{T\neq
0}\left(\frac{g_{X}^{\sigma}(T)}{g_{X}(T)}\right)^{1/2}\geq 1\,,\quad\forall
X,T\,,$

where $g_{X}^{\sigma}(T):=\sup_{W\neq
0}\left\\{\sigma(Y,T)^{2}/g_{X}(Y)\right\\}$, and
$\sigma(Y,T):=t\cdot\eta-y\cdot\tau$.

  3. (3)

Temperateness\- We say that $g$ is temperate if there exist $\overline{C}>0$
and $J\in\mathbb{N}$ such that

$\left(\frac{g_{X}(T)}{g_{Y}(T)}\right)^{\pm
1}\leq\overline{C}(1+g_{Y}^{\sigma}(X-Y))^{J}\,,\quad\forall X,Y,T\,.$ (2.1)

For $g$ being a fixed Hörmander metric, and $M$ being a positive function on
the phase space we say that $M$ is a $g$-weight, if the following are
satisfied:

  1. (1)

$g$\- continuous, if there exists $\tilde{C}>0$ such that

$g_{X}(X-Y)\leq\frac{1}{\tilde{C}}\implies\left(\frac{M(X)}{M(Y)}\right)^{\pm
1}\leq\tilde{C}\,,\quad\forall X,Y\,.$

  2. (2)

$g$-temperate, if there exist $\tilde{C}>0$ and $N\in\mathbb{N}$ such that

$\left(\frac{M(X)}{M(Y)}\right)^{\pm
1}\leq\tilde{C}(1+g_{Y}^{\sigma}(X-Y))^{N}\,,\quad\forall X,Y\,.$

For $M,g$ being as above, we define the set of symbols $S(M,g)$ as the set of
smooth functions $a$ on the phase space such that for any integer $k$ there
exists $C_{k}>0$, such that for all
$X,T_{1},...,T_{k}\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ we have

$|a^{(k)}(X;T_{1},...,T_{k})|\leq C_{k}M(X)\prod_{i=1}^{k}g_{X}^{1/2}(T_{i}).$
(2.2)

The notation $a^{(k)}$ stands for the $k^{th}$ derivative of $a$ and
$a^{(k)}(X;T_{1},...,T_{k})$ denotes the $k^{th}$ derivative of $a$ at $X$ in
the directions $T_{1},...,T_{k}$. For $a\in S(M,g)$ we define

$\|a\|_{k,S(M,g)}:=\inf\\{C_{k}:C_{k}\quad\text{satisfies}\quad(\ref{inwhk})\\}\,.$

The family of seminorms $\parallel\cdot\parallel_{k,S(M,g)}$ endows $S(M,g)$
with the topology of a Fréchet space.

The family of seminorms $\parallel\cdot\parallel_{k,S(M,g)}$ endows $S(M,g)$
with the topology of a Fréchet space.

Finally let us discuss some basic definitions of the Schatten-von Neumann
classes of operators; for a more detailed exposition of the theory we refer to
[15], [34], [37], [36].

Let $H$ be a separable Hilbert space over $\mathbb{C}$ endowed with the inner
product $(\cdot,\cdot)$, and let $T$ be some compact (linear) operator from
$H$ to itself. We denote by $|T|:=(T^{*}T)^{1/2}$ the absolute value of $T$,
and by $s_{n}(T)$ the singular values of $T$; that is the eigenvalues of
$|T|$, that correspond to the eigenfunctions of the latter given by the
spectral theorem. The operator $T$ belongs to the Schatten-von Neumann class
of operators $S_{p}(H)$, where $1\leq p<\infty$, if

$\|T\|_{S_{p}}:=\left(\sum_{k=1}^{\infty}(s_{k}(T))^{p}\right)^{\frac{1}{p}}<\infty\,.$

The space $S_{p}$ is a Banach space if endowed with the natural norm
$\|\cdot\|_{S_{p}}$. In particular, the Banach space $S_{1}(H)$ is the space
of trace-class operators, while for $T\in S_{1}$ the quantity

$\textnormal{Tr}(T):=\sum_{n=1}^{\infty}(T\phi_{n},\phi_{n})\,,$

where $(\phi_{n})$ is an orthonormal basis in $H$, is well-defined and shall
be called the trace $\textnormal{Tr}(T)$ of $T$. Moreover, the space
$S_{2}(H)$ is identified with the space of Hilbert-Schmidt operators on $H$.

##  3\. The class of operators and the metric

In this section we start with the study of the specific class of operators
that we consider here that are regarded to be of the form $A(D)+V(x)$. We
first introduce a suitable class of functions for the terms $A(\xi),V(x)$ of
their symbols.

######  Definition 3.1.

Let $\Gamma:{\mathbb{R}}^{n}\rightarrow\mathbb{R}$ be a continuous function.
We say that $\Gamma$ is a $\tau$-function, for some $\tau>0$, if the following
conditions are satisfied:

  1. (a)

there exists $q>0$ such that $\Gamma(x)+\frac{q}{4}>0$ for all
$x\in{\mathbb{R}}^{n}$;

  2. (b)

there exist $\tau>0$ such that

$C_{1}|(\frac{q}{2}+\Gamma(x))^{\frac{1}{2\tau}}-(\frac{q}{2}+\Gamma(y))^{\frac{1}{2\tau}}|\leq|x-y|,\,\mbox{
for all }|x|,|y|\geq R\,,$

for some $C_{1}>0,R>0$.

######  Example 1.

As some examples of functions satisfying Definition 3.1 we can consider
potentials of the form:

  * (i)

$V(x)=\langle x\rangle^{2\kappa}+\,$ lower order, with $\kappa>0$ and choosing
$\tau=\kappa$;

  * (ii)

$V(x)=|x|^{2k}+\,$ lower order, where $k$ is an integer $\geq 1$, and choosing
$\tau=k$.

Similarly for the symbol $A(\xi)$ of the operator $A(D)$ we can consider:

  * (i)

$A(\xi)=\langle\xi\rangle^{2\gamma}+\,$ lower order, with $\tau=\gamma>0$;

  * (ii)

$A(\xi)=|\xi|^{2\ell}+\,$ lower order, where $\ell$ is an integer $\geq 1$ and
$\tau=\ell$.

