2021

[1]Indrit Nallbani

[1]Institute of Informatics, Istanbul Technical University, Katar Caddesi,
Istanbul, 34467, Istanbul, Turkey

2]Department of Computer Science, University of Maryland Baltimore County,
1000 Hilltop Cir, Baltimore, 21250, Maryland, USA

3]Department of Biomedical sciences, Istanbul Medeniyet University, Dumlupınar
D100 Karayolu, Istanbul, 34720, Istanbul, Turkey

# ResVGAE: Going Deeper with Residual Modules for Link Prediction

<EMAIL_ADDRESS>Reyhan Kevser Keser<EMAIL_ADDRESS>Aydin Ayanzadeh
<EMAIL_ADDRESS>Nurullah Çalık<EMAIL_ADDRESS>Behçet Uğur
Töreyin<EMAIL_ADDRESS>*  [  [

###### Abstract

Graph autoencoders are efficient at embedding graph-based data sets. Most
graph autoencoder architectures have shallow depths which limits their ability
to capture meaningful relations between nodes separated by multi-hops. In this
paper, we propose Residual Variational Graph Autoencoder, ResVGAE, a deep
variational graph autoencoder model with multiple residual modules. We show
that our multiple residual modules, a convolutional layer with residual
connection, improve the average precision of the graph autoencoders.
Experimental results suggest that our proposed model with residual modules
outperforms the models without residual modules and achieves similar results
when compared with other state-of-the-art methods.

###### keywords:

Residual Variational Graph Autoencoders, Graph Embedding, Residual
Connections, Graph Convolutional Layers, Link Prediction

##  1 Introduction

Learning-based feature extraction approaches have led to better performance in
machine learning tasks, such as computer vision, machine translation, and
object detection. Most real-world data sets have proven to be very successful
in producing data representations that are successfully used in several tasks,
such as fraud detection Wang et al. [2019], recommendation systems Feng et al.
[2019], churn prediction Liu et al. [2018] and predicting earthquakes using
graph processes Ruiz et al. [2019]. Graph neural networks (GNN) can
efficiently exploit the relationship between data set instances in non-
Euclidean space. Different variants of graph autoencoders, Shi et al. [2020],
Yu et al. [2021], Weng et al. [2020], Davidson et al. [2018], Perozzi et al.
[2014a], have been very successful in capturing meaningful representations for
node classification Grover and Leskovec [2016], link prediction Vilnis and
McCallum [2015] and graph classification Tu et al. [2018] tasks.

Figure 1: End-to-end framework of ResVGAE. Given two distant nodes $X$ and
$Y$, the model predicts a link (blue dashed line in the output) between them.
This is made possible efficiently, thanks to the residual modules (residual
connections with their consecutive Graph Convolutional Layers (GCLs). Here, we
show a model with three modules, namely, $GCL_{1}$, $GCL_{2}$, and $GCL_{3}$.

In recent years, we see a proliferation of graph embedding techniques for
improving graph convolutional networks and their applications in graph-
structured data sets Scardapane et al. [2018]. Perozzi et al. propose
DeepWalk, a framework that embeds graph nodes based on the information
acquired from truncated random walks. The framework is robust and can learn
meaningful node representations for large-scale data sets Perozzi et al.
[2014b], whereas Grover et al. presents a framework that maps node features in
a low-dimensional space while preserving the networks neighborhoods of nodes
Grover and Leskovec [2016]. Tang et al. propose another classical approach
that is capable of network embedding while preserving first and second-order
proximities of the nodes Tang et al. [2015]. Another method that embeds graph
data sets is Deep Variational Network Embedding in Wasserstein Space (DVNE), a
network embedding framework that uses a 2-Wasserstein distance as a similarity
measure between the latent node distributions. The proposed framework
preserves the first-order and second-order node proximities in the network Zhu
et al. [2018]. GraphSAGE is a framework that generates inductive node
embeddings for previously unseen data by leveraging node features. In this
work, the authors train a set of aggregator functions that learn from the
neighbor’s node features Hamilton et al. [2017], whereas Pan et al. propose an
adversarially regularized variational graph autoencoder (ARVGA), a framework
where the latent representations are enforced to match prior distributions
using an adversarial training scheme Pan et al. [2018]. Several theoretical
works have pointed out the occurrence of the over-smoothing phenomenon, the
process in which node representations become indistinguishable when nodes
interact with other nodes they are not directly connected, Chen et al. [2020],
Alon and Yahav [2021], Mavromatis and Karypis [2021], Zhou et al. [2020]. This
limits the ability of the models to stack many GNN layers. We want deep graph
networks to capture long-distance interactions between far nodes for link
prediction tasks. As shown in Fig.1, to capture meaningful information between
distant nodes $X$ and node $Y$, our model embeds the nodes into a similar low-
dimensional vector space. If the node embeddings are similar, then our model
predicts a link between them. In this example, nodes $X$ and $Y$ are three
hops apart so ResVGAE has three residual modules.

We propose a model architecture with residual modules that combines residual
connections He et al. [2016] and variational graph autoencoders (cf. Fig.2) in
order to alleviate the over-smoothing phenomenon.

The overall contributions of our study are as follows:

  * •

We propose a graph variational autoencoder model with residual modules and
compare it with the other state-of-the-art models for link prediction tasks.

  * •

We measure the accuracy of adding residual modules on similar graph-based
autoencoders with different depths.

  * •

We study the over-smoothing phenomena by implementing our proposed
architecture from one up to eight residual modules.

The rest of the paper is organized as follows: Section 2 covers the proposed
model architectures. In Section 3, comparative results of the proposed ResVGAE
model are presented for the link prediction task on three benchmark data sets.
We conclude the paper and propose new research paths for future work in
Section 4.

Figure 2: Model architecture of ResVGAE. Residual connections start after the
first Hidden Layer ($HL$) since the input and the output size of layers with
residual modules must be the same. The encoder takes the adjacency matrix $A$
and the feature matrix $X$ as inputs and outputs the node embeddings $Z$. The
decoder takes as input the embedding matrix $Z$ and outputs the reconstructed
adjacency matrix $\hat{A}$. The blocks in blue indicate the graph
convolutional layers that embed 32-dimensional node feature vectors into a
matrix. Similarly, yellow blocks constitute the graph convolutional layers
that embed 16-dimensional hidden layer features into the output Z matrix. The
upper and lower branches of the encoder represent variational graph
autoencoder and graph autoencoder architectures, respectively.

##  2 Method

Graph autoencoders are deep learning frameworks whose inputs are instance
features and adjacency matrices and whose output is the reconstructed
adjacency matrix. These frameworks mainly consist of two components, namely,
the encoder and the decoder. The encoder part transforms the input into a
lower-dimensional embedding while the decoder part transforms the embedding
into the reconstructed adjacency matrix. Let $G=(V,E)$ denote a graph $G$,
where $V$ is the set with $N$ nodes, and $E$ is the set of edges. Moreover,
let $v_{i}\in V$ denote a node and $e_{ij}=(v_{i},v_{j})$ denote an edge
between two nodes $v_{i}$ and $v_{j}$. We define the adjacency matrix $A$ as:

$A:=\left\\{\begin{array}[]{ll}A_{ij}=1,&\text{e}_{ij}\in E\\\
A_{ij}=0,&\text{e}_{ij}\notin E\\\ \end{array}\right.$ (1)

where $A$ is a symmetric matrix. Let $X$ be the feature matrix of the nodes.
For all the encoder architectures, we use the graph convolutional layer
proposed by Kipf and Welling [2017]. The layer follows the propagation rule:

$H^{(l+1)}=\varphi\left(\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}H^{(l)}W^{(l)}\right)$
(2)

where $\varphi$ is the sigmoid activation function, $\tilde{D}$ is the degree
matrix, $\tilde{A}$ is the normalized adjacency matrix with self-loops and
$W^{(l)}$ is the layer’s weight matrix. In a multi-layer model $H^{(0)}=X$ and
$H^{(l)}$ is the feature map of the $l^{th}$ layer.

Variational graph encoders transform the graph into a lower-dimensional
embedding, using graph convolutional layers and a sampling layer. These graph
convolutional layers transform the graph into the desired lower-dimensional
space and produces mean and standard deviation values using two layers. Then,
the sampling layer takes these mean and deviation values to generate samples
from the prior distribution. Hence the generated samples constitute the
embedding of the graph.

The final embeddings are encoded as a distribution over the latent space as:

$q(Z\mid X,A)=\prod_{i=1}^{N}q\left(z_{i}\mid X,A\right)$ (3)

where

$q\left(z_{i}\mid
X,A\right)=\mathcal{N}\left(\mu_{i},\operatorname{diag}\left(\sigma^{2}\right)\right)$
(4)

Here, $z_{i}$ is the embedding vectors for node $i$; $\mu_{i}$ and $\sigma$
are the graph convolutional layer vector outputs of the encoder.

Using this structure provides a graph embedding with the desired distribution
(cf. Fig.2).

For the graph autoencoder models, the decoder is a non-probabilistic model
that reconstructs the adjacency matrix by computing the inner-product of the
latent representations of two-node embeddings :

$\hat{A}=\varphi\left(ZZ^{T}\right)$ (5)

In all the models, $\varphi(\cdot)$ denotes the sigmoid activation function.

For the graph variational autoencoder models, the decoder is a probabilistic
model that reconstructs the adjacency matrix by computing the probabilistic
inner product of the latent representation of nodes:

$p(A\mid Z)=\prod_{i=1}^{N}\prod_{j=1}^{N}p\left(A_{ij}\mid
z_{i},z_{j}\right)$ (6)

where

$p\left(A_{ij}\mid z_{i},z_{j}\right)=\varphi\left(z_{i}^{T}z_{j}\right)$ (7)

Vanilla models are trained by minimizing reconstruction loss:

$\mathcal{L}=\mathbb{E}_{q(Z\mid X,A)}[\log p(A\mid Z)]$ (8)

and variational models are optimized by maximizing the variational lower bound
while minimizing reconstruction loss:

$\mathcal{L}=\mathbb{E}_{q(Z\mid X,A)}[\log p(A\mid Z)]-\mathrm{KL}[q(Z\mid
X,A)\|p(Z)]$ (9)

where $\mathrm{KL}[q(\cdot)\|p(\cdot)]$ is the Kullback-Leibler divergence
between $q(\cdot)$ and $p(\cdot)$ . The posterior distribution is approximated
by a Gaussian distribution using the reparametrization trick Kingma and
Welling [2014]. Our proposed method, ResVGAE is a variational graph
autoencoder with multiple residual modules. The decoder and loss are the same
as variational graph autoencoders. To improve the performance of the
variational graph autoencoder, we propose utilizing residual modules in the
encoder similar to (2). A residual module can be represented as:

$H^{(l+1)}=\varphi\left(\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}H^{(l)}W^{(l)}\right)+H^{(l)}$
(10)

The input and the output sizes of the hidden layers must be the same to add
residual modules. The number of residual modules can be determined depending
on the depth of exploration among nodes for link prediction.

##  3 Experimental Results

We evaluate the model’s performance on three benchmark data sets: Cora
McCallum et al. [2000], CiteSeer Giles et al. [1998] and PubMed Namata et al.
[2012]. Cora and CiteSeer are networks of computer science publications where
the nodes represent the publications and the edges represent the citations.
The PubMed data set is also a citation network that contains a set of articles
related to diabetes disease.

We compare ResVGAE against three baseline algorithms: Deep Walk (DW)Perozzi et
al. [2014a], Spectral Clustering (SC) Tang and Liu [2011] and Adversarially
Regularized Graph Autoencoder (ARVGE) Pan et al. [2018]

All models and data sets in this paper have been used for link prediction
tasks and Table 1 gives a detailed summary of the data sets we used.

Table 1: Data set statistics

Data set | # Nodes | # Edges | # Classes | # Features
---|---|---|---|---
CitesSeer | 3,327 | 4,732 | 6 | 3,703
Cora | 2,708 | 5,429 | 7 | 1,433
PubMed | 19,717 | 44,338 | 3 | 500

We evaluate all the models based on the same baseline as in Kipf and Welling
[2016] and we use AP and AUC scores to report the average precision of 10 runs
with random train, validation, and test splits of the same size, and all the
models are trained for 200 epochs. The validation and test sets contain 5% and
10% of the total edges, respectively. The embedding dimensions of the node
features for the hidden layers is 32 and the embedding dimension of node
features for the output layer is 16. The models are optimized using the Adam
optimizer with a learning rate of 0.01.

In our experiments, we use average precision (AP) and area under ROC curve
(AUC) metrics to measure the accuracy of our models. We use PyTorch Geometric
Fey and Lenssen [2019] and our code will be available upon acceptance.

Figure 3: The AP scores of the models with multiple numbers of layers for the
Cora data set. ResGAE has the highest AP score followed by ResVGAE and the
graph autoencoders without the residual modules. Residual modules
substantially improve the performance of the variational autoencoder models in
comparison to models without them. The performance gap increases for a higher
number of layers.

Experiments indicate that, for shallow models, all proposed models achieve
similar average precision scores. Here we see that all models with one
residual module embed the node features in a very similar way. The score
differs when we use models with deeper networks.

In models with eight graph convolutional layers, we see that models with
residual modules have higher average precision scores than models without
residual modules. In Fig. 3, we run all the models from one up to eight graph
convolutional layers. Here we see that as the models become deeper their
average precision decreases significantly. The models with residual modules
achieve higher scores than the models without residual modules.

