# A Multi-Heuristic Search-based Motion Planning
for Automated Parking ††thanks: *Work done while at Virtual Vehicle Research
GmbH.

1st Bhargav Adabala  Virtual Vehicle Research GmbH
Graz, Austria
<EMAIL_ADDRESS>2nd Zlatan Ajanović  Virtual Vehicle Research GmbH
Graz, Austria
0000-0003-2214-0007

###### Abstract

In unstructured environments like parking lots or construction sites, due to
the large search-space and kinodynamic constraints of the vehicle, it is
challenging to achieve real-time planning. Several state-of-the-art planners
utilize heuristic search-based algorithms. However, they heavily rely on the
quality of the single heuristic function, used to guide the search. Therefore,
they are not capable to achieve reasonable computational performance,
resulting in unnecessary delays in the response of the vehicle. In this work,
we are adopting a Multi-Heuristic Search approach, that enables the use of
multiple heuristic functions and their individual advantages to capture
different complexities of a given search space. Based on our knowledge, this
approach was not used previously for this problem. For this purpose, multiple
admissible and non-admissible heuristic functions are defined, the original
Multi-Heuristic A* Search was extended for bidirectional use and dealing with
hybrid continuous-discrete search space, and a mechanism for adapting scale of
motion primitives is introduced. To demonstrate the advantage, the Multi-
Heuristic A* algorithm is benchmarked against a very popular heuristic search-
based algorithm, Hybrid A*. The Multi-Heuristic A* algorithm outperformed
baseline in both terms, computation efficiency and motion plan (path) quality.

###### Index Terms:

Motion Planning, Automated Driving, Multi-Heuristic Search, A* Search

##  I Introduction

Robot motion planning problems can be elegantly formulated as path planning in
higher-dimensional configuration space [1]. However, finding a solution is
computationally challenging due to a large continuous search space and
kinodynamic constraints. The sampling-based approaches for motion planning
have been extensively studied in robotics [2]. Instead of explicitly
constructing the collision-free configuration space, which is time-consuming
to compute, these algorithms probe the free space and search with a sampling
strategy. The algorithms stop when a path connecting the initial and final
poses is found. According to the sampling type, the sampling-based path-
finding algorithms can be classified into two categories: random-sampling-
based algorithms and orderly-sampling-based algorithms.

The most popular random-sampling-based algorithm is the Rapidly-exploring
Random Tree (RRT) [3]. RRT can be considered as a special case of Monte Carlo
Tree Search (MCTS) [4]. The most notable orderly sampling-based algorithm is
A* [5], including many of its extensions. It was initially developed to plan a
path for the Shakey robot and it was further generalized and used for many
different domains since then.

Figure 1: Motion Planning for Autonomous Parking.

The orderly sampling-based algorithms tend to be more efficient than random
sampling-based planners, especially when the dimensions of the state spaces
are fewer than six [6]. Random sampling-based planners inherently come with
the disadvantages of being highly non-deterministic and converging towards
solutions that are far from the optimum and suffer from bug trap problems [7].
The variants of RRT such as RRT* also provide comparable solutions, but they
are computationally expensive and the efficiency depends on the size of the
search space [8]. Furthermore, if no collision-free path to the goal exists,
orderly sampling-based algorithms can report this failure much more quickly
than random sampling-based ones. However, original A* deals with discrete
state-space and requires discretization of the configuration space. Continuous
state-space and kinodynamic constraints can be satisfied by constructing a
lattice in the form of a regular grid [9]. Another approach is to use motion
primitives [10]. To avoid the problem of rounding states generated using
motion primitives to the grid, Hybrid-State A* [11] might be used.

Sampling-based motion planning approaches were extensively used for autonomous
vehicle path planning in unstructured environments during the 2007 DARPA Urban
Challenge, both A*-based [12], [11] and RRT-based [13]. Several extensions for
these approaches have been introduced in recent years, namely A*-based [14,
15, 16], RRT*-based [17] and optimization-based [18]. An overview of recent
developments and open challenges is presented in [19].

Besides motion planning for wheeled vehicles in unstructured environments
(e.g. autonomous parking), different variants of search-based planning were
recently used for planning footsteps for humanoid robots [20], robot
manipulation [21], underwater vehicles [22], the aggressive flight of UAVs
[23], as well as for special use-cases in automated driving such as energy-
efficient driving [24], driving in complex scenarios in structured urban
environments [25], unstructured and partially observable environments [15] and
performance driving including drifting maneuvers [26][27].