More generally we define the following class of operators we are going to
consider.

######  Definition 3.2.

The $(\gamma,\kappa)$-class Let $\gamma,\kappa>0$. We say that an operator of
the form $A(D)+V(x)$ belongs to the $(\gamma,\kappa)$-class, if $A(\xi)$ is a
$\gamma$-function, and $V(x)$ is a $\kappa$-function.

######  Example 2.

Special examples of operators in the $(\gamma,\kappa)$-class.

  * (i)

Anharmonic oscillators of the form

$(-\Delta)^{\ell}+|x|^{2k},$

where $k,\ell$ integers $\geq 1$, belong to the $(\ell,\kappa)$-class.

Spectral properties for this class of operators have been recently analysed by
the authors in [8] within the setting of Weyl-Hörmander calculus. Another
example of operators in the $(\ell,\kappa)$-class are those expressed by the
formula $-\frac{d^{2\ell}}{dx^{2\ell}}+x^{2k}+p_{1}(x)$, where $p_{1}$ is a
suitable polynomial of order $2k-1$. Those have been studied by Helffer and
Robert (cf. [18], [19]).

  * (ii)

Relativistic Schrödinger operators of the form

$\sqrt{I-\Delta}+V(x),$

where $V(x)$ satisfies the Definition 3.1 for a suitable $\kappa>0$, belong
the the $(1,\kappa)$-class.

  * (iii)

Anharmonic oscillators with fractional potential

$(-\Delta)^{\ell}+\langle x\rangle^{2\kappa},$

where $\ell$ is an integer $\geq 1$ and $\kappa>0$, are in the
$(\ell,\kappa)$-class.

We associate to an operator $A(D)+V(x)$ in the $(\gamma,\kappa)$-class, the
following metric:

$g^{(A,V)}=\frac{dx^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\kappa}}}+\frac{d\xi^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\gamma}}}$
(3.1)

We choose a constant $q>0$ in (3.1) to be such that $q+V(x)+A(\xi)\geq 1$, for
all $x,\xi\in\mathbb{R}^{n}$.

Our first aim is to prove that the metric introduced in (3.1) is a Hörmander
metric. To this end, let us first recall the auxiliary lemma that will be used
later in the proof of the continuity property. For the proof of it, the
interested reader, can consult [21, Theorem 18.4.2].

######  Lemma 3.3.

Let $g$ be a Riemannian metric on the phase space. The following statements
are equivalent:

  1. (i)

$g$ is continuous.

  2. (ii)

There exists a constant $C\geq 1$ such that

$g_{X}(X-Y)\leq C^{-1}\,\,\mbox{ implies }\,\,g_{Y}\leq Cg_{X}.$

  3. (iii)

here exists a constant $C\geq 1$ such that

$g_{X}(Y)\leq C^{-1}\,\,\mbox{ implies }\,\,g_{X+Y}\leq Cg_{X}.$

We now consider the metric $g^{(A,V)}$ in detail.

######  Theorem 3.4.

The metric $g=g^{(A,V)}$ defined by (3.1) is a Hörmander metric.

###### Proof.

Regarding the uncertainty parameter we have

$\lambda_{g}(X)=(q+V(x)+A(\xi))^{\frac{\kappa+\gamma}{2\kappa\gamma}}\,,$

and it is clear that, due the choice of $q$ in (3.1), we have
$\lambda_{g}(X)\geq 1$ for all $X$.

For the proof of the continuity of the metric $g^{(A,V)}$, we will use the
characterisation (ii) in Lemma 3.3; i.e., we aim to prove that

$\frac{|x-y|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\kappa}}}+\frac{|\xi-\eta|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\gamma}}}\leq
C^{-1}\implies$
$\frac{|t|^{2}}{(q+V(y)+A(\eta))^{\frac{1}{\kappa}}}+\frac{|\tau|^{2}}{(q+V(y)+A(\eta))^{\frac{1}{\gamma}}}\\\
\leq
C\left(\frac{|t|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\kappa}}}+\frac{|\tau|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\gamma}}}\right),$

for all $X,Y,T\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$.

The proof can be reduced to proving that

$\frac{|x-y|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\kappa}}}\,,\,\,\frac{|\xi-\eta|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\gamma}}}\leq
C^{-1}\implies\frac{(q+V(x)+A(\xi))^{\frac{1}{\kappa}}}{(q+V(y)+A(\eta))^{\frac{1}{\kappa}}}\leq
C.$

Or, even simpler, to proving the following implication

$\frac{|x-y|}{(q+V(x)+A(\xi))^{\frac{1}{2\kappa}}}\,,\,\,\frac{|\xi-\eta|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{2\gamma}}}\leq
C^{-1}\implies\frac{q+V(x)+A(\xi)}{q+V(y)+A(\eta)}\leq C.$ (3.2)

We will assume that the LHS of (3.2) holds for a constant $C>0$ to be chosen
later on. Since

$C_{1}|(\frac{q}{2}+V(x))^{\frac{1}{2\kappa}}-(\frac{q}{2}+V(y))^{\frac{1}{2\kappa}}|\leq|x-y|,\,\mbox{
for all }|x|,|y|\geq R;$

and analogously for $A(\xi)$, we have

$C_{1}(\frac{q}{2}+V(x))^{\frac{1}{2\kappa}}\leq
C^{-1}(q+V(x)+A(\xi))^{\frac{1}{2\kappa}}+C_{1}(\frac{q}{2}+V(y))^{\frac{1}{2\kappa}},$
(3.3)

for all $|x|,|y|\geq R$, and

$C_{1}(\frac{q}{2}+A(\xi))^{\frac{1}{2\gamma}}\leq
C^{-1}(q+V(x)+A(\xi))^{\frac{1}{2\gamma}}+C_{1}(q+A(\eta))^{\frac{1}{2\gamma}}.$
(3.4)

for all $|\xi|,|\eta|\geq R$. We note that due to the continuity of $A$ and
$V$ we can also assume that the above inequalities hold for $|x|,|y|\leq 2R$
and $|\xi|,|\eta|\leq 2R$.