Plots in Figs. 4 and 5 indicate the steady drop in average precision values
for all the models on CiteSeer and PubMed data sets, respectively. Moreover,
for deeper models, the results suggest that deep models with residual modules
outperform the models without residual modules.

Figure 4: The AP scores of the models with a different number of GCL layers
for the CiteSeer data set. Models with residual modules score higher than the
models without residual modules. The highest scoring model is ResVGAE followed
by ResGAE for the number of layers seven and eight. Figure 5: The AP scores of
the models with a different number of GCL layers for the PubMed data set.
Residual modules with more layers result in a higher AP score.  Table 2:
Performance comparison of models with and without residual modules. Our
proposed architectures, ResVGAE and ResGAE, achieve higher AUC and AP scores
on Cora, CiteSeer, and PubMed data sets when compared to their variational
graph autoencoder counterparts without residual modules.

Method | Cora | CiteSeer | PubMed
---|---|---|---
| AUC | AP | AUC | AP | AUC | AP
ResVGAE (Ours) | $\textbf{85.93}\pm\textbf{1.02}$ | $\textbf{88.55}\pm\textbf{0.65}$ | $\textbf{86.90}\pm\textbf{1.85}$ | $\textbf{80.30}\pm\textbf{0.40}$ | $\textbf{91.28}\pm\textbf{0.80}$ | $\textbf{93.50 }\pm\textbf{0.68}$
VGAE | $81.97\pm 0.79$ | $85.51\pm{2.54}$ | $83.70\pm 0.77$ | $79.00\pm 0.60$ | $85.30\pm 0.95$ | $82.60\pm 0.60$
ResGAE (Ours) | $\textbf{86.13}\pm\textbf{0.844}$ | $\textbf{88.85}\pm\textbf{0.31}$ | $\textbf{87.10}\pm 1\textbf{.99}$ | $\textbf{84.40}\pm\textbf{1.95}$ | $\textbf{93.00}\pm\textbf{0.46}$ | $\textbf{91.40}\pm\textbf{0.29}$
GAE | $82.65\pm 0.79$ | $86.46\pm 0.32$ | $82.60\pm 1.28$ | $81.60\pm 1.17$ | $90.60\pm 1.70$ | $87.80\pm 0.90$

Improvement in terms of average precision values is evident for models with
residual models tested on Cora, CiteSeer, and PubMed data sets (Table 2). More
specifically, ResVGAE outputs higher AUC and AP scores than VGAE, and the
difference in score is significant when compared with GAE for CiteSeer and
PubMed data sets. For the Cora data set, we see that ResGAE performs better
than ResVGAE and the other graph autoencoders without residual modules.

Furthermore, in Table 3, we compare our model with other baseline models on
link prediction tasks. Our model outperforms Spectral Clustering and DeepWalk
algorithms in all three datasets and scores a very similar result to the ARVGE
algorithm on the Cora and PubMed datasets.

Tests on publicly available data sets reveal the importance of architectures
with residual modules for exploring the interactions among far nodes for link
prediction tasks. In models without residual modules, variational graph
autoencoder scores higher in the CiteSeer data sets whereas graph autoencoder
outperforms graph variational autoencoder in the Cora and PubMed data sets.

Table 3: Performance comparison of models with other baseline models.

Models | Cora | Citeseer | PubMed |
---|---|---|---|---
| AUC | AP | AUC | AP | AUC | AP
DW | $83.1\pm 0.01$ | $85.0\pm 0.00$ | $80.5\pm 0.02$ | $83.6\pm 0.01$ | $84.4\pm 0.00$ | $84.1\pm 0.00$
SC | $84.6\pm 0.01$ | $88.5\pm 0.00$ | $80.5\pm 0.01$ | $85.0\pm 0.01$ | $84.2\pm 0.02$ | $87.8\pm 0.01$
ARVGE | ${92.4}\pm{0.004}$ | $92.6\pm 0.004$ | ${92.4}\pm{0.003}$ | ${93.0}\pm{0.003}$ | $96.5\pm 0.001$ | $96.8\pm 0.001$
ResVGAE | ${87.93}\pm{1.90}$ | $90.23\pm 1.35$ | ${88.7}\pm{1.53}$ | ${84.47}\pm{0.53}$ | $94.5\pm 0.64$ | $94.8\pm 0.20$

##  4 Conclusion

In this paper, we introduce ResVGAE, a variational graph autoencoder with
residual modules to mitigate the over-smoothing phenomenon occurring in graph
neural networks with deep architectures. Results indicate that the proposed
model scores similar average precisions for shallow models and outperforms
variational graph autoencoders without residual modules for link prediction
task among nodes with multi-hop separation.

As a future research direction, we plan to test our models’ performance on
larger benchmark data sets and focus on other tasks such as graph
classification, node classification, and graph clustering. Furthermore, we
intend to experiment with the effect of the inclusion of residual modules with
different distributions for the variational graph autoencoders.

Acknowledgments

This work is supported by The Scientific and Technological Research Council of
Turkey (TÜBİTAK) under Project No. 121E378 and by Istanbul Technical
University (ITU) Vodafone Future Lab under Project No. ITUVF20180901P04.

## Declarations

  * •

Funding: Noted in the acknowledgments section

  * •

Conflict of interest/Competing interests Not applicable

  * •

Ethics approval: Not applicable

  * •

Consent to participate: Not applicable

  * •

Consent for publication: All authors’ consent

  * •

Availability of data and materials: Upon request

  * •

Code availability: Available upon acceptance and can be shared with editors at
any time

  * •

Authors’ contributions: All authors consent

## References

  * Wang et al. [2019] Jianyu Wang, Rui Wen, Chunming Wu, Yu Huang, and Jian Xion.  Fdgars: Fraudster detection via graph convolutional networks in online app review system.  In _Companion Proceedings of The 2019 World Wide Web Conference_ , pages 310–316, 2019.
  * Feng et al. [2019] Chenyuan Feng, Zuozhu Liu, Shaowei Lin, and Tony QS Quek.  Attention-based graph convolutional network for recommendation system.  In _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 7560–7564. IEEE, 2019.
  * Liu et al. [2018] Xi Liu, Muhe Xie, Xidao Wen, Rui Chen, Yong Ge, Nick Duffield, and Na Wang.  A semi-supervised and inductive embedding model for churn prediction of large-scale mobile games.  In _2018 IEEE International Conference on Data Mining (ICDM)_ , pages 277–286. IEEE, 2018.
  * Ruiz et al. [2019] Luana Ruiz, Fernando Gama, and Alejandro Ribeiro.  Gated graph convolutional recurrent neural networks.  In _2019 27th European Signal Processing Conference (EUSIPCO)_ , pages 1–5. IEEE, 2019.
  * Shi et al. [2020] Han Shi, Haozheng Fan, and James T Kwok.  Effective decoding in graph auto-encoder using triadic closure.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, pages 906–913, 2020.
  * Yu et al. [2021] Bin Yu, Yu Zhang, Yu Xie, Chen Zhang, and Ke Pan.  Influence-aware graph neural networks.  _Applied Soft Computing_ , 104:107169, 2021.
  * Weng et al. [2020] Ziqiang Weng, Weiyu Zhang, and Wei Dou.  Adversarial attention-based variational graph autoencoder.  _IEEE Access_ , 8:152637–152645, 2020.
  * Davidson et al. [2018] Tim R. Davidson, Luca Falorsi, Nicola De Cao, Thomas Kipf, and Jakub M. Tomczak.  Hyperspherical variational auto-encoders.  _34th Conference on Uncertainty in Artificial Intelligence (UAI-18)_ , 2018.
  * Perozzi et al. [2014a] Bryan Perozzi, Rami Al-Rfou, and Steven Skiena.  Deepwalk: Online learning of social representations.  In _Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining_ , pages 701–710. ACM, 2014a.
  * Grover and Leskovec [2016] Aditya Grover and Jure Leskovec.  node2vec: Scalable feature learning for networks.  In _Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining_ , pages 855–864, 2016.
  * Vilnis and McCallum [2015] Luke Vilnis and Andrew McCallum.  Word representations via gaussian embedding.  _CoRR_ , abs/1412.6623, 2015.
  * Tu et al. [2018] Ke Tu, Peng Cui, Xiao Wang, Fei Wang, and Wenwu Zhu.  Structural deep embedding for hyper-networks.  In _AAAI_ , pages 426–433, 2018.  URL https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16797.
  * Scardapane et al. [2018] Simone Scardapane, Steven Van Vaerenbergh, Danilo Comminiello, and Aurelio Uncini.  Improving graph convolutional networks with non-parametric activation functions.  In _2018 26th European Signal Processing Conference (EUSIPCO)_ , pages 872–876, 2018.  10.23919/EUSIPCO.2018.8553465.
  * Perozzi et al. [2014b] Bryan Perozzi, Rami Al-Rfou, and Steven Skiena.  Deepwalk: Online learning of social representations.  In _Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining_ , pages 701–710, 2014b.
  * Tang et al. [2015] Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei.  Line: Large-scale information network embedding.  In _Proceedings of the 24th International Conference on World Wide Web_ , WWW ’15, page 1067–1077, Republic and Canton of Geneva, CHE, 2015\. International World Wide Web Conferences Steering Committee.  ISBN 9781450334693.  10.1145/2736277.2741093.  URL https://doi.org/10.1145/2736277.2741093.
  * Zhu et al. [2018] Dingyuan Zhu, Peng Cui, Daixin Wang, and Wenwu Zhu.  Deep variational network embedding in wasserstein space.  In _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, pages 2827–2836. ACM, 2018.
  * Hamilton et al. [2017] Will Hamilton, Zhitao Ying, and Jure Leskovec.  Inductive representation learning on large graphs.  _Advances in neural information processing systems_ , 30, 2017.
  * Pan et al. [2018] Shirui Pan, Ruiqi Hu, Guodong Long, Jing Jiang, Lina Yao, and Chengqi Zhang.  Adversarially regularized graph autoencoder for graph embedding.  _arXiv preprint arXiv:1802.04407_ , 2018.
  * Chen et al. [2020] Deli Chen, Yankai Lin, Wei Li, Peng Li, Jie Zhou, and Xu Sun.  Measuring and relieving the over-smoothing problem for graph neural networks from the topological view.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, pages 3438–3445, 2020.
  * Alon and Yahav [2021] Uri Alon and Eran Yahav.  On the bottleneck of graph neural networks and its practical implications.  In _International Conference on Learning Representations_ , 2021.  URL https://openreview.net/forum?id=i80OPhOCVH2.
  * Mavromatis and Karypis [2021] Costas Mavromatis and George Karypis.  Graph infoclust: Maximizing coarse-grain mutual information in graphs.  In _Pacific-Asia Conference on Knowledge Discovery and Data Mining_ , pages 541–553. Springer, 2021.
  * Zhou et al. [2020] Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun.  Graph neural networks: A review of methods and applications.  _AI Open_ , 1:57–81, 2020.
  * He et al. [2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 770–778, 2016.
  * Kipf and Welling [2017] Thomas N. Kipf and Max Welling.  Semi-supervised classification with graph convolutional networks.  In _5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings_ , 2017.  URL https://openreview.net/forum?id=SJU4ayYgl.
  * Kingma and Welling [2014] Diederik P. Kingma and Max Welling.  Auto-Encoding Variational Bayes.  In _2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings_ , 2014.
  * McCallum et al. [2000] Andrew Kachites McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymore.  Automating the construction of internet portals with machine learning.  _Information Retrieval_ , 3(2):127–163, 2000\.
  * Giles et al. [1998] C Lee Giles, Kurt D Bollacker, and Steve Lawrence.  Citeseer: An automatic citation indexing system.  In _ACM DL_ , pages 89–98, 1998.
  * Namata et al. [2012] Galileo Namata, Ben London, Lise Getoor, Bert Huang, and UMD EDU.  Query-driven active surveying for collective classification.  In _10th International Workshop on Mining and Learning with Graphs_ , volume 8, 2012.
  * Tang and Liu [2011] Lei Tang and Huan Liu.  Leveraging social media networks for classification.  _Data Mining and Knowledge Discovery_ , 23(3):447–478, 2011.
  * Kipf and Welling [2016] Thomas Kipf and Max Welling.  Variational graph auto-encoders.  _ArXiv_ , abs/1611.07308, 2016.
  * Fey and Lenssen [2019] Matthias Fey and Jan E. Lenssen.  Fast graph representation learning with PyTorch Geometric.  In _ICLR Workshop on Representation Learning on Graphs and Manifolds_ , 2019.


following table, which has the same structure as that of Subsection <ref>.

Solution     Parameters     Conditions on $Q$     Green's operator

$m\in\C$, $\Re(k)\ge0$, $k\neq0$     $Q\in\mathscr{L}^{(\infty)}_0$     Backward  $G^0_\leftarrow$

2*$q_{-m}(\cdot)$     2*$\Re(m)\ge\frac{1-\delta}{2}$, $k=0$     $Q\in\mathscr{L}^{(\infty)}_{\delta}$, $\delta\ge1$, if $m\neq0$     2*Backward  $G^0_\leftarrow$

$Q\in\mathscr{L}^{(\infty)}_{1,\mathrm{ln}}$, if $m=0$

$q_{0,\mathrm{ln}}(\cdot)$     $m=0$, $k=0$     $Q\in\mathscr{L}^{(\infty)}_{1,\mathrm{ln}^2}$     Backward  $G^0_\leftarrow$

tableDistinguished solutions of the perturbed eigenequation with a prescribed behavior near $\infty$.