This paper presents a motion planning approach for wheeled vehicles in
unstructured environments based on Multi-Heuristic Search [28]. Based on our
knowledge, this is the first application of a Multi-Heuristic Search for
motion planning in automated parking scenarios. This is achieved by using a
combination of geometric and orderly sampling-based approaches in order to
achieve maximum coverage of the complexities within the search space. For this
purpose, two heuristic functions are defined to get an accurate estimate of
the cost-to-go and prune the unnecessary search nodes for faster path
computation. The geometric approach is used to solve the simplified problem
(without obstacles) by modeling the physical constraints of the vehicle and is
used as a second heuristic function in a Multi-Heuristic A* algorithm, the
first being the path length to the destination while considering obstacles but
neglecting some physical constraints like turning radius. With this approach,
both non-holonomic and holonomic constraints are combined to provide an
optimal solution to the motion planning problem. Two approaches for the
solution have been developed, one using Forward Search and the other using Bi-
directional Search. Additionally, adaptive motion primitive arc length was
developed to avoid the search getting stuck in depression regions
indefinitely. An earlier version of this work is presented in the ICAPS 2020
PlanRob workshop [29].

##  II Autonomous Parking as Motion Planning Problem

The autonomous parking problem tackled in this work represents a fully
observable problem where intelligent infrastructure provides a connected
vehicle with information about the structure and area of the parking lot. The
driver enters the parking and drives the vehicle into a designated drop-off
area. The autonomous parking algorithm overtakes a control and guides and
maneuvers the vehicle into the assigned parking slot automatically based on
the information from the infrastructure. This concept was demonstrated by
Bosch and Daimler as Automated Valet Parking (AVP) [30].

The planning problem this work aims to solve can be stated as follows:

”Find a solution in real-time that autonomously navigates a non-holonomic
vehicle without any collisions, from a given start position to a desired goal
position within the parking layout based on the input of a two-dimensional
obstacle map, or report the non-existence of such a solution.”

To fully define the problem, the environment (obstacles), the vehicle model,
and Key Performance Indicators (KPIs) must be defined.

###  II-A Environment

Figure 2: Parking with perpendicular parking slots

Figure 2, shows an example of the parking layout structure which is considered
for the motion planning problem of autonomous parking. There might be two most
prominent parking orientations, perpendicular or parallel. However, the
algorithm should be general to work on different parking arrangements. Each
parking slot is enumerated with a parking slot ID. The number of parking slots
and dimensions of the open space are configurable. The red cross symbols in
the figure highlight the goal position for each parking slot within the
layout. The entry point to the parking lot is fixed at
$(x_{s},y_{s},{\theta}_{s})=(0,10,0)$. The parking position for the respective
parking slots ID is read from a pre-computed map. The computation of goal
position within each parking slot is based on the vehicle dimensions as the
control reference would be different for the individual vehicle due to
differences in length and width.

###  II-B Vehicle Model

Due to vehicle geometry and physics, there are constraints on the vehicle
motion that restrict the allowable velocities. The first-order constraints,
that consider the first derivative of the position (velocity), are often
called kinematic constraints. Including the dynamics of a vehicle results in
second-order differential constraints, which allows the modeling of
acceleration. The planning with such models is called kinodynamic planning. As
the focus of this work is on low-velocity parking maneuvers, higher-order
constraints are not included, only kinematic constraints are considered.

(a) Single-track vehicle model.

(b) 3 Disk approximation for vehicle geometry.

Figure 3: Vehicle models.

A simple yet useful model of a car is the single track model, also known as
the bicycle model, shown in Figure 3(a). It does not consider dynamics, but it
is useful to model lower-velocity driving. Consider the case where a car with
wheelbase $L$ moves forward with velocity $v_{x}$ with a steering angle
$\alpha$ and assuming no wheel slip, then the car will move along a circle
with radius $R$. The kinematic constraints can then be derived by
trigonometry. Let x = $(x,y,\theta)$ denote the configuration state of the
car-like robot, where $x$ and $y$ denote the position and $\theta$ the heading
of the car. Vehicle motion is constrained to $\dot{x}/\dot{y}=\tan(\theta)$,
which together with the constraint $R={L}/{\tan(\alpha)}$ gives the following
first-order differential constraints:

$\dot{x}=v_{x}\cos(\theta)$ (1) $\dot{y}=v_{x}\sin(\theta)$ (2)
$\dot{\theta}=\frac{v_{x}}{L}\tan(\alpha)$ (3)