By taking powers $2\kappa$ and $2\gamma$ of (3.3) and (3.4) respectively,
there exists a constant $C_{2}>1$ only dependent on $\kappa$ and $\gamma$ such
that

$V(x)\leq C^{-2\kappa}C_{2}(q+V(x)+A(\xi))+C_{2}V(y),$ (3.5)

$A(\xi)\leq C^{-2\gamma}C_{2}(q+V(x)+A(\xi))+C_{2}A(\eta).$ (3.6)

By adding (3.5) and (3.6), taking $\kappa_{0}=\min\\{\kappa,\gamma\\}$ and $C$
large enough, we obtain

$q+V(x)+A(\xi)\leq\tilde{C}(q+V(y)+A(\eta))\,,$

for some $\tilde{C}>0$. Hence, we obtain

$\frac{q+V(x)+A(\xi)}{q+V(y)+A(\eta)}\leq\tilde{C}\,,$

and the last inequality shows the continuity of the metric $g^{(A,V)}$.

We now prove the temperateness. According to (2.1), we need to prove that
there exist $C>0$ and $N\in\mathbb{N}$ such that

$\frac{|t|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\kappa}}}+\frac{|\tau|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\gamma}}}\\\
\leq
C\left(\frac{|t|^{2}}{(q+V(y)+A(\eta))^{\frac{1}{\kappa}}}+\frac{|\tau|^{2}}{(q+V(y)+A(\eta))^{\frac{1}{\gamma}}}\right)\times\\\
\times\left(1+(q+V(x)+A(\xi))^{\frac{1}{\gamma}}|x-y|^{2}+(q+V(x)+A(\xi))^{\frac{1}{\kappa}}|\xi-\eta|^{2}\right)^{N},$

and

$\frac{|t|^{2}}{(q+V(y)+A(\eta))^{\frac{1}{\kappa}}}+\frac{|\tau|^{2}}{(q+V(y)+A(\eta))^{\frac{1}{\gamma}}}\\\
\leq
C\left(\frac{|t|^{2}}{(q+V(x)+A(\xi))^{\frac{1}{\kappa}}}+\frac{|\tau|^{2}}{(2+V(x)+A(\xi))^{\frac{1}{\gamma}}}\right)\times\\\
\times\left(1+(q+V(x)+A(\xi))^{\frac{1}{\gamma}}|x-y|^{2}+(q+V(x)+A(\xi))^{\frac{1}{\kappa}}|\xi-\eta|^{2}\right)^{N},$

for all $t,\tau\in{\mathbb{R}}^{n}$.

We will only prove the first inequality since the second one can be proven in
a similar way.

We now observe that we can reduce the proof of the first inequality to the
proof of the following two inequalities:

$\frac{(q+V(y)+A(\eta))^{\frac{1}{\kappa}}}{(q+V(x)+A(\xi))^{\frac{1}{\kappa}}}\leq\\\
\leq
C\left(1+(q+V(x)+A(\xi))^{\frac{1}{\gamma}}|x-y|^{2}+(q+V(x)+A(\xi))^{\frac{1}{\kappa}}|\xi-\eta|^{2}\right)^{N},$

and

$\frac{(q+V(y)+A(\eta))^{\frac{1}{\gamma}}}{(q+V(x)+A(\xi))^{\frac{1}{\gamma}}}\leq\\\
\leq
C\left(1+(q+V(x)+A(\xi))^{\frac{1}{\gamma}}|x-y|^{2}+(q+V(x)+A(\xi))^{\frac{1}{\kappa}}|\xi-\eta|^{2}\right)^{N}.$

We note that it is enough to prove the first inequality, which, in turn, can
be reduced to the following one

$\frac{(q+V(y)+A(\eta))^{\frac{1}{2\kappa}}}{(q+V(x)+A(\xi))^{\frac{1}{2\kappa}}}\leq$
$\leq
C\left(2+(q+V(x)+A(\xi))^{\frac{1}{2\gamma}}|x-y|+(2+(q+V(x)+A(\xi))^{\frac{1}{2\kappa}}|\xi-\eta|\right)^{N}.$
(3.7)

Now (3.7) can be obtained from the inequalities below

$\frac{(\frac{q}{2}+V(y))^{\frac{1}{2\kappa}}}{(q+V(x)+A(\xi))^{\frac{1}{2\kappa}}}\leq\frac{(\frac{q}{2}+V(y))^{\frac{1}{2\kappa}}}{(\frac{q}{2}+V(x))^{\frac{1}{2\kappa}}}\leq
C\left(2+(q+V(x))^{\frac{1}{2\gamma}}|x-y|\right)^{N},$ (3.8)
$\frac{(\frac{q}{2}+A(\eta))^{\frac{1}{2\kappa}}}{(q+V(x)+A(\xi))^{\frac{1}{2\kappa}}}\leq\frac{(\frac{q}{2}+A(\eta))^{\frac{1}{2\kappa}}}{(\frac{q}{2}+A(\xi))^{\frac{1}{2\kappa}}}\leq
C\left(2+(\frac{q}{2}+A(\xi))^{\frac{1}{2\kappa}})|\xi-\eta|\right)^{N}.$
(3.9)

In order to verify (3.8) we observe that, by choosing an integer $N$ such that
$\frac{N}{2\gamma}\geq 1$, and applying the assumption (b) of Definition 3.1
we obtain for a suitable $C>0$:

$(\frac{q}{2}+V(y))^{\frac{1}{2\kappa}}\leq
C\left(|x-y|+(\frac{q}{2}+V(x))^{\frac{1}{2\kappa}}\right)\leq$ $\leq
C\left(1+(\frac{q}{2}+V(x))^{\frac{1}{2\gamma}}|x-y|\right)^{N}(1+V(x))^{\frac{1}{2\kappa}}$

The proof of inequality (3.9) follows similarly, and the poof of Theorem 3.4
is complete. ∎

We have shown that $g=g^{(A,V)}$ is a Hörmander metric,thus the $S(M,g)$
classes are defined for any $g$-weight $M$ and a Weyl-Hörmader calculus is
available to our disposal.

We now make some essential observations regarding our symbol $V(x)+A(\xi)$.

######  Theorem 3.5.

Let $g=g^{(A,V)}$ be the metric defined by (3.1). Then

  * (i)

$q+V(x)+A(\xi)$ is a $g$-weight.