§ WRONSKIANS

In this section we study the wronskians of distinguished solutions constructed in the previous two sections.

§.§ The Jost function

Suppose that $Q \in \mathscr{L}^{(\infty)}_0 $ and $Q \in \mathscr{L}^{(0)}_\varepsilon$ with $\varepsilon\ge\max\big(0,-2\Re(m)\big)$ if $m \neq 0$, $Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}}$ if $m=0$. Let $\Re(k)\geq0$, $k\neq0$.
Then, by Propositions <ref> and <ref>,
both $ u_{m}( \cdot , k )$ and $ v_{m}( \cdot , k )$ are well defined.
Their  Wronskian

\begin{equation}
\Wr_{m}(k) :=
\Wr \big ( v_{m}( \cdot , k ) , u_{m}( \cdot , k ) \big )
\end{equation}

will be called the Jost function. Assuming in addition that $Q \in \mathscr{L}^{(\infty)}_\delta$ for some $\delta\ge1$, we also set for $k=0$ and $\Re(m)\ge\frac12(1-\delta)$, $m\neq 0$,

\begin{equation}
\Wr_{m}(0) :=
\frac12 \Gamma(m)  \Wr \big ( q_{-m}( \cdot ) , u_{m}( \cdot , 0 ) \big )
\end{equation}

Using the regularity properties of $u_m(x,k)$ and $v_m(x,k)$, we obtain the regularity of the map $(m,k)\mapsto \Wr_m(k)$ on suitable domains.

$ $

* If
Q \in \mathscr{L}^{(0)}_\varepsilon\cap
\mathscr{L}^{(\infty)}_0
for some $\varepsilon>0$,
then  the map $(m,k)\mapsto
\Wr_{m}(k)$ is regular on

\begin{align}
\Big\{\Re(m)\geq-\frac\varepsilon2 \Big \} \times \Big \{
\Re(k)\geq0,\, k\neq0\Big\}.
\end{align}

* If
$Q \in \mathscr{L}^{(0)}_0\cap  \mathscr{L}^{(\infty)}_0,
then the map $(m,k)\mapsto
\Wr_{m}(k)$ is regular on

\begin{align}
\Big\{\Re(m)\geq0,\ m\neq0  \Big \} \times \Big \{ \Re(k)\geq0,\, k \neq 0\Big\}.
\label{porto1ka} \end{align}

* If
Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}} \cap \mathscr{L}^{(\infty)}_{1}$,
then the map $(m,k)\mapsto
\Wr_{m}(k)$ is regular on

\begin{align}  \label{porto1ka_bis}
\Big\{\Re(m)\geq0 \Big\} \times \Big \{\Re(k)\geq0\Big\} \setminus \big\{ m=0 \big \} \times \big \{ k = 0 \big \} .
\end{align}

Analyticity and continuity are consequences of the analyticity and
continuity of the maps $u_{m}( x , k ) $, $w_{m}( x , k )$, $q_m(x)$ and
their derivatives (see Propositions <ref>, <ref> and <ref>).

Continuity at $k=0$ in (<ref>) uses in addition the limit (<ref>).

The next proposition gives a convenient representation of the Jost function.

Suppose that
Q \in \mathscr{L}^{(\infty)}_0
$. Let $m\in\mathbb{C}$. Suppose that $Q \in \mathscr{L}^{(0)}_\varepsilon$ with $\varepsilon=\max(0,-2\Re(m))$ if  $m\neq0$, or
$Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}^2}$ if $m=0$.
Then for $\Re(k)\geq0$, $k\neq0$

\begin{align}
\Wr_{m}(k) =
\langle u_{m}^0(\cdot,k) | Q v_{m}(\cdot,k) \rangle.
\label{kzl}
\end{align}

Consider first $m\neq0$.
By Proposition <ref> with $\varepsilon=\max(0,-2\Re(m))$,  for
$ x\to0$ we have

\begin{align}\label{uu1a+a}
\\\label{uu1a+b}
\partial_x  u_m(x,k)&=\partial_x u^0_m(x,k)+o(x^{-\frac12+|\Re(m)|}).
\end{align}

By Proposition <ref>(i) with $\varepsilon=0$ we have

\begin{align}
v_m(x,k) &=v^0_m(x,k) + \langle u^0_m |Qv_m \rangle v^0_m(x,k) + o( x^{\frac12-|\mathrm{Re}(m)|} ) , \label{eq:aaab-1_3_bis} \\
\partial_x v_m(x,k)&  =\partial_x v^0_m(x,k)
+ \langle u^0_m |Qv_m \rangle \partial_x v^0_m(x,k) + o( x^{-\frac12-|\mathrm{Re}(m)|} ) . \label{eq:aaab0_3_bis}
\end{align}

Now (<ref>), (<ref>)–(<ref>) and (<ref>)–(<ref>) yield

\begin{align*}
\Wr(v_m,u_m;x)
1+\langle u^0_m|Qv_m\rangle\Big)
\end{align*}

Moreover it follows from (<ref>) and  (<ref>) that

\begin{align*}
\lim_{x\to0}\Wr(v_m^0,u_m;x)&=\Wr(v^0_m,u^0_m;x)+o(x^0) = 1 + o(x^0).
\end{align*}

Then note that $\Wr(v_m,u_m;x)$
does not depend on $x$
and use the definition (<ref>) of $\mathscr{W}_m(k)$.

In the case $m=0$,
we use Proposition <ref> (iii).
Then we repeat the same arguments, using in addition

\begin{equation*}
\Wr(u^0_0,u_0;x)=\Wr(u^0_0,u^0_0;x)+o(x^0),\quad\Wr(u^0_0,u^0_0;x)=0.
\end{equation*}

This ends the proof of (<ref>).

The asymptotic behavior of the Jost function can be deduced from Proposition <ref> together with the following lemma.

Let $m\in\mathbb{C}$. Suppose that $Q \in \mathscr{L}^{(0)}_\varepsilon\cap
\mathscr{L}^{(\infty)}_0$ with $\varepsilon=\max(0,-2\Re(m))$ if  $m\neq0$, or
$Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}^2}\cap
\mathscr{L}^{(\infty)}_0$ if $m=0$.

\begin{align}
\langle u_m^0 | Qv_m \rangle = o( |k|^0 )+
\mathcal{O(}{|k|^{-1+\varepsilon}}), \qquad |k| \to \infty,\quad\Re(k)\geq0. \label{eq:l1}
\end{align}

Assume that $m \neq 0$.
We have the estimate

\begin{equation}
|u^0_m(x,k)|\lesssim \mu_k(x)^{\frac12+\Re(m)}\eta_k(x),
\end{equation}

uniformly in $k$. Next, in the estimate of Proposition <ref>, we first note that the big exponential on the right hand side
is uniformly bounded for large $k$. Therefore, uniformly for large enough $|k|$,

\begin{equation}
|v_m(x,k)|\lesssim  \mu_k(x)^{\frac12-|\Re(m)|}\eta_{-k}(x).
\end{equation}

\begin{align*}
& \big | \langle u^0_m | Qv_m \rangle \big | \\
&=  \Big | \int_0^\infty u^0_m(y) Q(y) v_m( y ) \mathrm{d} y \Big | \\
&\lesssim   \int_0^\infty \mu_k(x)^{1-\varepsilon}
| Q(x)|\mathrm{d} x\\
&\lesssim \Big ( \int_0^{\frac1{|k|}} x^{1-\varepsilon} |Q(x)|\mathrm{d} x+
|k|^{-1+\varepsilon} \int_{\frac1{|k|}}^1|Q(x)|\mathrm{d} x
+  |k|^{-1+\varepsilon}\int_1^\infty|Q(x)|\mathrm{d} x\Big).
\end{align*}

First note that

\begin{align*}
\int_0^{\frac1{|k|}} x^{1-\varepsilon} |Q(x)|\mathrm{d} x = o( 1 ) \quad \text{ and } \quad  \int_1^\infty|Q(x)|\mathrm{d} x = \mathcal{O}( 1 ).
\end{align*}

Therefore, the first term on the right  is $ \smallO{
|k|^0}$ and the third is $ \mathcal{O}(|k|^{-1+\varepsilon})$.

If $1>\varepsilon\geq0$, then applying Lemma <ref> with $h(y)=y^{1-\varepsilon}$ and
$x=\frac{1}{|k|}$, we obtain

\begin{align*}
\int_{\frac1{|k|}}^1|Q(x)|\mathrm{d} x = o(|k|^{1-\varepsilon}).
\end{align*}

So the middle term is $ o(
|k|^0 )$.

If $\varepsilon\geq1$, then

\begin{align*}
\int_{\frac1{|k|}}^1|Q(x)|\mathrm{d} x = o(|k|^{1-\varepsilon}).
\end{align*}

So the middle term is $ \mathcal{O}(
|k|^{-1+\varepsilon} )$.

Finally, if $m=0$, the proof is identical, the only difference being that

\begin{align*}
&  \big | \langle u^0_0 | Qv_0 \rangle \big |  \\
&\lesssim   \Big ( \int_0^{\frac1{|k|}} x ( 1 -  | \mathrm{ln}(|k|x) | ) |Q(x)|\mathrm{d} x+
|k|^{-1}\int_{\frac1{|k|}}^1|Q(x)|\mathrm{d} x
+|k|^{-1}\int_1^\infty|Q(x)|\mathrm{d} x\Big).
\end{align*}

We deduce from the previous lemma that, for $\Re(m)>-1$,
$\Wr_m$ cannot constantly vanish except on a discrete set. Corollary <ref> implies Proposition <ref> from the introduction.

In addition to the assumptions of Proposition <ref>,
suppose that $\Re(m)>-1$.

\begin{align}\label{asim}
\lim_{|k|\to\infty} \Wr_{m}(k) = 1,
\quad \Re(k)\geq0.
\end{align}

\begin{equation*}
\big \{ k \in \mathbb{C} , \, \mathrm{Re}(k)>0 , \, \Wr_{m}(k) = 0
\big \}
\quad\text{ is discrete.}
\end{equation*}

Under the condition $\Re(m)>-1$ the right hand side of (<ref>) becomes
$ o( |k|^0)$. Hence
(<ref>) follows by (<ref>).

The fact
that $ \{ k \in \mathbb{C} , \, \mathrm{Re}(k)>0 , \, \Wr_{m}(k) = 0
\} $ is discrete is then a consequence of the analyticity of $\Wr_m$
stated in Proposition <ref>.

Note that Corollary <ref> is the second place in our paper
where the condition $\Re(m)>-1$ appears (see also Theorem <ref>). This condition will play an
important role in Section <ref> about closed realizations.

§.§ Wronskians – refined results

If $u_{-m}$ is ill-defined, we can often use $u_{-m}^{[n]}$

Let $\Re(k)\geq0$, $k\neq0$.
Suppose that  $      Q \in \mathscr{L}^{(0)}_\varepsilon \cap
\mathscr{L}^{(\infty)}_0$,  $\varepsilon\ge0$.
Let  $n$ be a nonnegative integer,
$\frac{\varepsilon}{2}(n+1)\geq \Re(m) \geq0$, $m\not\in\mathbb{N}$.

\begin{align}\label{wrono}
\Wr\big( u_{-m}^{[n]}(\cdot,k),u_m(\cdot,k)\big)&=\frac{2\sin(m\pi)}{\pi}.
\end{align}

Hence there exists a constant $C_m^{[n]}(k)$ such that

\begin{align}\label{wroni-0}
\end{align}

First we check (<ref>), which follows from (<ref>), using also Propositions <ref> and <ref>. Then we write

\begin{align}
v_m(\cdot,k)=B_m^{[n]} (k)u_{-m}^{[n]}(\cdot,k)+C_m^{[n]} (k)u_m(\cdot,k),
\end{align}

and take the Wronskian of both sides with $u_m(\cdot,k)$. This allows us to compute
$B_m^{[n]} (k)$ and yields (<ref>).

Suppose that the assumptions of Proposition <ref> are
satisfied.  Then the map

\begin{equation*}
\Big \{ 0 \le \Re(m) \leq\frac{\varepsilon}{2}(n+1) ,\quad m\neq 0\Big \} \times \Big \{ \Re(k) \ge 0 , \, k
\neq 0 \Big \}
\ni(m,k)\mapsto \Wr ( v_m(\cdot,k) , u_{{-}m}^{[n]}( \cdot ,k ) )
\end{equation*}

is regular.

This follows from Propositions <ref> and <ref>.

§.§ Green's functions for  perturbed Bessel operators

As for every 1-dimensional Schrödinger operator, we can define the
canonical bisolution and various Green's functions for perturbed
Bessel operators. The solutions that we constructed allow us to
give explicit expressions for these Green's functions.