Setting the maximum steering angle $|\alpha|\leq{\alpha}_{\mathrm{max}}$
results in a minimum turning radius $R_{\mathrm{min}}$. The model clearly
represents the non-holonomic behavior as it is impossible to move sideways
without violating the no-slip condition. Restricting the allowed velocities
and steering angles to the finite set $\mathcal{U}_{v_{x}}=\\{0,1\\}$ and
$\mathcal{U}_{\alpha}=\\{-{\alpha}_{\mathrm{max}},0,{\alpha}_{\mathrm{max}}\\}$
results in the Dubin’s car that can only stop and move forward at unit speed
[31] and setting it to $\mathcal{U}_{v_{x}}=\\{-1,0,1\\}$ results in the Reed-
Shepp car that can also reverse at a unit speed [32]. Even though these models
are very simplified they have efficient analytic solutions for optimal paths
between any two states that can be useful for designing heuristic functions
for search-based motion planners.

A time-discretized single-track model is easily obtained using Euler forward
or higher-order methods. By integrating the differential equations forward in
time, a simulated path or trajectory resulting from a given input can be
obtained and used to construct motion segments within a planning framework.

###  II-C Collision Detection

To avoid collisions with obstacles in the environment, vehicle geometry should
be considered. The geometry of a car, approximate rectangular shape, can be
reasonably approximated with overlapping circular disks. This simplifies the
collision detection significantly as it is sufficient to check if the
obstacles fall within the boundaries of the disk, which is represented by the
radii of the disks. The problem of covering rectangles with equal-sized disks
in an optimal way has been studied in the literature [33].

As stated in [34], a rectangle of length $l$ and width $w$ can be covered by
$n$ circles of radius $r$ calculated as:

${r=\sqrt{\frac{l^{2}}{n^{2}}+\frac{w^{2}}{4}}}$ (4)

placed at a distance of $d$ calculated as:

${d=2\sqrt{{r^{2}}-\frac{w^{2}}{4}}}$ (5)

In practical applications, the above approximation may lead to under-
utilization of available collision-free space, especially in the case of
environments where tight or narrow maneuvering is required, e.g. parking lots.
Besides, the stated approach assumes the control reference to be at the center
of the rectangular shape which is not true in the case of car-like robots
which are mostly either front-wheel or rear-wheel driven.

In this work, an approach proposed in [34] is adapted to fit a practical
application of a car-like robot. The bounding disks can be arranged as shown
in Figure 3(b). By this method, the bounding disks are arranged compactly to
fit the geometry of the vehicle allowing better utilization of the free space
even for tight maneuvers in narrow spaces. Moreover, all the calculations are
based on the standard dimensions available from any production car design, and
parametrizing the same makes the approach generic for any vehicle under
consideration.

###  II-D KPI Definition for Benchmark

To compare the performance and quality of solutions generated by motion
planning algorithms, the following Key Performance Indicators (KPIs) have been
used.

Performance Paramters

Number of Expanded States: For a given configuration space, the number of
expanded states reflects the guidance power of the heuristic functions in
pruning the unwanted branches of the search. The lesser the number of expanded
states, the better the heuristic.

Execution Time: The execution time depends on the implementation of the
vehicle model, the definition of motion primitives, and the algorithm itself.
It is a measure of the time that the algorithm needs to return a solution
using the defined attribute functions.

Number of Iterations: The iteration counter quantifies how quickly the
algorithm converges to either finding a solution or reporting that there
exists no solution.

Solution Path Quality Parameters

Path Length: The path length is computed as the accumulated sum of Euclidean
distance between two points on the final trajectory. It quantifies the
efficiency of the generated solution as shorter path lengths are preferred.

Reverse Path Length: The reverse path length indicates the quality of the
algorithm to foresee a wrong branch. The longer reverse path length indicates
that the vehicle had to move a lot in a backward direction in order to correct
its path or in some cases the algorithm prefers to move the vehicle more in a
backward direction rather than forward. In any case, longer reverse path
lengths are not preferred.

Direction Changes: Each direction change during driving indicates a stop-and-
go situation, which will be annoying for a human driver. Even though it is an
autonomous vehicle the quality solution shall be close to an experienced human
driver, i.e., avoid multiple direction changes.

##  III Motion Planning Approach

The motion planning approach presented in this paper is based on Multi-
Heuristic A* search, extended with a concept from the Hybrid A* Algorithm,
employed in a bi-directional search fashion. The kinodynamic feasibility of
the solution is provided by motion primitives based on the vehicle model.
Several admissible heuristic functions enable efficient optimal planning.