  * (ii)

There exists a regular weight $m$ equivalent to $M(X)=q+V(x)+A(\xi)$, i.e., we
have $\left(\frac{m(Y)}{M(Y)}\right)^{\pm 1}\leq C$, for all $Y$, and $m\in
S(m,g)$. Consequently, $S(m,g)=S(M,g)$.

###### Proof.

(i) A look at the proof of the continuity and temperateness of the metric $g$
shows that $q+V(x)+A(\xi)$ is a $g$-weight. We also give a different and more
general argument showing this fact.

For the metric $g=g^{(A,V)}$ the uncertainty parameter is given by

$\lambda_{g}(X)=(q+V(x)+A(\xi))^{\frac{\kappa+\gamma}{2\kappa\gamma}}.$

In general the uncertainty parameter of a Hörmander metric $G$ is a $G$-weight
(cf. [21] ). So, by taking the power $\frac{2\kappa\gamma}{\kappa+\gamma}$ of
$\lambda_{g}$, we obtain that $q+V(x)+A(\xi)$ is a $g$-weight. (ii) For any
weight $M$ one can construct an equivalent regular weight $\widetilde{M}$ as a
consequence of the existence of suitable $g$-partitions of unity (cf. [22],
[4], [3]). Since $q+V(x)+A(\xi)$ is a $g$-weight, in particular, there exists
a regular weight $m$ equivalent to $M=q+V(x)+A(\xi)$. It is clear that
$S(m,g)=S(M,g)$. ∎

For some purposes it is enough to work with the equivalent weight
$\widetilde{M}$. In our case with the perspective of establishing Schatten-von
Neumann properties we will assume some further regularity conditions that will
be introduced in Section 5.In particular, it will be assumed that
$A(\xi)+V(x)$ is a symbol in the class $S(q+V(x)+A(\xi),g^{(A,V)})$, a fact
that holds for the main examples we are going to present.

######  Example 3.

As an example we observe the relativistic Schrödinger operator
$\sqrt{I-\Delta}+V(x)$ with $V$ satisfying the Definition 3.1 for some
$\kappa>0$ and $V\in S(q+V(x)+A(\xi),g^{(A,V)})$ with $A(D)=\sqrt{I-\Delta}$.
In this case we have $\gamma=\frac{1}{2}$ and

$\langle\xi\rangle+V(x)\in
S(\lambda_{g}^{\frac{\kappa}{\kappa+\frac{1}{2}}},g^{(A,V)}),$

i.e. the relativistic Schrödinger operator is of order
$\frac{\kappa}{\kappa+\frac{1}{2}}$ with respect to $g^{(A,V)}$.

##  4\. An intrinsic definition of the classes $S(\lambda_{g}^{m},g)$

Let $g=g^{(A,V)}$ be the Hörmander metric defined by (3.1). Since we have
already a guarantee of the existence of a corresponding pseudo-differential
calculus, we can now define the class $S(\lambda_{g}^{m},g)$, for
$m\in\mathbb{R}$, in an equivalent and intrinsic way without referring
explicitly to the metric $g$.

######  Definition 4.1.

Let $m\in\mathbb{R}$, and let $A(\xi),V(x)$ be a $\gamma$-function and a
$\kappa$-function, respectively. For $a\in
C^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$, we will say that
$a\in\Sigma_{A,V}^{m}$, if for some $q>0$ large enough, and for all multi-
indices $\alpha,\beta$ there exists a constant $C_{\alpha\beta}$ such that

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leq
C_{\alpha\beta}(q+V(x)+A(\xi))^{m-\frac{|\beta|}{2\kappa}-\frac{|\alpha|}{2\gamma}}\,,$
(4.1)

for all $x,\xi\in\mathbb{R}^{n}$.

We point out that the definition above corresponds indeed to the one for the
metric $g^{(A,V)}$:

######  Proposition 4.2.

Let $m\in\mathbb{R}$ and let $g=g^{(A,V)}$ be the metric defined by (3.1).
Then

$\Sigma_{A,V}^{m}=S(\lambda_{g}^{m(\frac{2\kappa\gamma}{\kappa+\gamma})},g).$

###### Proof.

Observe that for $g=g^{(A,V)}$ we have

$q+V(x)+A(\xi)\sim\lambda_{g}^{\frac{2\kappa\gamma}{\kappa+\gamma}}.$

Therefore, by (4.1), we have

$S(\lambda_{g}^{m(\frac{2\kappa\gamma}{\kappa+\gamma})},g)\supset\Sigma_{A,V}^{m}.$

On the other hand, if $a\in
C^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})\in
S(\lambda_{g}^{m(\frac{2\kappa\gamma}{\kappa+\gamma})},g)$, then by Definition
2.2 and by taking canonical directional derivatives for every pair of multi-
indices $\alpha,\beta$ there exists a constant $C_{\alpha\beta}>0$ such that

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leq
C_{\alpha\beta}(q+V(x)+A(\xi))^{m-\frac{|\beta|}{2\kappa}-\frac{|\alpha|}{2\gamma}}\,,$

for all $x,\xi\in\mathbb{R}^{n}$.
Therefore
$S(\lambda_{g}^{m(\frac{2\kappa\gamma}{\kappa+\gamma})},g)\subset\Sigma_{A,V}^{m}$,
and this completes the proof. ∎

We now consider the composition formula in the setting of the classes
$\Sigma_{A,V}^{m}$ as a consequence of the corresponding one in the $S(M,g)$
calculus.

######  Theorem 4.3.

Let $m_{1},m_{2}\in\mathbb{R}$. If
$a\in\Sigma_{A,V}^{m_{1}},b\in\Sigma_{A,V}^{m_{2}}$. There exists
$c\in\Sigma_{A,V}^{m_{1}+m_{2}}$ such that $a(x,D)\circ b(x,D)=c(x,D)$ and

$c(x,\xi)\sim\sum\limits_{\alpha}(2\pi
i)^{-|\alpha|}\partial_{\xi}^{\alpha}a(x,\xi)\partial_{x}^{\alpha}b(x,\xi),$

i.e., for all $N\in\mathbb{N}$

$c(x,\xi)-\sum\limits_{|\alpha|<N}(2\pi
i)^{-|\alpha|}\partial_{\xi}^{\alpha}a(x,\xi)\partial_{x}^{\alpha}b(x,\xi)\in\Sigma_{k,\ell}^{m_{1}+m_{2}-N\left(\frac{\kappa+\gamma}{2\kappa\gamma}\right)}.$

The $L^{2}$ boundedness of operators in the class $\Sigma_{A,V}^{0}$ is
celarly obtained from the corresponding one for the $S(1,g)$ class.