As usual, when defining Green's operators we will always
assume that $\Re(k)\geq0$ (although sometimes an extension to a larger
domain is possible).  Let
$Q \in \mathscr{L}^{(\infty)}_0 \cap \mathscr{L}^{(0)}_0$ for $m\neq0$,  and
Q \in \mathscr{L}^{(\infty)}_0 \cap \mathscr{L}^{(0)}_{0,\ln}$, for $m=0$.
The canonical bisolution associated with the operator $L_{m^2}+k^2$ is

\begin{equation}
G_{m^2,\leftrightarrow}(-k^2; x , y ) =
\frac{1 }{\Wr_{m}(k)}\big ( v_{m}( x , k ) u_{m}( y , k ) - u_{m}( x , k ) v_{m}( y , k ) \big ),
\label{qrel}\end{equation}

where  $v_m(\cdot,k)$, $u_m(\cdot,k)$ are the solutions to
(<ref>) constructed in Propositions <ref> and
<ref>, and $\Wr_m(k)$ is the Jost function defined in
(<ref>). The expression (<ref>) is well-defined when $\Wr_m(k)\neq0$ and has a limit at $k=0$. For $m=0$ we can use

\begin{align*}
&=\frac{1}{\Wr_0(k)}\big(- p_0(x,k)u_0( y,k ) +
u_0( x ,k) p_0( y,k )\big).
\end{align*}

From the canonical bisolution, we can construct in the usual way the forward Green's
operator $G_{m^2,\rightarrow}(-k^2)$ and the backward Green's operator
$G_{m^2,\leftarrow}(-k^2)$ of $L_{m^2}+k^2$:

\begin{align}\label{right}
G_{m^2,\rightarrow}(-k^2;x,y)& : =
\theta(x-y) G^0_{m^2,\leftrightarrow}(-k^2; x , y ),\\\label{left}
G_{m^2,\leftarrow}(-k^2;x,y) & :=
-\theta(y-x) G^0_{m^2,\leftrightarrow}( -k^2;x , y ).
\end{align}

Green's operators defined by
specifying boundary conditions at zero and at infinity will be called
They will often be bounded on  $L^2]0,\infty[$ and
coincide with the resolvents of various closed realizations of
$L_{m^2}$. However, they  are of
interest even when they do not define bounded operators and  do
not correspond to closed realizations of

The most natural two-sided Green's operator corresponds to pure boundary
conditions. In the unperturturbed case they are usually called  homogeneous boundary
conditions, but in the perturbed case this name seems no
longer appropriate. It can be defined for
$Q \in \mathscr{L}^{(\infty)}_0 \cap \mathscr{L}^{(0)}_\varepsilon$,  $m\neq0$,
$-\frac \varepsilon2\leq\Re(m)$, and
Q \in \mathscr{L}^{(\infty)}_0 \cap
\mathscr{L}^{(0)}_{0,\ln}$, $m=0$. Moreover, if $\Re(m)\leq0$ we neeed to assume
$k\neq0$. Then if  $\Wr_m(k)\neq0$ we set

\begin{equation}    G_{m,\bowtie}( -k^2 ; x , y ) :=
\frac{1}{ \Wr_{m}(k)    }
\left \{
\begin{array}{lr}
, & x < y , \\
v_{m}(x,k)u_{m}(y,k) , & y<x.
\end{array}
\right.
\label{green-bowtie}
\end{equation}

We also have Green's operators with mixed boundary conditions.
The cleanest situation we have under the assumption
$Q \in \mathscr{L}^{(\infty)}_\varepsilon \cap \mathscr{L}^{(0)}_0$,  $m\neq0$,
$|\Re(m)|\leq\frac \varepsilon2$,
Then if $k\neq0$, $\kappa\in \C\cup\{\infty\}$ and
$ \Wr_m(k)+\kappa\frac{\Gamma(1-m)}{\Gamma(1+m)}\frac{k^{2m}}{2^{2m}}\Wr_{-m}(k)\neq0$, we define

\begin{align}\label{bowtie-kappa}
&  G_{m,\bowtie,\kappa}( -k^2 ; x , y )
\\:=&
\frac{1}{ \big( \Wr_m(k)+\kappa\frac{\Gamma(1-m)}{\Gamma(1+m)}\frac{k^{2m}}{2^{2m}}\Wr_{-m}(k)\big) }
\left \{
\begin{array}{lr}
( u_m + \kappa \frac{\Gamma(1-m)}{\Gamma(1+m)}u_{-m} )(x,k)v_{m}(y,k)
, & x < y , \\
v_{m}(x,k) (u_m + \kappa \frac{\Gamma(1-m)}{\Gamma(1+m)} u_{-m} )(y,k) , & y<x.
\end{array}
\right.\notag\end{align}

Note that

\begin{equation}
G_{m,\bowtie,\kappa}( -k^2 ; x , y )=
G_{-m,\bowtie,\kappa^{-1}}( -k^2 ; x , y ).\end{equation}

Similarly, for
Q \in \mathscr{L}^{(\infty)}_0 \cap
\mathscr{L}^{(0)}_{0,\ln}$,
$m=0$, if
$\nu \Wr_0(k)+\Wr( v_0 (\cdot,k), p_{0}(\cdot,k) )
\neq 0$,
then  we define

\begin{align}\label{bowtie-zero}
&    G^\nu_{0,\bowtie}( -k^2 ; x , y ) \\:=&
\frac{1}{\big(\nu\Wr_0(k)+\Wr( v_0(\cdot,k) ,
p_{0} (\cdot,k))\big)
\left \{
\begin{array}{lr}
( \nu  u_0 +  p_{0} )(x,k)v_{0}(y,k)
, & x < y , \\
v_{0}(x,k) ( \nu  u_0 +  p_{0} )(y,k) , & y<x.
\end{array}
\right.\notag\end{align}

Without the assumption
$\Re(m)\geq-\frac \varepsilon2$ we are not guaranteed the existence of
$u_m$, and hence we are not sure whether Green's function with pure
boundary conditions can be extended. However, we can use the boundary
conditions given by $u_{-m}^{[n]}$.
Choose $\varepsilon>0$, $Q \in \mathscr{L}^{(\infty)}_\varepsilon \cap \mathscr{L}^{(0)}_0$,  $m\neq0$, and a nonnegative integer $n$.
Then for $-\frac{\varepsilon}{2}(n+1)\leq\Re(m)<0$ if
$k\neq0$, $\kappa\in \C\cup\{\infty\}$ and
\Wr\big(v_{m}(\cdot,k), u_{m}^{[n]}(\cdot,k)\big)+\kappa\frac{\Gamma(1-m)}{\Gamma(1+m)}\frac{k^{2m}}{2^{2m}}\Wr_{-m}(k)\neq0$,
we can define

\begin{align} \label{bowtie-kappa-n}&
G_{m,\bowtie,\kappa}^{[n]}( -k^2 ; x , y )\\ := & \frac{1}{ \Wr\big(v_{m}(\cdot,k), u_{m}^{[n]}(\cdot,k)\big)+\kappa\frac{\Gamma(1-m)}{\Gamma(1+m)}\frac{k^{2m}}{2^{2m}}\Wr_{-m}(k) }
\left \{
\begin{array}{lr}
(  u_{m}^{[n]}+\kappa\frac{\Gamma(1-m)}{\Gamma(1+m)} u_{-m})(x,k)v_{m}(y,k)
, & x < y , \\
v_{m}(x,k) ( u_{m}^{[n]} +\kappa\frac{\Gamma(1-m)}{\Gamma(1+m)} u_{-m})(y,k) , & y<x.
\end{array}
\right.\notag
\end{align}

§ CLOSED REALIZATIONS OF LG

In this section we describe realizations of  $L_{m^2}$ as closed operators
on $L^2]0,\infty[$. We will see that under certain assumptions on the perturbation $Q$ one can impose the boundary condition at $0$ in a similar way as in the unperturbed case. If we fix $Q$, it is also natural to organize these  operators in holomorphic families, analogous to the holomorphic families studied in  [12].

In the first  two subsections we recall
the basics of the theory of 1d Schrödinger operators and their boundary conditions.

§.§ 1-dimensional Schrödinger operators on the halfline

We will  follow [13], other references include [20, 21].

Suppose that $]0,\infty[\ni x\mapsto V(x)$ is a function in
$L_\loc^1]0,\infty[$, possibly complex valued. Consider the expression

\[L:=-\partial_x^2+V(x),\]

viewed as a linear map from $AC^1]0,\infty[$ to $L_\loc^1]0,\infty[$. Let us describe the four most obvious closed realizations
of $L$ on $L^2]0,\infty[$.

First define

\begin{align*}
\cD(L^{\max})
& := \big \{ f \in L^2]0,\infty[ \, \cap \, AC^1]0,\infty[\quad \mid\ L f \in L^2]0,\infty[ \big \} ,\\
\cD(L^{\min})&:=\text{the closure of }\{f\in\cD(L^{\max})\ \mid\
f=0\text{ near }0\text{ and }\infty\},\\
\cD(L^0)&:=\text{the closure of }\{f\in\cD(L^{\max})\ \mid\
f=0\text{ near }0\},\\
\cD(L^\infty)&:=\text{the closure of }\{f\in\cD(L^{\max})\ \mid\
f=0\text{ near }\infty\}.
\end{align*}

$\cD(L^{\max})$ is treated as a Hilbert space with the norm given by
We define

\begin{equation*}
L^{\max}:=L\big|_{\cD(L^{\max})},\quad L^{\min}:=L\big|_{\cD(L^{\min})},\quad
\end{equation*}

Then $L^{\max}$, $L^{\min}$, $L^0$ and $L^\infty$ are closed operators satisfying

\begin{equation*}
L^{\max}\supset  L^\infty\supset L^{\min},\quad
L^{\max}\supset  L^0\supset L^{\min}.
\end{equation*}

Let us quote some general results.
The following proposition is well-known, it is e.g. proven as Proposition 5.15 of [13]:

\begin{align}
\limsup_{c\to\infty}\int_c^{c+1}|V(x)|\mathrm{d} x<\infty,
\label{bounda1}
\end{align}

\begin{equation}\label{bounda}
L^{\max}=L^\infty,\quad L^0=L^{\min}.
\end{equation}

Thus, there is no need to set boundary conditions at infinity.

By  <cit.>, we have

Suppose that (<ref>) holds.
Then we have the following alternative:

\begin{align*}
&    \text{(i) either }
\cD(L^{\max})/\cD(L^{\min})=0\\
&\quad    \text{ and }
\dim\big\{f\in \Ker(L-\lambda )\ \mid\ f\text{ is square integrable near }0\big\}\leq1\quad \text{ for all $\lambda \in\C$ };\notag\\
&  \text{(ii) or }
\cD(L^{\max})/\cD(L^{\min})=2\\
& \quad  \text{ and }
\dim\{ f\in\Ker(L^{\max}-\lambda )\ \mid\ f\text{ is square integrable near }0\}=2\quad\text{ for all $\lambda \in\C$}. \quad \notag
\end{align*}

Until the end of this subsection
we suppose that alternative (ii) of Proposition <ref> holds.
Fix $\lambda\in\C$ and $\xi\in C_\mathrm{c}[0,\infty[$ equal $1$ near
$0$. Then by <cit.>  we have a direct sum decomposition

\begin{equation}\label{wroo1}
\cD(L^{\max})=\cD(L^{\min})\oplus\{\xi f\ |\

where $\mathcal{N}(L)$ denotes all functions in $AC^1[0,\infty[$
annihilated by $L$.

We are interested in operators $L^\bullet$ lying “in the middle” between
$L^{\min}$ and $L^{\max}$, that is satisfying

\begin{equation*}
L^{\min}\subset L^\bullet\subset L^{\max},
\end{equation*}

where both inclusions are of codimension $1$.
All such operators correspond to one-dimensional
subspaces of $ \cD(L^{\max})/\cD(L^{\min})$. To specify
such a subspace it is enough to choose

\begin{equation}
\cD(L^{\max}),\quad r\notin\cD(L^{\min}),\label{rdom}
\end{equation}

and to define

\begin{align}\label{roro1}
\cD(L^r):&=\cD(L^{\min})+\mathbb{C} r,\\ L^r:&=L^{\max}\big|_{\cD(L^r)}.
\end{align}

§.§ Boundary functionals

We continue to analyze general 1d Schrödinger operators.
Until the end of this subsection we assume (<ref>).
We will give a method to describe boundary conditions which is often
more practical than (<ref>).

First recall the concept of Wronskian  (<ref>). If $f,g\in \cD(L^{\max})$,  then $f,g\in AC^1]0,\infty[\subset C^1]0,\infty[$,
hence their Wronskian at
$x\in]0,\infty[$, denoted
$\Wr(f,g;x)$, is well defined. Interestingly, the
Wronskian extends to the boundary $x=0$, as follows  e.g. from <cit.>:

For $f,g\in\cD(L^{\max})$

\begin{equation*}
\lim_{x\searrow0}\Wr(f,g;x)=:\Wr(f,g;0)
\end{equation*}

exists and defines a continuous
bilinear form.
If in addition (<ref>) holds, then

\begin{align*}
\mathcal{D} ( L^{\mathrm{min}} ) &= \Big \{ f \in \mathcal{D} ( L^{\mathrm{max}} ) \ \mid
\ \Wr( f , g ; 0 ) = 0 \text{ for all } g \in \mathcal{D} ( L^{\mathrm{max}} ) \Big \}.
\end{align*}

Let us define the boundary space

\begin{equation*}
\cB:= \big(\cD(L^{\max})/\cD(L^{\min})\big)',
\end{equation*}

where the prime denotes the dual.