###  III-A Hybrid A* Algorithm

The Hybrid A* algorithm was developed as a practical path-planning algorithm
that can generate smooth paths for an autonomous vehicle operating in an
unstructured environment and used in the DARPA Urban Challenge by the Stanford
University team [11]. The hybrid A* algorithm is based on the A* algorithm,
with the key difference being, that state transitions occur in continuous
rather than in a discrete space. By considering the non-holonomic constraints
of the robotic vehicle, the algorithm generates feasible transitions which can
be executed by the actuator module.

The three-dimensional state space $\mathcal{X}$ (represented by $x,y$ position
and $\theta$ heading angle of the vehicle) is associated with a discrete grid
of reasonable resolution such that each continuous state is associated with
some grid cell to enable the use of discrete search algorithm. Continuous
states are rounded to the grid for association in order to prune the branches,
by keeping only the best trajectory coming to the grid cell. The expansion
still uses the actual continuous value that is not rounded to the grid.
Similar to the original A*, if the current state being expanded is not the
goal state, new successors are generated for all possible actions
$u\in\mathcal{U}(\textbf{x})$. The cost-to-come is only computed for successor
states that are not in the Closed list. If the state is not in the Open list
it is directly pushed to the Open list. If the state is already in the Open
list, and the cost-to-come is smaller than the cost for a state with the same
index that is in the Open list then the pointer to the parent, the cost-to-
come and the cost-to-go are updated. After that, the key is decreased using
the newly computed cost.

###  III-B Multi Heuristic A* Algorithm

The performance of the A* algorithm depends on the quality of the heuristic
function used to guide the search. It is hard to design a single heuristic
function that captures all the complexities of the problem. Furthermore, it is
hard to ensure that heuristics are admissible (provide lower bounds on the
cost-to-go) and consistent, which is necessary for an A*-like search to
provide guarantees on completeness and bounds on sub-optimality.

In [35] authors introduced an approach of alternation between different
heuristic functions for satisficing (i.e. non-optimal) planning. Multi-
Heuristic A* (MHA*) [28] overcomes the dependency on a single heuristic
function in optimal planning too. MHA* can use multiple inadmissible heuristic
functions in addition to a single consistent heuristic simultaneously to
search in a way that preserves guarantees on completeness and bounds on sub-
optimality. This enables us to effectively combine the guiding powers of
different heuristic functions and simplifies dramatically the process of
designing heuristic functions by a user because these functions no longer need
to be admissible or consistent [28].

MHA* has two variants: Independent Multi-Heuristic A* (IMHA*) which uses
independent cost-to-come and cost-to-go values for each search, and Shared
Multi-Heuristic A* (SMHA*) which uses different cost-to-go values but a single
cost-to-come value for all the searches. With this shared approach, SMHA* can
guarantee the sub-optimality bounds with at most two expansions per state. In
addition, SMHA* is potentially more powerful than IMHA* in avoiding depression
regions as it can use a combination of partial paths found by different
searches to reach the goal [28].

In SMHA* approach the optimal path for a given state is shared among all the
searches so that if a better path to a state is discovered by any of the
searches, the information is updated in all the priority queues. This allows
the algorithm to expand each state at most twice, which significantly improves
the computational time.

###  III-C Our Planning Algorithm

1

2function  _$\texttt{SMHA}^{\ast}$(_$x_{\mathrm{I}}$ , $X_{\mathrm{G}}$,
$\mathcal{O}$,$h_{i}$_)_:

3 return _Collision free trajectory from $x_{\mathrm{I}}$ to $x\in
X_{\mathrm{G}}$_

4

5function  _CombinePath(_$(x_{\mathrm{start}},\dots,x_{\mathrm{f}}^{\prime})$
,$(x_{\mathrm{f}}^{\prime\prime},\dots,x_{\mathrm{G}})$_)_:

/* Use an analytic function to connect paths */

6 return _Combined trajectory from $x_{\mathrm{start}}$ to
$x_{\mathrm{goal}}$_

7

8begin

/* Forward Search */

9 $X_{\mathrm{G}}\leftarrow\\{x\;|\;\lVert x-x_{\mathrm{G}}\rVert\leq
d_{\mathrm{fw1}}\\}$;

10 $(x_{\mathrm{start}},\dots,x_{\mathrm{f}}^{\prime})$ $\leftarrow$
$\texttt{SMHA}^{\ast}(x_{\mathrm{start}},X_{\mathrm{G}},\mathcal{O},h_{i})$;

/* Reassign Start and Goal Positions */

11 $x_{\mathrm{start}}^{\prime}$ $\leftarrow$ $x_{\mathrm{goal}}$;

12 $X_{\mathrm{G}}^{\prime}\leftarrow\\{x\;|\;\lVert
x-x_{\mathrm{f}}^{\prime}\rVert\leq d_{\mathrm{fw2}}\\}$;