######  Theorem 4.4.

Let $a\in\Sigma_{A,V}^{0}$. Then, $a(x,D)$ extends to a bounded operator
$a(x,D):L^{2}({\mathbb{R}}^{n})\rightarrow L^{2}({\mathbb{R}}^{n})$.

##  5\. Schatten-von Neumann classes and spectral properties

We are now ready to study some properties of negative powers of operators of
the form $q+A(D)+V(X)$ in relation with their behaviour in the Schatten-von
Neumann classes of operators. The main aim of this section is to provide a
simple proof for the main term of the spectral asymptotics of these operators.
In the context of the Weyl-Hörmander calculus it is useful to recall the
following result by Toft [38]. See also [7] for further developments.

######  Theorem 5.1.

Let $g$ be a split Hörmander metric, and let $M$ be a $g$-weight. For $1\leq
r<\infty$, if $h_{g}^{\frac{k}{2}}M\in
L^{r}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$ for some $k\geq 0$, then, for
any $a\in S(M,g)$, we have

$a_{t}(x,D)\in S_{r}(L^{2}({\mathbb{R}}^{n})),\,\,\mbox{ for all
}t\in\mathbb{R}.$

In order to obtain the desired spectral properties for the negative powers of
our operator $q+A(D)+V(x)$, we are going to assume two additional conditions,
that hold for relevant cases, and in particular for our main examples.
Precisely, the following definition introduces these extra assumptions:

######  Definition 5.2.

We define the following conditions on the pair of functions $(A(x),V(\xi))$:

  * (C1)

$A,V$ are $C^{\infty}$ functions on ${\mathbb{R}}^{n}$ and
$A(\xi)+V(x)\in\Sigma_{A,V}^{1}.$

  * (C2)

There exist $\delta>0$ and $C>0$ such that $q+V(x)+A(\xi)\geq
C(1+|x|+|\xi|)^{\delta}$.

  * (C3)

There exists $\mu>0$ such that

$\int\limits_{{\mathbb{R}}^{n}}\int\limits_{{\mathbb{R}}^{n}}(q+V(x)+A(\xi))^{-\mu}<\infty.$
(5.1)

######  Remark 5.3.

Let us highlight, that condition (C2) implies condition (C3); indeed, one can
choose $\mu>\frac{2n}{\delta}$ in (C3), where $\delta$ is the one appearing in
(C2). In particular, given condition (C1), condition (C2) implies, one the one
hand, that the symbol $q+V(x)+A(\xi)$ is $g$-elliptic with respect to the
metric $g=g^{(A,V)}$ in the sense of [5], and, on the other hand, the
integrability condition (5.1). However, condition (5.1) can also be achieved
for $\mu\leq\frac{2n}{\delta}$. This is why we chose to refer to condition
(C3) explicitly, and not to include it in condition (C2). In particular, as we
will see for the fractional relativistic Schrödinger operators
$(I-\Delta)^{\gamma}+\langle x\rangle^{2k}$, one can choose in condition (C3)
some $\mu>\mu_{0}$, where $\mu_{0}=\frac{n(\kappa+\gamma)}{2\kappa\gamma}.$
For this type of operators the natural choice for $\delta$ is
$\delta=\min\\{2\kappa,2\gamma\\}$ in order to guarantee (C2). Now, comparing
the lower bounds, $\frac{2n}{\delta}$ and
$\frac{n(\kappa+\gamma)}{2\kappa\gamma}$, for $\mu$, we see that, for
$\kappa\neq\gamma$, we have that

$\frac{2n}{\delta}=\frac{2n}{\min\\{2\kappa,2\gamma\\}}>\frac{n(\kappa+\gamma)}{2\kappa\gamma}.$

In the case where $\kappa=\gamma$, the two lower bounds are identical.

Therefore the exponent $\mu_{0}=\frac{n(\kappa+\gamma)}{2\kappa\gamma},$ gives
a better condition for $\mu$ in (C3) and moreover one can show that it is
sharp; see Theorem 5.9 and Remark 5.10.

Regarding the condition (C1), we give a mild sufficient condition guaranteeing
such membership.

######  Lemma 5.4.

Let $m\in\mathbb{R}$, and let $A,V$ be a $\gamma$-function and a
$\kappa$-function, respectively. Let $B=B(x),W=W(\xi)$ be $C^{\infty}$
complex-valued functions defined on ${\mathbb{R}}^{n}$ such that:

(i) for all multi-indices $\beta$ there exists a constant $C_{\beta}$ such
that

$|\partial_{x}^{\beta}B(x)|\leq
C_{\beta}(q+V(x)+A(\xi))^{m-\frac{|\beta|}{2\kappa}}\,,\mbox{ for all
}x\in\mathbb{R}^{n}\,;$ (5.2)

(ii) for all multi-indices $\alpha$ there exists a constant $C_{\alpha}$ such
that

$|\partial_{\xi}^{\alpha}W(\xi)|\leq
C_{\alpha}(q+V(x)+A(\xi))^{m-\frac{|\alpha|}{2\gamma}}\,,\mbox{ for all
}\xi\in\mathbb{R}^{n}.$ (5.3)

Then, $W(\xi)+B(x)\in\Sigma_{A,V}^{m}$.