Let $r$ be as in (<ref>).
Let $\phi\neq0$ be    a boundary functional (that
is, an element of $\cB$)  such that
$\phi(r)=0$. Obviously,

\begin{align*}
\cD(L^r):&=\{f\in\cD(L^{\max})\mid \phi(f)=0\}
\end{align*}

is equivalent to (<ref>).

The boundary functional $\phi$
can be simply written as

\begin{equation}\label{wroo}
\phi=c\Wr(r,\cdot;0),\end{equation}

where $c\neq0$.
Using (<ref>) and (<ref>) we obtain

Suppose that the alternative (ii) of  Proposition <ref>
holds. Fix $\lambda\in\C$.
Then we have a natural isomorphism of $\cB$ and $\mathcal{N}(L-\lambda)$:

\begin{equation}\label{isom}
\cB=\{\Wr(f,\cdot;0)\ |\ f\in\mathcal{N}(L-\lambda)\}.\end{equation}

We will say that a function $f\in C^1]0,\infty[$ possesses the
Wronskian at zero on $\cD(L^{\max})$

\begin{equation*}
\Wr(f,g;0):=\lim_{x\searrow0}\Wr(f,g;x),\quad g\in\cD(L^{\max} ),
\end{equation*}

exists. Proposition <ref> says that each  function in
$\cD(L^{\max})$ possesses the Wronskian at zero  on $\cD(L^{\max})$.

In practice, it may be difficult to make explicit an element $r$ in
$\cD(L^{\max})$ describing the functional $\phi$ by (<ref>).
Instead, we can often find a
simpler function $r_1$, not necessarily in $\cD(L^{\max})$,
which also possesses the Wronskian at zero  on $\cD(L^{\max})$ and
such that

\begin{equation}
\Wr(r,\cdot;0)=\Wr(r_1,\cdot;0).\end{equation}
instead of (<ref>)  the domain of $L^r$ can be equivalently
characterized as :

\begin{align}\label{roro2}
\cD(L^r):&=\{f\in\cD(L^{\max})\mid \Wr(r_1,f;0)=0\}.
\end{align}

§.§ The maximal and minimal realizations of Lg

As everywhere in this paper, we assume
that $]0,\infty[\ni x\mapsto Q(x)$ belongs to $L_\loc^1]0,\infty[$.
For  $m \in \mathbb{C}$, set

\begin{equation*}
V_{m^2}(x):= \Big(m^2-\frac14\Big)\frac{1}{x^2}  + Q(x).
\end{equation*}

We consider the differential expression

\begin{equation*}
L_{m^2}:=-\partial_x^2+  V_{m^2}(x).
\end{equation*}

as a linear map on $AC^1]0,\infty[$.
By applying the definitions of Subsection
<ref>, we can introduce
the closed operators
$L_{m^2}^{\max}$, $L_{m^2}^{\min}$ such that

\begin{equation*}
\big ( L^{\mathrm{min}}_{m^2}  \big )^\# = L^{\mathrm{max}}_{m^2} , \qquad \big ( L^{\mathrm{max}}_{m^2} \big )^\# = L^{\mathrm{min}}_{m^2} .
\end{equation*}

Until the end of this section we assume that $Q \in \mathscr{L}^{(\infty)}_0$.

Let $m\in\C$.
Suppose that

\begin{align*}
Q \in \mathscr{L}^{(0)}_0 , & \text{ if } m \neq 0 , \qquad Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}} ,  \text{ if } m = 0 .
\end{align*}

Then the  following holds:

* If $1\leq | \mathrm{Re}( m )|$, then $L^{\mathrm{min}}_{m^2} = L^{\mathrm{max}}_{m^2}$.

* If $| \mathrm{Re}( m ) |  < 1$, then $\mathcal{D} ( L^{\mathrm{min}}_{m^2} )$ is a closed subspace of $\mathcal{D} ( L^{\mathrm{max}}_{m^2} )$ of codimension $2$.

the condition (<ref>) holds. Therefore,
only the boundary conditions at zero matter.

We can assume that $\Re(m)\ge0$ and $\Re(k)\geq0$.
For  $m\neq0$, in
the space $\Ker(L_{m^2}+k^2)$ all
elements proportional to $u_{m}(\cdot,k)$    behave as
and all other elements of $\Ker(L_{m^2}+k^2)$,
by Proposition <ref>, behave as $x^{\frac12-m}$.
For $m=0$ they behave respectively as  $x^{\frac12}$ and
$x^{\frac12}\mathrm{ln}(x)$. Both are   square integrable iff
$|\Re(m)|<1$. Hence

\begin{equation}
\begin{split}\label{pau1}
&\dim\big\{f\in\Ker(L_{m^2}^{\max}+k^2)\ \mid\ f\text{ is square integrable near }0\big\}\leq1 \text{ for all $\Re(k)\geq0$ } \\
&\text{ iff }\ |\Re(m)|\geq1;
\end{split}
\end{equation}

\begin{equation}
\begin{split}
\label{pau2}
&\dim\{f\in\Ker(L_{m^2}^{\max}+k^2)\ \mid \ f\text{ is square integrable near }0\}=2 \text{ for all $\Re(k)\geq0$}\\
&\text{ iff }\ |\Re(m)|<1 .
\end{split}
\end{equation}

Now  we apply Proposition <ref>: $(i)$
corresponds to (<ref>)   and $(ii)$
corresponds to (<ref>).

§.§ Closed realizations of the unperturbed Bessel operator

Let us recall the basic theory of closed realizations of $L_{m^2}^0$.
We will essentially follow [16],
except that we will put  the superscript $0$ on the symbols of various operators.

Let $\cB_{m^2}^0$ denote  the boundary space of $L_{m^2}^0$. Below we
give natural bases of $\cB_{m^2}^0$:

\begin{align}
\Wr(x^{\frac12-m},\cdot;0),&\quad  \Wr(x^{\frac12+m},\cdot;0),\quad
\Wr(x^{\frac12},\cdot;0),&\quad
\Wr(x^{\frac12}\mathrm{ln}(x),\cdot;0),\quad m=0.\label{real5}
\end{align}

Note that for $|\Re(m)|<1$,

\begin{align}
\Wr(x^{\frac12-m},x^{\frac12+m})&=2m, \label{eq:wronskian_mneq0} \\
\Wr(x^{\frac12},x^{\frac12}\mathrm{ln}(x))&=1, \label{eq:wronskian_m=0}
\end{align}

which implies the linear independence of (<ref>) and (<ref>).

Let us describe the basic families of closed realizations of
Bessel operators. We will use two kinds of definitions of their
In what follows,  $\xi$ is a smooth compactly supported function equal to $1$ near $x=0$.

We have the family of realizations with pure boundary conditions
defined for  $\Re(m)>-1$:

\begin{align}\label{dom1}
\Dom(H_{m}^0)  :=& \, \cD(L_{m^2}^0)+\mathbb{C} x^{\frac12+m}\xi(x)\\
=& \, \Big\{f\in \Dom(L_{m^2}^{0,\max})\mid
\, \Wr(x^{\frac12+m},f;0)=0\Big\},\label{wron1}
\\\label{real1}
H_{ m}^0:=& \, L_{ m^2}^0\big|_{\Dom(H_{m}^0)}.
\end{align}

We have also two families with mixed boundary conditions: The first is the generic family defined for $-1<\Re(m)<1$,

\begin{align}\label{dom2}
\Dom(H_{ m,\kappa}^0) &:
\kappa x^{\frac12-m}\big)\xi(x)\\
&= \Big\{f\in \Dom(L_{m^2}^{0,\max})\mid
\Wr\big(x^{\frac12+m} + \kappa x^{\frac12-m},f;0
\big)=0\Big\},\label{wron2}
\\\label{real2}
\Dom(H_{m,\infty}^0) & := \Dom(H_{ -m}^0),\quad
H_{m,\kappa}^0:=L_{ m^2}^0\big|_{\Dom(H_{m,\kappa}^0)}.
\end{align}

The second family corresponds to $m=0$ and depends on

\begin{align}\label{dom3}
\Dom(H_{0}^{0,\nu}) & :
+ \nu x^{\frac12}\big)\xi(x)\\
& = \Big\{f\in \Dom(L_{0}^{0,\max})\mid
\Wr\big(\nu x^{\frac12} + x^{\frac12}\mathrm{ln}(x),f;0
\big)=0\Big\},\label{wron3}
\\\label{real3}
\Dom(H_{ 0}^{0,\infty}) &: = \Dom(H_{0}^{0}),\quad
H_{ 0}^{0,\nu}:=L_{0}^0\big|_{\Dom(H_{0}^{0,\nu})}.
\end{align}

The families of closed operators defined in (<ref>),
(<ref>) and (<ref>) are clearly independent of the cutoff $\xi$. They
are holomorphic with respect to the parameters $m,\kappa,\nu$. They are
situated between $L_{m^2}^{0,\min}$ and  $L_{m^2}^{0,\max}$.

§.§ Boundary functionals for perturbed Bessel operators

In this subsection, as well as in Subsections <ref>, <ref> and <ref>, we
analyze boundary conditions near zero and the corresponding closed
realizations of perturbed Bessel operators.  For definiteness, throughout these four
subsections we assume that $Q \in \mathscr{L}^{(\infty)}_0$.

It does not seem practical to
define   boundary conditions for perturbed Bessel operators
analogously as in (<ref>), (<ref>) and (<ref>). In fact, even after imposing
stronger conditions on $Q$, such as in Proposition <ref>, it
is not easy to describe explicitly sufficiently well  the behavior of
elements in $\cD(L^{\mathrm{max}}_{m^2})$  near zero.
In particular,
conditions of Theorem <ref>
in general do not allow us to conclude that $x^{\frac12\pm m}\xi(x)$ belongs
to  $\cD(L_{m^2}^{\max})$ and $x^{\frac12}\mathrm{ln}(x)\xi(x)$ to

Fortunately, we can use the method of (<ref>), (<ref>)
and (<ref>) involving  the Wronskian at $0$.
The precise choice of a boundary functional is in general
more complicated than in the unperturbed case, as we explain

Let us first describe some properties of the minimal domain.
In the next proof, we denote by $G_\rightarrow=G_{m^2,\rightarrow}(-k^2)$ the forward Green's
operator associated to $L_{m^2}$ defined in (<ref>).

$ $

* Let $0\leq\Re(m)<1$, $m\neq0$,
$Q \in \mathscr{L}^{(0)}_0$ and   $h\in\cD(L_{m^2}^{\min})$. Then

\begin{align}\label{iuy1}
\partial_xh(x)=o(x^{\frac12}).
\end{align}

* Let $m=0$,
$Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}}$ and
$h\in\cD(L_{m^2}^{\min})$. Then

\begin{align}
\label{iuy2}
\partial_xh(x)=o(x^{\frac12}\mathrm{ln}(x)).\end{align}

Let $h\in\cD(L_{m^2}^{\min})$ and $c>0$. Let $\Re(k)\ge0$ be such that $\Wr_m(k)\neq0$ ($k$ exists by Corollary <ref>). Using e.g. <cit.> we know that there exists
$f\in L^2]0,c[$ such

\begin{equation*}
h\big|_{]0,c[}=G_\rightarrow(-k^2) f\big|_{]0,c[}.
\end{equation*}

By e.g. <cit.>

\begin{align*}
|G_\to(-k^2) f(x)|&\leq \frac12\big(|w_m(x,k)|\|u_m(\cdot,k)\|_x+
| u_m(x,k)|\|
|\partial_xG_\to(-k^2) f(x)|&\leq \frac12\big(|\partial_x w_m(x,k)|\|u_m(\cdot,k)\|_x+
|\partial_x u_m(x,k)|\|
\end{align*}

where $\|f\|_x:=(\int_0^x|f(y)|^2\d y)^{\frac12}$. By Proposition <ref>, for small $x$,

\begin{align*}
&| w_m(x,k) | \lesssim x^{\frac12-\Re(m)} , \quad | \partial_x w_m(x,k) | \lesssim x^{-\frac12-\Re(m)} , \quad \text{if }m \neq 0 , \\
&| w_0(x,k) | \lesssim x^{\frac12}|\mathrm{ln}(x)| , \quad | \partial_x w_m(x,k) | \lesssim x^{-\frac12}|\mathrm{ln}(x)| , \quad \text{if }m = 0 ,
\end{align*}

while, by Proposition <ref>,

\begin{align*}
&| u_m(x,k) | \lesssim x^{\frac12+\Re(m)} , \quad | \partial_x u_m(x,k) | \lesssim x^{-\frac12+\Re(m)} .
\end{align*}

This yields the estimates (<ref>)–(<ref>).

[2]

$ $

* Let $0\leq\Re(m)<1$, $m\neq0$ and $Q\in\mathscr{L}^{(0)}_0$. Let $g \in AC^1]0,\infty[$ be such that $g(x)=o(x^{\frac12+\Re(m)})$ and $\partial_xg(x)=o(x^{-\frac12+\Re(m)})$. Then

\begin{equation*}
\Wr(g,f;0)=0 \quad \text{for all } f \in \Dom ( L^{\mathrm{max}}_{m^2} ).
\end{equation*}

* The same holds if $m=0$,
$Q\in\mathscr{L}^{(0)}_{0,\mathrm{ln}}$ and $g \in
AC^1]0,\infty[$ satisfies $g(x)=o(x^{\frac12} | \mathrm{ln}(x)
|^{-1} )$ and $\partial_xg(x)=o(x^{-\frac12} | \mathrm{ln}(x) |^{-1}

Fix any $k\in\C$. Every $f \in \Dom (
L^{\mathrm{max}}_{m^2} )$ can be written as $f=\xi f_0+f_1$ where
$f_0\in\mathcal{N}(L_{m^2}^{\max}+k^2)$ and $f_1
\in\cD(L_{m^2}^{\min})$. Now (i)
follows by  Lemma <ref>(i)
and Proposition <ref>(i), and (ii) follows by
Lemma <ref>(ii)
and Proposition <ref>(ii).