###### Proof.

The assumptions (i) and (ii) ensure that $B$ and $W$ belong to
$\Sigma_{A,V}^{m}$ according to Definition 4.1. Hence, by the general theory,
we also have $W(\xi)+B(x)\in\Sigma_{A,V}^{m}$. ∎

######  Example 4.

We note that the pairs $(A(x),V(\xi))$ listed in Example 1, also satisfy the
conditions given by Definition 5.2. Indeed, the condition (C1) follows from
Lemma 5.4, while the condition (C2) holds by choosing
$\delta=\min\\{2\kappa,2\gamma\\}$. Condition (C3) then follows from condition
(C2) as explained above. For instance, let us consider the fractional
relativistic Schrödinger operator $(I-\Delta)^{\gamma}+\langle
x\rangle^{2\kappa}$, where $\gamma>0$. We take
$A(\xi)=\langle\xi\rangle^{2\gamma},V(x)=\langle x\rangle^{2\kappa}$ and
observe that $V(x),A(\xi)$ satisfy (i) and (ii), respectively, in Lemma 5.4
with $m=1$. Therefore $\langle\xi\rangle^{2\gamma}+\langle
x\rangle^{2\kappa}\in\Sigma_{\langle\xi\rangle^{2\gamma},\langle
x\rangle^{2\kappa}}^{1}$.

We can now formulate some consequences for Schatten-von Neumann classes:

######  Theorem 5.5.

Let $g=g^{(A,V)}$ be the metric defined in (3.1), $1\leq r<\infty$, and assume
that (C3) holds for some $\mu>0$ as in (5.1). Then, for $a\in
S((q+V(x)+A(\xi))^{-\frac{\mu}{r}},g)$, we have

$a_{t}(x,D)\in S_{r}(L^{2}({\mathbb{R}}^{n})),\,\,\mbox{ for all
}t\in\mathbb{R}.$

###### Proof.

We observe that condition (5.1) on $\mu$ implies that

$(q+V(x)+A(\xi))^{-\frac{\mu}{r}}\in
L^{r}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n}).$

Hence, an application of Theorem 5.1 with
$M=(q+V(x)+A(\xi))^{-\frac{\mu}{r}},$ and for $k=0$, yields that for $a\in
S((q+V(x)+A(\xi))^{-\frac{\mu}{r}},g)$, we have

$a_{t}(x,D)\in S_{r}(L^{2}({\mathbb{R}}^{n})),\,\,\mbox{ for all
}t\in\mathbb{R}\,,$

proving the theorem. ∎

We now recall some basic properties regarding the Hilbert-Schmidt and trace
class operators, see e.g. [26].

######  Remark 5.6.

If $a^{w}$ extends to a Hilbert-Schmidt operator on $L^{2}(\mathbb{R}^{n})$,
then one has

$\|a^{w}\|_{S_{2}}=(2\pi)^{-\frac{n}{2}}\|a\|_{L^{2}(\mathbb{R}^{2n})}\,,$

while if $a^{w}$ extends to a trace-class operator on $L^{2}(\mathbb{R}^{n})$,
then one has

$\textnormal{Tr}(a^{w})=(2\pi)^{-n}\int_{\mathbb{R}^{2n}}a(x,\xi)dx\,d\xi\,.$

We can now obtain some consequences for the negative powers of our
Hamiltonians.

######  Corollary 5.7.

Let $g=g^{(A,V)}$ be the metric defined by (3.1).

  1. (a)

Let $a\in S((q+V(x)+A(\xi))^{-\nu},g)$. Then, the following hold true:

    1. (i)

For all $\nu\geq\frac{\mu}{2}$, where $\mu$ is the one appearing in (5.1), and
for all $t\in\mathbb{R}$, the operator $a_{t}(x,D)$ extends to a Hilbert-
Schmidt operator on $L^{2}(\mathbb{R}^{n})$, and in particular we have

$\|a_{t}(x,D)\|_{S_{2}}=(2\pi)^{-\frac{n}{2}}\|a\|_{L^{2}(\mathbb{R}^{2n})}\,.$

    2. (ii)

For all $\nu\geq\mu$,where $\mu$ is the one appearing in (5.1), and for all
$t\in\mathbb{R}$, the operator $a_{t}(x,D)$, extends to a trace-class operator
on $L^{2}(\mathbb{R}^{n})$, and in particular we have

$\textnormal{Tr}(a_{t}(x,D))=(2\pi)^{-n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}a(x,\xi)\,dxd\xi\,.$

  2. (b)

Let us additionally assume that the pair $(A(x),V(\xi))$ satisfies the
conditions (C1), (C2), and (C3) for some $\mu>0$. Then, for $1\leq r<\infty$
and $\nu\geq\frac{\mu}{r}$, we have

$T^{-\nu}\equiv(q+A(D)+V(x))^{-\nu}\in S_{r}(L^{2}({\mathbb{R}}^{n}))\,.$

  3. (c)

We keep the pair $(A(x),V(\xi))$ satisfying the conditions (C1), (C2) and (C3)
for some $\mu>0$. Then, the following hold true:

    1. (i)

The operator $T^{-\frac{\mu}{2}}$ extends to a Hilbert-Schmidt operator on
$L^{2}(\mathbb{R}^{n})$, and there exists a constant $C>0$ such that

$\|T^{-\frac{\mu}{2}}\|_{S_{2}}\leq
C\|(q+V(x)+A(\xi))^{-\frac{\mu}{2}}\|_{L^{2}(\mathbb{R}^{2n})}\,.$

    2. (ii)

The operator $T^{-\mu}$ extends to a trace-class operator on
$L^{2}(\mathbb{R}^{n})$, and there exists a constant $C>0$ such that

$|\textnormal{Tr}(T^{-\mu})|\leq
C\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}(q+V(x)+A(\xi))^{-\mu}\,dxd\xi\,.$

###### Proof of Corollary 5.7.

The proof of part (a) follows by Theorem 5.5 for $r=1,2$ and by Remark 5.6,
since, as mentioned above, condition (C2) implies condition (C3). We now prove
part (b). We note that, on the one hand, we have $A(D)=A^{w}(x,D)$, while on
the other hand, the multiplication operator $V(x)$ has the potential $V(x)$
itself as is Kohn-Nirenberg symbol, with its Weyl symbol being equal to
$V(x)$\+ lower order. The last is due to the asymptotic formula for comparing
the corresponding symbols with respect to different $t$-quantizations (cf.
formula (2.3.29) of Theorem 2.3.18 in [22]). Moreover, the $g$-ellipticity of
the symbol $q+V(x)+A(\xi)$ ensures that, the negative powers
$(q+A(D)+V(x))^{-\nu}$ for all $\nu>0$, are well defined, and their symbols
belong to the class $S((q+V(x)+A(\xi))^{-\nu},g)$. Now an application of
Theorem 5.5 to the symbol of $T^{-\nu}$ concludes the proof of part (b). The
proof of part (c) follows from part (b), and Remark 5.6, since, arguing as in
the proof of part (b), we see that the symbol $\sigma_{1}$ (resp.
$\sigma_{2}$) of $T^{-\frac{\mu}{2}}$ (resp. $T^{-\mu}$) is in the class
$((q+V(x)+A(\xi))^{-\frac{\mu}{2}},g)$ (resp. $S((q+V(x)+A(\xi))^{-\mu},g))$.
The latter means that we, there exists some constant $C$ such that

$|\sigma_{1}(x,\xi)|\leq
C(q+V(x)+A(\xi))^{-\mu}\,,\quad(\text{resp.}\,\quad\sigma_{2}\leq
C(q+V(x)+A(\xi))^{-\frac{\mu}{2}})\,.$

The proof is now complete. ∎

We now derive some consequences for the singular values of the operators we
considered. In the sequel we denote by $\lambda_{j}(K)$ the eigenvalues, in
decreasing order, of the compact operator $K$ on the space $H$ as above, and
by $s_{j}(K)$ the singular values of it.

######  Corollary 5.8.

Let $g=g^{(A,V)}$ be the metric defined by (3.1) and let $1\leq r<\infty$.

  1. (i)

If $a\in S((q+V(x)+A(\xi))^{-\nu},g)$, assume (C3) holds for some $\mu>0$ as
in (5.1) and let $\nu\geq\frac{\mu}{r}$. Then

$s_{j}(a_{t}(x,D))=o(j^{-\frac{1}{r}})\,,\quad\text{as}\quad
j\rightarrow\infty\,,$

for every $t\in\mathbb{R}$. Consequently, also

$\lambda_{j}(a_{t}(x,D))=o(j^{-\frac{1}{r}})\,,\quad\text{as}\quad
j\rightarrow\infty\,,$

for every $t\in\mathbb{R}$.

  2. (ii)

If we assume that the pair $(A,V)$ satisfy conditions (C1) and (C2), then as a
particular case of (i), we get that for $T^{-\nu}=(q+A(D)+V(x))^{-\nu}$, we
have that

$\lambda_{j}(T^{-\nu})=o(j^{-\frac{1}{r}})\,,\quad\text{as}\quad
j\rightarrow\infty\,,$ (5.4)

for all $\nu\geq\frac{\mu}{r}$, where $1\leq r<\infty$.

###### Proof.

As Corollary 5.7 implies we have $a_{t}(x,D)\in S_{r}({\mathbb{R}}^{n})$ for
$\nu$ in the range as in the statement. Hence, as it is well known, one can
get for the membership of the above class that

$s_{j}(a_{t}(x,D))=o(j^{-\frac{1}{r}})\,,\quad\text{as}\quad
j\rightarrow\infty\,.$

Moreover from the Weyl inequality one has

$\lambda_{j}(a_{t}(x,D))=o(j^{-\frac{1}{r}})\,,\quad\text{as}\quad
j\rightarrow\infty\,.$

This proves (i), while similar arguments can be used to prove (ii) having into
account that $T^{-\nu}$ is positive definite. ∎

We now derive an immediate consequence on the rate of growth of the
eigenvalues. First we note that by Corollary 5.8 (ii), for $\nu=1$ and
$r\geq\mu$ we get

$\lambda_{j}((q+A(D)+V(x))^{-1})=\operatorname{o}(j^{-\frac{1}{r}}),\,\mbox{
as }j\rightarrow\infty.$ (5.5)

From this we obtain the following estimate for the rate of growth of the
eigenvalues of $A(D)+V(x)$:

For every $L\in\mathbb{N}$ there exists $L_{0}\in\mathbb{N}$ such that

$Lj^{\frac{1}{r}}\leq\lambda_{j}(A(D)+V(x)),\,\,\mbox{ for }j\geq L_{0}.$
(5.6)

Summarising we have obtained the following: Let $r\geq\mu$. Then, for every
$L\in\mathbb{N}$ there exists $L_{0}\in\mathbb{N}$ such that

$Lj^{\frac{1}{r}}\leq\lambda_{j}(A(D)+V(x)),\,\,\mbox{ for }j\geq L_{0}.$
(5.7)

Thus, the eigenvalues $\lambda_{j}(A(D)+V(x))$ have a growth of order at least

$j^{\frac{1}{r}},\mbox{ as }j\rightarrow\infty.$ (5.8)

We now consider a special class of pairs $(A,V)$ satisfying the following
condition :

Let $A,V$ satisfy Definition 3.1 respectively for $\gamma,\kappa$.

(C*) There exist constants $C_{1},C_{2}>0$ such that:

$A(\xi)\geq C_{1}|\xi|^{2\gamma}\,\,\,{\rm for}\,\,\,|\xi|\geq
C_{2}\,;\,\,V(x)\geq C_{1}|x|^{2\kappa}\,\,\ {\rm for}\,\,|x|\geq C_{2}\,.$

We observe that condition (C*) implies (C2). Moreover, will see that for this
type of pairs we can obtain a sharp $\mu$ satisfying (C3). We also note that
all pairs $(A,V)$ taken from the list we provided in the Example 1 satisfy
(C*).

######  Theorem 5.9.

Let $(A,V)$ be a pair satisfying condition (C*). Then we have

$\mathcal{I}_{\mu}:=\int\limits_{{\mathbb{R}}^{n}}\int\limits_{{\mathbb{R}}^{n}}(q+A(\xi)+V(x))^{-\mu}dxd\xi<\infty,$
(5.9)

provided $\mu>\frac{n(\kappa+\gamma)}{2\kappa\gamma}$.

Consequently, if additionally (C1) holds, and $T^{-\nu}=(q+A(D)+V(x))^{-\nu},$
we have

$T^{-\nu}\in S_{r}(L^{2}({\mathbb{R}}^{n})),$

provided $\nu>\frac{n(\kappa+\gamma)}{2r\kappa\gamma}$ for $1\leq r<\infty$.