In what follows, we will denote by $\cB_{m^2}$ the space of boundary
functionals of $L_{m^2}$ with a given perturbation $Q$.
We will describe
convenient bases of
In other words, we will find
pairs of linearly independent functionals on $\cD(L_{m^2}^{\max})$
that vanish on  $\cD(L_{m^2}^{\min})$.

Note that cases (i)(a) and (iii)(a) of the next theorem have quite weak assumptions on the
perturbation, however  their non-principal boundary functionals depend
on an arbitrary parameter $a$.

$ $

* Let $0<\Re(m)<1$.
* Assume $Q \in \mathscr{L}^{(0)}_0$.
Let $a>0$ be small enough as in Proposition

\begin{equation}\label{ia}
\Wr( u_{-m}^{\bowtie(a)}(\cdot,0),\cdot;0),\quad  \Wr(x^{\frac12+m},\cdot;0) ,
\end{equation}

is a basis of $\cB_{m^2}$.

* Suppose that the assumption
is strengthened to $Q \in
\mathscr{L}^{(0)}_\varepsilon$ for some $\varepsilon>0$ (but we drop
the assumption on $a$). Let $n$ be a non-negative integer such that $\Re(m)\le\frac{(n+1)\varepsilon}{2}$. Then

\begin{equation}\label{ib}
\Wr(  u_{{-}m}^{0[n]},\cdot;0),\quad \Wr(x^{\frac12+m},\cdot;0)
\end{equation}

is a basis of $\cB_{m^2}$.

* If we assume
$0<\Re(m)\leq\frac{\varepsilon }{2}$ and $Q \in
\mathscr{L}^{(0)}_\varepsilon$, then

\begin{equation}\label{ic}
\Wr(x^{\frac12-m},\cdot;0),\quad\Wr(x^{\frac12+m},\cdot;0)
\end{equation}

is a basis of $\cB_{m^2}$.

* Let $\Re(m)=0$, $m\neq0$. Assume $Q \in \mathscr{L}^{(0)}_0$. Then

\begin{equation}
\Wr(x^{\frac12-m},\cdot;0),\quad  \Wr(x^{\frac12+m},\cdot;0)
\label{iia}  \end{equation}

is a basis of $\cB_{m^2}$.

* Let $m=0$.

* Assume $Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}}$.   Let $a>0$ be small enough as in Proposition

\begin{equation}
\Wr(p_0^{\diamond(a)},\cdot;0),\quad \Wr( u_0(\cdot,0) ,\cdot;0)
\label{iiia}\end{equation}

is a basis of $\cB_{0}$.

* Suppose that the assumption on $Q$ is strengthened to $Q \in
\mathscr{L}^{(0)}_{0,\mathrm{ln}^2}$ (but we drop the condition on $a$). Then

\begin{equation}
\Wr(p_0,\cdot;0),\quad \Wr(x^{\frac12},\cdot;0)
\label{iiib}\end{equation}

is a basis of $\cB_{0}$.

* If the assumption is further strengthened to $Q \in \mathscr{L}^{(0)}_\varepsilon$ for some $\varepsilon>0$, then

\begin{equation}
\Wr(x^{\frac12}\mathrm{ln}(x),\cdot;0),\quad \Wr(x^{\frac12},\cdot;0)
\label{iiic}\end{equation}

is a basis of $\cB_{0}$.

(i)  Recall that  in Propositions <ref> and
<ref> we introduced the
u_m(\cdot,0), u_{-m}^{\bowtie(a)}(\cdot,0)$
spanning $\mathcal{N}(L_{m^2})$.
Therefore,  by (<ref>),

\begin{equation}\label{wrow0}
\Wr( u_{-m}^{\bowtie(a)}(\cdot,0),\cdot;0),\quad
\Wr(u_m(\cdot,0),\cdot;0)\end{equation}

is a basis of $\cB_{m^2}$.
Using (<ref>)–(<ref>) and  Lemma <ref>(i)
we see that

\begin{equation}\label{wroni}
\Wr( u_m(\cdot,0),\cdot;0)=  \frac{1}{\Gamma(m+1)}\Wr(x^{\frac12+m},\cdot;0).\end{equation}

Therefore, we can replace $u_{ m}(\cdot,0)$ by $x^{\frac12+ m}$,
obtaining the basis (<ref>).

Assume now that $Q \in \mathscr{L}^{(0)}_\varepsilon$ for some
and suppose  that $n$ is a positive integer,  $0<\varepsilon<\frac{2}{n+1}$,
$0<\Re(m)\leq\frac{\varepsilon (n+1)}{2}$ and $Q \in
\mathscr{L}^{(0)}_\varepsilon$.  By Proposition <ref>, we have

\begin{align*}
\Wr( u_{{-}m}^{\bowtie(a)}(\cdot,0),\cdot;0)
=\Wr( u_{{-}m}^{0[n]},\cdot;0)+c_{m}^{(a)[n]}\Wr(u_m(\cdot,0),\cdot;0),
\end{align*}

for some constant $c_{m}^{(a)[n]}$ depending on the parameters. Therefore,
(<ref>) is also a basis of $\cB_{m^2}$.

If $2\Re(m)\leq\varepsilon$, then we can take $n=0$:

\begin{equation}
u_{{-}m}^{0[0]}(\cdot,0) =
\end{equation}

we can replace $\Wr( u_{-m}^{0[0]}(\cdot,0),\cdot;0)$ with
obtaing the basis (<ref>).

(ii) By
Proposition <ref>,  both $ u_m(\cdot,0)$ and
$u_{-m}(\cdot,0)$ are well defined and span $\mathcal{N}(L_{m^2})$.  By   (<ref>),
we obtain a basis of $\cB_{m^2}$

\begin{equation*}
\Wr(u_m(\cdot,0),\cdot;0),\quad  \Wr(u_{-m}(\cdot,0),\cdot;0).
\end{equation*}

(<ref>) is still valid so that
we can replace $u_{\pm m}(\cdot,0)$ by $x^{\frac12\pm m}$, obtaining
the basis (<ref>)

(iii) In Propositions <ref> and <ref> we constructed
functions $ u_0(\cdot,0)$ and $p_0^{\diamond(a)}(\cdot,0)$ spanning
It follows from   (<ref>) that (<ref>)
is a basis of $\cB_{0}$.

If we strengthen the assumption to
$Q\in\mathscr{L}^{(0)}_{0,\mathrm{ln}^2}$, then in Proposition <ref> we defined
$p_0(\cdot,0)\in\mathcal{N}(L_0)$.  The functions $p_0(\cdot,0)$ and $u_0(\cdot,0)$
span $\mathcal{N}(L_0)$.
Therefore, by   (<ref>),

\begin{equation*}
\Wr(u_0(\cdot,0),\cdot;0),\quad  \Wr(p_{0}(\cdot,0),\cdot;0).
\end{equation*}

is a basis of $\cB_{0}$.
Besides, by Proposition <ref> and  Lemma <ref>(ii)
we have

\begin{align*}
&   \Wr( u_0(\cdot,0),\cdot;0)=  \Wr(x^{\frac12},\cdot;0).
\end{align*}

Therefore, (<ref>)
is a basis of $\cB_{0}$.

If the assumption is further strengthened to
$Q\in\mathscr{L}^{(0)}_{\varepsilon}$ with $\varepsilon>0$, then
by  Proposition <ref> and   Lemma <ref>(ii)
we have

\begin{align*}
& \Wr(p_0,\cdot;0) =  \Wr(x^{\frac12} \mathrm{ln}(x),\cdot;0).
\end{align*}

Therefore, (<ref>)
is a basis of $\cB_{0}$.

Let $Q(x) = - \frac{ \beta }{ x }\one_{]0,1]}(x)$ as in Remark <ref>. Taking $n=1$ in the previous theorem, it follows from that remark that for $0<\Re(m)<1$,

\begin{equation*}
\Wr\Big(x^{\frac12-m}\Big ( 1 -  \frac{\beta x}{1-2m} \Big ),\cdot;0\Big),\quad \Wr(x^{\frac12+m},\cdot;0)
\end{equation*}

forms a basis of $\cB_{m^2}$.

The next table summarizes the bases of $\cB_{m^2}$ that we constructed, depending on the values of $-1<\Re(m)<1$ and on the conditions on $Q$.

2*$-1<\Re(m)<0$     2c|$\Re(m)=0$     2*$0<\Re(m)<1$

$m\neq0$     $m=0$

$Q \in \mathscr{L}^{(0)}_0$     $x^{\frac12-m}$, $u_{m}^{\bowtie(a)}$     $x^{\frac12-m}$, $x^{\frac12+m}$     ?     $u_{-m}^{\bowtie(a)}$, $x^{\frac12+m}$

$Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}}$     $x^{\frac12-m}$, $u_{m}^{\bowtie(a)}$     $x^{\frac12-m}$, $x^{\frac12+m}$     $p_0^{\diamond(a)}$, $x^{\frac12}$     $u_{-m}^{\bowtie(a)}$, $x^{\frac12+m}$

$Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}^2}$     $x^{\frac12-m}$, $u_{m}^{\bowtie(a)}$     $x^{\frac12-m}$, $x^{\frac12+m}$     $p_0$, $x^{\frac12}$     $u_{-m}^{\bowtie(a)}$, $x^{\frac12+m}$

2*$Q \in \mathscr{L}^{(0)}_{\varepsilon}$
$-\frac{(j+1)\varepsilon}{2} \le \Re(m)
\leq                                                  0$     2*$x^{\frac12-m}$, $x^{\frac12+m}$     2*$x^{\frac12}\mathrm{ln}(x)$, $x^{\frac12}$      $0 \leq\Re(m) \le  \frac{(j+1)\varepsilon}{2}$

2-2 5-5
$x^{\frac12-m}$, $ u_{m}^{0[j]}$               $ u_{-m}^{0[j]}$, $x^{\frac12+m}$

tableBases of the boundary space $\cB_{m^2}$ of $L_{m^2}$. In each case, we write $g_1$, $g_2$ if $\Wr(g_1,\cdot;0)$, $\Wr(g_2,\cdot;0)$ is a basis of $\cB_{m^2}$. To shorten notations, we write $u_{m}^{\bowtie(a)}$ for $u_{m}^{\bowtie(a)}(\cdot,0)$ and likewise for other functions. Note that $u_{m}^{\bowtie(a)}$ and $p_0^{\diamond(a)}$ depend on an arbitrary parameter $a$. Each line corresponds to a condition on $Q$, from the minimal one $Q \in \mathscr{L}^{(0)}_0$ to the strongest one $Q \in \mathscr{L}^{(0)}_{\varepsilon}$, with $\varepsilon>0$ (beside the condition $Q \in \mathscr{L}^{(\infty)}_0$ which we everywhere assume). In the last line, $j\in \{ 0 , \dots , n \}$, where $n$ is the smallest nonnegative integer such that $\frac{(n+1)\varepsilon}{2} \ge 1$.

As mentioned above, in the last line, for $j=0$, $ u_{m}^{0[0]}$ can be replaced by $x^{\frac12+m}$ and $ u_{-m}^{0[0]}$ can be replaced by $x^{\frac12-m}$. For growing values or $\Re(m)>0$, the picture is then that, to pass from the region $R_j:=\{\frac{j\varepsilon}{2} < \Re(m) \le  \frac{(j+1)\varepsilon}{2}\}$ to $R_{j+1}$, one needs to add a further term to $ u_{-m}^{0[j]}$ in order to still have an element of $\cB_{m^2}$. Of course, we could also use $ u_{-m}^{0[n]}$ in the whole region $0 < \Re(m) < 1$ for any $n$ such that $\frac{(n+1)\varepsilon}{2} \ge 1$, but then all the terms of order $o(x^{\frac12+m})$ in $ u_{-m}^{0[n]}$ are irrelevant.

§.§ The perturbed Bessel operator with pure boundary conditions

In this subsection we introduce the most natural family
of perturbed Bessel operators.  It is parallel to what in the
unperturbed case was called the family of homogeneous
Bessel operators. (In the perturbed case the homogeneity is no longer
true, therefore the name has to be changed).

Let $m\in\mathbb{C}$, $-1<\Re(m)$. We assume that

\begin{align*}
&  Q \in \mathscr{L}^{(0)}_\varepsilon, \quad
\varepsilon=\max\big(0,-2\Re(m)\big);
\\
& Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}^2} ,\quad m=0.
\end{align*}

We can then define

\begin{align*}
\Dom(H_{ m}) & := \Big\{f\in \Dom(L_{m^2}^{\max})\mid
\, \Wr(x^{\frac12+m},f;0)=0\Big\},\\
H_{ m}&:=L_{m^2}\big|_{\Dom(H_{m})},
\end{align*}

which we will call
the perturbed Bessel operator with pure boundary conditions.