###### Proof.

Since the functions $A,V$ are continuous, they are measurable, and by (C*)
there exists a constant $C>0$ such that:

$\mathcal{I}_{\mu}=\int\limits_{{\mathbb{R}}^{n}}\int\limits_{{\mathbb{R}}^{n}}(q+A(\xi)+V(x))^{-\mu}dxd\xi\geq
C\int\limits_{{\mathbb{R}}^{n}}\int\limits_{{\mathbb{R}}^{n}}(1+|x|^{2\kappa}+|\xi|^{2\gamma})^{-\mu}dxd\xi$
(5.10)

Now the integral

$\int\limits_{{\mathbb{R}}^{n}}\int\limits_{{\mathbb{R}}^{n}}(1+|x|^{2\kappa}+|\xi|^{2\gamma})^{-\mu}dxd\xi$

can be estimated on the subset
$B=\\{(x,\xi)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}:\xi_{i}>0\,,i=1,\dots,n\,\\}$.
Without loss of generality, we can assume that $\kappa\geq\gamma,$ . The
change of variable in $B$ given by
$(x,\xi)\rightarrow(x_{1},\dots,x_{n},\xi_{1}^{\frac{\kappa}{\gamma}},\dots,\xi_{n}^{\frac{\kappa}{\gamma}})$
lead us to

$\displaystyle\int\limits_{{\mathbb{R}}^{n}}\int\limits_{{\mathbb{R}}^{n}}(1+|x|^{2\kappa}+|\xi|^{2\gamma})^{-\mu}dxd\xi=$
$\displaystyle
C\int\limits_{B}(1+|x|^{2\kappa}+|\xi|^{2\kappa})^{-\mu}\xi_{1}^{\frac{\kappa}{\gamma}-1}\cdots\xi_{n}^{\frac{\kappa}{\gamma}-1}dxd\xi$
$\displaystyle\leq$
$\displaystyle\,C\int\limits_{B}(1+|x|^{2}+|\xi|^{2})^{-\mu\kappa}|\xi|^{(\frac{\kappa}{\gamma}-1)n}dxd\xi$
$\displaystyle\leq$
$\displaystyle\,C\int\limits_{B}(1+|x|^{2}+|\xi|^{2})^{-\mu\kappa}|(x,\xi)|^{(\frac{\kappa}{\gamma}-1)n}dxd\xi$
$\displaystyle\leq$ $\displaystyle\,C\int\limits_{B}\langle
X\rangle^{(\frac{\kappa}{\gamma}-1)n-2\mu\kappa}dxd\xi$ $\displaystyle<$
$\displaystyle\infty\,,$

provided $(\frac{\kappa}{\gamma}-1)n-2\mu\kappa<-2n.$ Now, the condition
$(\frac{\kappa}{\gamma}-1)n-2\mu\kappa<-2n$ for the convergence of the
integral is reduced to $\mu>\frac{n(\kappa+\gamma)}{2\kappa\gamma}$.

For the second part, since (C*) implies (C2), and (C3) holds with
$\mu>\frac{n(\kappa+\gamma)}{2\kappa\gamma}$ from the above estimate, an
application of Corollary 5.7 (b) concludes the proof of the theorem, since
$\nu>\frac{n(\kappa+\gamma)}{2r\kappa\gamma}$ implies
$r\nu>\frac{n(\kappa+\gamma)}{2\kappa\gamma}$. ∎

######  Remark 5.10.

It is clear that the condition $\mu>\frac{n(\kappa+\gamma)}{2r\kappa\gamma}$
is sharp, as it can be tested on the case of the symbol of the harmonic
oscillator and by using the classical integrability criteria for negative
powers of $\langle X\rangle$ of dimension $2n$.

######  Example 5.

Theorem 5.9 applied to the particular case of the fractional relativistic
Schrödinger operator $(I-\Delta)^{\gamma}+\langle x\rangle^{2k}$, where
$\gamma>0$, yields $((I-\Delta)^{\gamma}+\langle x\rangle^{2k})^{-\nu}\in
S_{r}(L^{2}(\mathbb{R}^{n}))$, for
$\nu>\frac{n(\kappa+\gamma)}{2r\kappa\gamma}$.

######  Remark 5.11.

Still regarding the example of the fractional relativistic Schrödinger
operator as above, we note that the condition in Example 5, can also be
obtained if one uses a different line of arguments: if $N(\lambda)$ is the
eigenvalue counting function of $(I-\Delta)^{\gamma}+\langle x\rangle^{2k}$,
then, by Theorem 3.2 of [6], we have
$N(\lambda)\lesssim\int\int_{\langle\xi\rangle^{2\gamma}+\langle
x\rangle^{2\kappa}<\lambda}dx\,d\xi$ for large values of $\lambda$. Indeed, by
the change of variable $\xi=\lambda^{\frac{1}{2\gamma}}\xi^{\prime}$ and
$x=\lambda^{\frac{1}{2\kappa}}x^{\prime}$ we can estimate

$\displaystyle N(\lambda)$ $\displaystyle\lesssim$
$\displaystyle\int\int_{\langle\xi\rangle^{2\gamma}+\langle
x\rangle^{2\kappa}<\lambda}dx\,d\xi\leq\int\int_{|\xi|^{2\gamma}+|x|^{2\kappa}<\lambda}dx\,d\xi$
$\displaystyle=$
$\displaystyle\lambda^{n\left(\frac{1}{2\gamma}+\frac{1}{2\kappa}\right)}\int\int\int_{|\xi^{\prime}|^{2\gamma}+|x^{\prime}|^{2\kappa}<1}dx^{\prime}\,d\xi^{\prime}\lesssim\lambda^{n\left(\frac{1}{2\gamma}+\frac{1}{2\kappa}\right)}\,.$

From the last estimate one can deduce that $((I-\Delta)^{\gamma}+\langle
x\rangle^{2\kappa})^{-\nu}\in S_{r}(L^{2}(\mathbb{R}^{n}))$ for
$\nu>\frac{n(\kappa+\gamma)}{2\kappa\gamma r}$, and our claim follows.

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