Using Theorem <ref> we see that the operator
$H_{m}$ is

\begin{align*}
L^{\mathrm{min}}_{m^2} = H_{m} = L^{\mathrm{max}}_{m^2},\quad \Re(m)\geq1;\\
L^{\mathrm{min}}_{m^2} \subset H_{m} \subset L^{\mathrm{max}}_{m^2} ,\label{pqo}\quad
-1< \Re(m)<1;
\end{align*}

and both inclusions are of codimension $1$.

Suppose that the assumptions on $Q$ stated at the beginning of this subsection
Let $\Re(k)>0$. Then
$k^2\notin \sigma(H_{m})$
if and only if $\Wr_{m}(k)\neq0$. Moreover,
the operator
$  G_{m,\bowtie}(-k^2)$ defined in (<ref>)  is
then bounded

\begin{equation*}
G_{m,\bowtie}(-k^2)=(k^2+  H_{m})^{-1}.
\end{equation*}

We use <cit.> together with the asymptotic
behavior near $0$ of $u_m(x,k)$ established in
Propositions <ref>.

the following theorem we fix a perturbation $Q$ and consider the
operator valued family
$   H_{m}$.
In the definition of regularity we use the
concept of a holomorphic family of closed operators
recalled in
Appendix <ref>.
Moreover, the continuity of a family of closed operators should be understood in the weak resolvent sense.

$ $

* Let $2>\varepsilon>0$ and suppose that
$  Q \in \mathscr{L}^{(0)}_\varepsilon$.

\begin{align}\label{kak1}
\Big\{-\frac{\varepsilon}{2} \leq \Re(m)\Big\}\ni m&\mapsto
\end{align}

is regular.

* Let
$Q \in \mathscr{L}^{(0)}_0. $

\begin{align}
\Big\{\Re(m)\geq0,\, m\neq0\Big\}\ni m&\mapsto
\end{align}

is regular.
If we strengthen the assumption  to
$Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}^2} $,
then we can include $m=0$ in (<ref>).

In view of Proposition <ref>, we can proceed as in the proof of Theorem 3.10 in [12]: It suffices to use Propositions <ref>, <ref> and <ref>.

Note that for $\Re(m)\geq1$ the operator $H_m$ is the unique
closed realization
of $L_{m^2}$. Theorem <ref> shows that the holomorphic

\begin{equation*}
\{\Re(m)>1\}\ni m\mapsto H_m
\end{equation*}

has an analytic continuation to a larger region,
(<ref>) or (<ref>), where the width of the additional strip  depends on  the assumption on the potential.

§.§ The perturbed Bessel operator with mixed boundary
conditions I

In this subsection we describe closed realizations of perturbed Bessel
operators with  mixed boundary conditions under sufficiently strong conditions on the perturbation,
which guarantee that these realizations are very similar to the
unperturbed case.

Let  $m\neq0$, $|\Re(m)|<1$, $\varepsilon=2|\Re(m)|$ and
$Q \in \mathscr{L}^{(0)}_\varepsilon$.
$\kappa\in\C\cup\{\infty\}$ we set

\begin{align*}
\Dom(H_{ m,\kappa}) & := \Big\{f\in \Dom(L_{m^2}^{\max})\mid
\Wr\big(x^{\frac12+m} + \kappa x^{\frac12-m},f;0
\big)=0\Big\},\quad\kappa\in\C, \nonumber \\
\Dom(H_{ m,\infty}) &: = \Big\{f\in \Dom(L_{m^2}^{\max})\mid
\Wr\big( x^{\frac12-m},f;0
\big)=0\Big\},\\
H_{ m,\kappa}&:=L_{ m^2}\big|_{\Dom(H_{ m,\kappa})}.
\end{align*}

If $m=0$ we assume $Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}^2} $.
$\nu\in\C\cup\{\infty\}$ we set

\begin{align*}
\Dom(H_{0}^{\nu}) &: = \Big\{f\in \Dom(L_{0}^{\max})\mid
\Wr\big(\nu x^{\frac12} + p_0
\big)=0\Big\},\quad\nu\in\C, \nonumber \\
\Dom(H_{ 0}^{\infty}) & := \Dom(H_{ 0}),\\
H_{ 0}^{\nu}&:=L_{ 0}\big|_{\Dom(H_{0}^{\nu})}.
\end{align*}

Note that if $Q \in \mathscr{L}^{(0)}_\varepsilon$ with $\varepsilon>0$, then

\begin{align*}
\Dom(H_{0}^{\nu}) &: = \Big\{f\in \Dom(L_{0}^{\max})\mid
\Wr\big(\nu x^{\frac12} + x^{\frac12}\mathrm{ln}(x)
\big)=0\Big\}.
\end{align*}

Clearly, the operators
$H_{m,\kappa}$,   $H_{0}^\nu$
are closed,

\begin{align}
L^{\mathrm{min}}_{m^2} \subset H_{m,\kappa} \subset
L^{\mathrm{max}}_{m^2} ,\label{pqo1a}\\
L^{\mathrm{min}}_0 \subset H_{0}^\nu \subset
L^{\mathrm{max}}_0 ,\label{pqo1a-}
\end{align}

and both inclusions in (<ref>) and (<ref>)
are of codimension $1$.

One can compute the resolvents of $H_{m,\kappa}$ in the same way as
for $H_m$.

Let $\Re(k)>0$.

* Let  $m\neq0$, $|\Re(m)|<1$, $\varepsilon=2|\Re(m)|$,
$Q \in \mathscr{L}^{(0)}_\varepsilon$ and
We have $k^2\notin \sigma(H_{m,\kappa})$
if and only if $
\Wr_m(k)+\kappa\frac{\Gamma(1-m)}{\Gamma(1+m)}\frac{k^{2m}}{2^{2m}}\Wr_{-m}(k)\neq0$.
Besides, the operator
$  G_{m,\bowtie,\kappa}(-k^2)$ defined in (<ref>)  is
then bounded  and

\begin{equation*}
G_{m,\bowtie,\kappa}(-k^2)=(k^2+  H_{m,\kappa})^{-1}.
\end{equation*}

* Let $m=0$,   $Q \in
\mathscr{L}^{(0)}_{0,\mathrm{ln}^2}$ and $\nu \in \C\cup\{\infty\}$.
We have $k^2\notin \sigma(H_{m}^\nu)$ if and only if
$ \Wr( w_0 , \nu  u_0 + p_{0} )
\neq 0$. Besides,  the operator $G^\nu_{0}(-k^2)$ with kernel defined in (<ref>)
is then bounded
\begin{equation*}
G^\nu_{0,\bowtie}(-k^2)=(k^2+  H^\nu_{0})^{-1}.
\end{equation*}

The argument is the same as in the proof of Proposition <ref>.

Let us fix a perturbation $Q$ and consider the operator valued
families with mixed boundary conditions.

$ $

* Let $2>\varepsilon > 0$ and $  Q \in \mathscr{L}^{(0)}_\varepsilon$.

\begin{align}\label{holi}
\Big\{(m,\kappa)\ |\ |\Re(m)|\leq\frac{\varepsilon}{2},\
\kappa\in \mathbb{C}\cup\{\infty\}, \ (m,\kappa)\neq(0,-1)\Big\}\ni (m,\kappa)&\mapsto
\end{align}

is regular.
Besides, $H_{m,\kappa}=H_{-m,\kappa^{-1}}$.

* Let $m=0$. Suppose that $Q \in \mathscr{L}^{(0)}_{0,\mathrm{ln}^2} $.  Then

\begin{align}
\mathbb{C}\cup\{\infty\}\ni\nu&\mapsto
\end{align}

is   analytic.

This follows as in Theorem <ref>.

Proposition 3.11(ii) in [12] (in the case $Q=0$) shows that $(m,\kappa)\mapsto H_{m,\kappa}$ cannot be extended by continuity at $(0,-1)$.

§.§ The perturbed Bessel operator with mixed boundary
conditions II

As discussed in Theorem <ref> we can define closed realizations of
under much weaker conditions on $Q$ than those in the previous
subsection. Let us choose a
nonnegative integer $n$. A natural method to
describe them is by using the boundary functionals defined by
$ u_{-m}^{0[n]}$.

Let $0\leq\Re(m)<1$, $m\neq0$, $\varepsilon=\frac{2\Re(m)}{n+1}$ and
$ Q \in \mathscr{L}^{(0)}_\varepsilon$.
$\kappa\in\C\cup\{\infty\}$ we set

\begin{align*}
\Dom(H_{ -m,\kappa}^{[n]}) & := \left\{f\in \Dom(L_{m^2}^{\max})\mid
\Wr\Big( u_{-m}^{0[n]} + \kappa \frac{\Gamma(1-m)}{\Gamma(1+m)} x^{\frac12+m},f;0
\Big)=0\right\},\quad\kappa\in\C, \nonumber \\
\Dom(H_{ -m,\infty}^{[n]}) &: = \Big\{f\in \Dom(L_{m^2}^{\max})\mid
\Wr\big( x^{\frac12+m},f;0
\big)=0\Big\},\\
H_{- m,\kappa}^{[n]}&:=L_{ m^2}\big|_{\Dom(H_{-m,\kappa})}.
\end{align*}

Clearly, the operators
are closed,

\begin{align}
L^{\mathrm{min}}_{m^2} \subset H_{-m,\kappa} ^{[n]}\subset
L^{\mathrm{max}}_{m^2} \label{pqo1a+}
\end{align}

and both inclusions in (<ref>)
are of codimension $1$.

Note that in the particular case $n=0$ we have $H_{ -m,\kappa}^{[0]}=H_{-m,\kappa}$.

Let $0\leq\Re(m)<1$, $m\neq0$, $\varepsilon=\frac{2\Re(m)}{n+1}$ and
$ Q \in \mathscr{L}^{(0)}_\varepsilon$.
We have $k^2\notin \sigma(H_{-m,\kappa}^{[n]})$
if and only if  $ \Wr( v_m (\cdot,k),  u_{-m}^{[n]}(\cdot ,k)) +
\kappa \frac{\Gamma(1-m)}{\Gamma(1+m)}\Wr_m(k)  \neq0$. Besides,
the operator
$  G_{m,\bowtie,\kappa}^{[n]}(-k^2)$ defined in (<ref>)
is then bounded and

\begin{equation*}
G_{m,\bowtie,\kappa}^{[n]}(-k^2)=(k^2+  H_{-m,\kappa}^{[n]})^{-1}.
\end{equation*}

The argument is the same as in the proof of Proposition <ref>.

The following theorem can be compared with Theorem  <ref>:

Let $1>\frac{\varepsilon(n+1)}{2} > 0$ and $  Q \in \mathscr{L}^{(0)}_\varepsilon$.

\begin{align}
\Big\{-\frac{\varepsilon(n+1)}{2}\geq-\Re(m)\geq0,m\neq0\Big\}\times(\mathbb{C}\cup\{\infty\})\ni (-m,\kappa)&\mapsto
\end{align}

is regular.

§.§ Scattering length

Suppose that

\begin{align*}
&Q\in\mathscr{L}^{(\infty)}_{\delta} , \quad \text{if }\ 0\leq\Re(m),\quad m\neq0,\quad \delta =1+2\Re(m); \\
&Q\in\mathscr{L}^{(\infty)}_{1,\mathrm{ln}^2}, \quad \text{if } m=0.
\end{align*}

(Note that we do not impose conditions on $Q$ near $0$ apart from the
local integrability)
By Propositions <ref> and
<ref>, under these assumptions the space
$\Ker(L_{m^2})$ possesses a distinguished basis

\begin{align*} q_{-m},\quad q_{m},&\quad m\neq0;\\
q_0,\quad q_{0,\mathrm{ln}},&\quad m=0,
\end{align*}

Therefore, the boundary space $\cB_{m^2}$ has a

\begin{align}\label{wrio1}
\Wr(q_{-m},\cdot;0),\quad \Wr(q_{m},\cdot;0),&\quad m\neq0;\\
\label{wrio2}
\Wr(q_0,\cdot;0),\quad \Wr(q_{0,\mathrm{ln}},\cdot;0),&\quad m=0.
\end{align}

Suppose that  $H_\bullet$ is one of the realizations of the
Bessel operator such that

\begin{equation*}
L_{m^2}^{\min}\subsetneq H_\bullet\subsetneq L_{m^2}^{\max}.
\end{equation*}

As we discussed above, to define $H_\bullet$ we need to fix  a
boundary functional. So far, we tried to express boundary functionals
in terms of the asymptotics of functions near zero, as  in Subsection <ref>.

In quantum mechanics one often prefers to describe realizations of
perturbed Bessel
operators using (<ref>) and (<ref>). We say that the scattering length of $H_\bullet$ is
$a\in\mathbb{C}$ if

\begin{align}
\cD(H_\bullet)&=\{f\in\cD(L_{m^2}^{\max})\ |\
\Wr(q_m-aq_{-m},f;0)=0\},\quad m\neq0;\\
\cD(H_\bullet)&=\{f\in\cD(L_{m^2}^{\max})\ |\
\Wr(q_{0,\mathrm{ln}}-aq_0,f;0)=0\},\quad m=0.
\end{align}

\begin{align}
\cD(H_\bullet)&=\{f\in\cD(L_{m^2}^{\max})\ |\
\Wr(q_{-m},f;0)=0\},\quad m\neq0;\\
\cD(H_\bullet)&=\{f\in\cD(L_{m^2}^{\max})\ |\
\Wr(q_0,f;0)=0\},\quad m=0,
\end{align}

then we say that the scattering length of $H_\bullet$ is $a=\infty$.

§ INTEGRAL OPERATORS

In this appendix we recall a well-known property of Volterra operators that was used in Section <ref>.
We start with the following easy lemma:

Suppose that $(x,y)\mapsto K(x,y)$ is a measurable function on $]0,\infty[\times]0,\infty[$ such that

\begin{equation*}
\sup_{x}\int_0^\infty |K(x,y)|\mathrm{d}y=:C<\infty.
\end{equation*}

Then the operator $K$ defined by

\begin{equation*}
Kf(x):=\int_0^\infty K(x,y)f(y)\mathrm{d}y
\end{equation*}

is bounded on $L^\infty]0,\infty[$ and $\|K\|\leq C$.

Given an integral operator $K$ as in the previous lemma and $a>0$, the operator $K^{(a)}$ is defined as an
operator on  $L^\infty]0,a[$ by the kernel

\begin{equation*}
\end{equation*}

The operator $K^{(a)}$ is called the compression of $K$ to $]0,a[$.

We will say that the operator $K$ with the kernel $K(x,y)$ is a
forward, respectively backward  Volterra operator if $K(x,y)=0$ for $x<y$,
respectively $x>y$. The following proposition can be proven by an induction argument.

Suppose that $K$ is a forward Volterra operator and

\begin{equation*}
\int_0^\infty |K(x,y)|  \mathrm{d}y=:C(x)<\infty.\]
Then for any $a>0$, $K^{(a)}$ is bounded
and $\one^{(a)}+K^{(a)}$ is invertible on
$L^\infty]0,a[$. Besides,
for all $x>0$,  $n\in\mathbb{N}$,
\begin{equation*}
|   ( K^nf)(x)|\leq\frac1{n!}C(x)^n\mathop{\mathrm{ess\, sup}}_{y<x} |f(y)|.
\end{equation*}
so that for  $f\in  L^\infty]0,a[$ the  series
\begin{equation*}
(\one+K)^{-1}f(x)=\sum_{n=0}^\infty(-K)^n f(x)
\end{equation*}
is convergent.
\end{proposition}

\section{Holomorphic families of closed operators}
\label{Holomorphic families of closed operators}

In this appendix we recall the concept of a holomorphic family of
operators on a complex Banach space $\mathcal{H}$.

Let  $\Theta$ be an open subset of $\mathbb{C}^d$.
We say that a family $\{ B(z) \}_{ z \in \Theta }$ of bounded
operators on $\cH$ is a {\em holomorphic family of bounded
if for any $f,g\in\cH$
\begin{equation}\label{rwr}
\Theta\ni z\mapsto (f|B(z)g)\end{equation}
is holomorphic. Note that this is equivalent to a weaker condition:
$\{ B(z) \}_{ z \in \Theta }$ is locally bounded on $\Theta$ and
there exists a dense subspace $\mathcal{D} \subset \mathcal{H}$ such
that, for all $f , g \in \mathcal{D}$, the map \eqref{rwr} is holomorphic.

One can also introduce another concept: that of {\em holomorphic  families of
closed operators}. We will not give here its general definition, which
can be found e.g. in [26, 7, 18] and will not be
used here.
We will restrict ourselves to defining this concept for families that
have nonempty resolvent set.

More precisely,
suppose that $\{ H(z) \}_{ z \in \Theta }$ is a function with values
in closed operators on $\cH$.
Suppose that  for any $z_0 \in \Theta$, there exist $\lambda  \in
\mathbb{C}$ and a neighborhood $\Theta_0 \subset \Theta$ of $z_0$ such
that, for all $z \in \Theta_0$, $\lambda $ is in the resolvent set of
$H(z)$. Then we say that  $\{ H(z) \}_{ z \in \Theta }$ is holomorphic
if for all such $\Theta_0$ the map $\Theta_0 \ni z \mapsto ( H(z) - \lambda  )^{-1} \in \mathcal{L}( \mathcal{H} )$ is holomorphic  as a family of bounded operators.

\section{Technical lemma}

The following easy lemma was used several times in the main part of the manuscript.
\begin{lemma}\label{lm:integr}
Let $a>0$ or $a=\infty$. Let $f\in L^1_{\mathrm{loc}}]0,a[$  and $h:]0,a[\to\mathbb{R}$ be a positive increasing function such that $h(x)\to0$ as $x\to0$ and
\begin{equation*}
\int_0^a h(x) |f(x)| \mathrm{d}x < \infty.
\end{equation*}
\begin{equation*}
\int_x^a f(y) \mathrm{d}y = o \big ( h(x)^{-1} \big ) , \quad x \to 0.
\end{equation*}
\end{lemma}
\begin{proof}
\begin{equation*}
w(x,y) := h(x) |f(y)| \mathds{1}_{[x,a[}(y).
\end{equation*}
Clearly, for a.e. $y\in]0,a[$, $w(x,y) \to 0$ as $x\to0$. Moreover, since $h$ is increasing,
\begin{equation*}
w(x,y) \le h(y) |f(y)| ,
\end{equation*}
for all $x,y\in]0,a[$. Since $y\mapsto h(y) |f(y)|$ is integrable on $]0,a[$ by assumption, the dominated convergence theorem implies that
\begin{equation*}
\int_0^a w(x,y) \mathrm{d}y \to 0 , \quad x \to 0.
\end{equation*}
This proves the lemma.
\end{proof}

\renewcommand{\abstractname}{Acknowledgements}

\begin{abstract}
We thank K. Yajima for useful comments. J.D. acknowledges the support from the National Science Centre (NCN
Project Nr. 2018/31/G/ST1/01166).
\end{abstract}

\begin{thebibliography}{10}

[1] Alekseeva, V.S., Ananieva, A.Yu., \textit{On extensions of the Bessel operator on a
finite interval and a half-line}, J. Math. Sci. 187, 1–8 (2012)
[2]
Anan’eva, A.Yu., Budyka, V.S., \textit{On the spectral theory of the Bessel operator
on a finite interval and the half-line}. Differ. Equ. 52(11), 1517–1522 (2016)

[3] R. Adami and A. Teta, \textit{On the Aharonov-Bohm Hamiltonian}, Lett. Math. Phys. \textbf{43}, (1998), 43–54.

[4]
Boitsev, A.A., Brasche, J.F., Malamud, M.M., Neidhardt, H., Popov, I.Y.,
\textit{Boundary triplets, tensor products and point contacts to reservoirs},
Annales Henri Poincaré  \textbf{19}, (2018),  2783–2837.

[5]   A. Bottino,
\textit{A retrospective look at Regge poles},

[6] W.Bulla, F. Gesztesy, \textit{Deficiency indices and singular boundary conditions in quantum mechanics},
J. Math. Phys. \textbf{26}(10), (1985), 2520–2528.

[7]
L. Bruneau, J. Derezi{\'n}ski and V. Georgescu, \emph{Homogeneous Schr\"odinger operators on half-line}, Annales Henri Poincar\'e \textbf{12}, (2011), 547--590.

[8] K.M. Case, \textit{Singular Potentials}, Phys. Rev. \textbf{80}, 797 (1950).

[9] M. Correggi,  D. Fermi,
\textit{Magnetic Perturbations of Anyonic and Aharonov-Bohm Schrödinger Operators},
J. Math. Phys. \textbf{62}(3), (2021), 032101.

[10] E.A. Coddington, N. Levinson, Theory of Ordinary
\textit{McGraw-Hill, New York}, (1955).

[11] V. De Alfaro, T. Regge, Potential Scattering. \textit{North Holland Publishing Co., Amsterdam}, (1965).

[12]
J. Derezi{\'n}ski, J. Faupin, Q.N. Nguyen and S. Richard, \emph{On radial Schr\"odinger operators with a Coulomb potential: general boundary conditions}, Adv. Oper. Theory \textbf{5} (2020), 1132--1192.

[13]
J. Derezi{\'n}ski and V. Georgescu, \emph{$1$-dimensional Schr{\"o}dinger operators with complex potentials}, Annales Henri Poincar\'e \textbf{21} (2020), 1947--2008.

[14]
J. Derezi\'{n}ski and V. Georgescu, \emph{On the domains of Bessel operators}, Annales Henri Poincare 22 (2021), 3291-3309,

[15]
J. Derezi\'{n}ski and O. Grocholski, Momentum approach to the $1/r^2$ potential as a toy model of the Wilsonian renormalization pdf file, J. Math. Phys. 63, 013504 (2022)

[16]
J. Derezi{\'n}ski and S. Richard, \emph{On Schr\"odinger operators with Inverse Square Potentials on the Half-line}, Annales Henri Poincar\'e \textbf{18}, (2017), 869--928.

[17]
J. Derezi{\'n}ski and S. Richard, \emph{On radial Schr\"odinger operators with a Coulomb potential}, Annales Henri Poincar\'e \textbf{19}, (2018), 2869--2917.

[18] J. Derezi{\'n}ski, M. Wrochna,
\emph{Continuous and holomorphic functions with values in closed operators}, J. Math. Phys. \textbf{55}, (2014), 083512.

[19] W. M. Derkach, M. M. Malamud, \emph{Theory of
extensions of symmetric operators and boundary problems}, National
Academy of Sciences of Ukraine, Kiiv 2017 (Russian)

[20] N. Dunford, J.~T. Schwartz, Linear Operators
vol. III, Spectral Operators, Chap. XX.1.1 Spectral Differential
Operators of Second Order, \textit{Wiley Interscience}, (1971).

[21] D.E. Edmunds, W.D. Evans, Spectral Theory
and Differential Operators, \textit{Oxford University Press}, 2nd ed.,

[22]
F. Gesztesy, \textit{On the one-dimensional Coulomb Hamiltonian}, J. Phys. A \textbf{13}(3), (1980), 867–875.

[23]
F. Gesztesy, M.M.H. Pang, J. Stanfill, \textit{On domain properties of Bessel-type operators}, arXiv:2107.09271, (2021).

[24]
F. Gesztesy, M. Zinchenko, Sturm–Liouville Operators, Their Spectral
Theory, and Some Applications,
Volume I, in preparation.

[25] D.M. Gitman, I.V. Tyutin, B.L. Voronov, Self-adjoint extensions in quantum mechanics.  General theory  and applications to Schr\"odinger and Dirac equations with singular  potentials, vol. 62 of Progress in Mathematical
Physics. \textit{Birkh\"auser/Springer, New York}, (2012).

[26] T. Kato, Perturbation Theory for Linear
Operators, \textit{Springer}, (1966).

[27] T. Kato, \textit{Holomorphic families of Dirac operators},  Math. Zeitschrift \textbf{183}, (1983), 399--406.

[28] H. Kova\v{r}ik, F. Truc, \textit{Schr\"odinger operators
on a half-line with inverse square
potentials}, Math. Model. Nat. Phenom. \textbf{9}(5), (2014), 170–176.

[29]
B.M. Levitan, I.S. Sargsjan, Introduction to spectral theory: selfadjoint ordinary differential operators. Translations of Mathematical Monographs, Vol. 39. \textit{American Mathematical Society, Providence, R.I.}, (1975).

[30] E. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason,
The Mathematics of the Bose Gas and its Condensation, \textit{Birkh\"auser}, (2000).

[31]
V.A. Marchenko, Sturm-Liouville operators and applications. Revised edition. \textit{AMS Chelsea Publishing, Providence, RI}, (2011).

[32]
K. Meetz, \textit{Singular potentials in nonrelativistic quantum mechanics}, Nuovo Cimento \textbf{34}, (1964), 690–708.

[33]  M.A. Naimark, \textit{Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second
order on a semi-axis}, Amer. Math. Soc. Transl. Ser. 2 \textbf{16}, (1960), 103--193.

[34] K. Pankrashkin, S. Richard, \textit{Spectral and
scattering theory for the Aharonov-Bohm
operators}, Rev. Math. Phys. \textbf{23}(1), (2011), 53–81.

[35] M. Reed, B. Simon,
Methods of Modern Mathematical Physics, II. Fourier
Analysis, Self-Adjointness. \textit{Academic Press, London}, (1975).

[36]
A.M. Savchuk, A.A. Shkalikov, \textit{Sturm-Liouville Operators with
Singular Potentials}, Mathematical Notes \textbf{66}, (1999), 741--753.

[37]
A.M.  Savchuk, A.A. Shkalikov, \textit{Sturm-Liouville Operators with
Distribution Potentials},
Trans. Mosc. Math. Soc.
% Trudy Moskovskogo Matematicheskogo Obshchestva,
\textbf{64}, (2003) 143--192.

[38]
E.C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, vol. I, 2nd edn., \textit{Oxford University Press, Oxford}, (1962).

[39]
G.N. Watson, A Treatise on the Theory of Bessel  Functions. \textit{Cambridge University
Press, Cambridge, England; The Macmillan Company, New York}, (1944).

[40]
J. Weidmann, Spectral theory of ordinary differential operators. \textit{Lecture Notes in Mathematics, 1258. Springer-Verlag, Berlin}, (1987).

[41] D. R. Yafaev, Mathematical Scattering Theory: Analytic Theory,
\textit{Mathematical Surveys and Monographs}, vol. 158, (2010)

\end{thebibliography}

\end{document}