/* Backward Search */

13 $(x_{\mathrm{G}},\dots,x_{\mathrm{f}}^{\prime\prime})$ $\leftarrow$
$\texttt{SMHA}^{\ast}(x_{\mathrm{start}}^{\prime},X_{G}^{\prime},\mathcal{O},h_{i})$;

/* Combine Paths */

14
$path\leftarrow\texttt{CombinePath}((x_{\mathrm{start}},\dots,x_{\mathrm{f}}^{\prime}),(x_{\mathrm{f}}^{\prime\prime},\dots,x_{\mathrm{G}}))$;

15

16

17return _path_

Algorithm 1 Bi-Directional Multi-Heuristic Search

The presented planning algorithm is based on MHA* Search with features of
Hybrid A* search and adaptive motion primitives. It uses MHA* in a forward and
backward manner as shown in Algorithm 1. For this purpose, multiple admissible
and non-admissible heuristic functions are defined. The framework has three
main functions SharedMultiHeuristicA*, GeneratePath, and CombinePath.
SharedMultiHeuristicA* searches the configuration space according to the
algorithm defined in [28]. It is expanded with hybrid A* features and uses
motion primitives. It is used in both forward and backward steps. GeneratePath
executes the search using SharedMultiHeuristicA* while continuously checking
if the EucledianDistance from the current state to the goal state is greater
than a configurable parameter $d_{\mathrm{fw}}$. If the search has reached the
closest state defined by $d_{\mathrm{fw}}$ then the function returns the path
from the start position to the closest point to the goal. Due to the
discretization of the continuous space, in CombinePath an analytic function is
required to combine the two paths generated by GeneratePath function. In this
work, we used Reeds-Shepp curves for this purpose.

####  III-C1 Bi-Directional Search

The presented algorithm is using the Bi-directional Search, by searching for
the path in two steps. In the first step, the search expands in the forward
direction (towards the goal) from the start state to reach the state close to
the goal position. In the second step, the search proceeds backward from the
goal position toward the closest point reached by the forward search. The
solution paths generated by the forward search step and backward search step
are then joined by the analytical expansion using Reeds-Sheep curves.

####  III-C2 Motion Primitives

The motion primitives refer to the motion sequence that is triggered by an
action request and corresponds to a basic move that is possible by the
vehicle, sampled from a continuous control space. In this work, motion
primitives are generated by applying one of the six control actions defined by
combinations of $\mathcal{U}_{v_{x}}=\\{-1,1\\}$ and
$\mathcal{U}_{\alpha}=\\{-{\alpha}_{\mathrm{max}},0,{\alpha}_{\mathrm{max}}\\}$.
These represent maximum steering left while driving in the forward direction,
no steering while driving in the forward direction, maximum steering right
while driving in the forward direction, maximum steering left while driving in
the backward direction, no steering while driving in the backward direction,
maximum steering right while driving in the backward direction. Finer
resolution is also possible, however that increases the branching factor and
computational complexity.

Figure 4: Motion Primitives.

As shown in Figure 4, each of these control actions is applied for a certain
amount of time, resulting in an arc of a circle with a lower bound turning
radius $R_{\mathrm{min}}$. This will ensure that the resulting paths are
always drivable, as the actual vehicle model is used to expand the state, even
though they might result in excessive steering actions. An adaptive sizing of
motion primitives is applied, wherein the arc length used for the execution of
motion primitives is adapted dynamically to adjust to the environment. A
shorter arc length is used near obstacles and a longer arc length in free
space. This approach improves maneuverability in tight spaces. Using a shorter
length in all cases promises higher levels of resolution completeness, as the
likelihood to reach each state is increasing but reduces the computational
efficiency.

####  III-C3 Heuristics

A heuristic function $\mathit{h}$ is used to estimate the cost needed to
travel from some state $x$ to the goal state $x_{g}$ (cost-to-go). As it is
shown in [5], if the heuristic function is underestimating the optimal cost-
to-go, A* search provides the optimal solution. For the shortest path search,
the usual heuristic function is the Euclidean distance.

In general, SMHA* algorithm supports $n$ number of heuristics with $n>1$. In
this work, to restrict the complexity and to be comparable with Hybrid A*
which is used as a reference for benchmarking, two heuristic functions have
been used. The two heuristics capture different aspects of the problem as
explained in the sections below.

Non-Holonomic without Obstacles This heuristic function takes into account the
non-holonomic constraints of the vehicle while neglecting the influence of the
environment (obstacles). The most suitable candidate functions are either
Dubins or Reeds-Shepp curves. These curves are the paths of minimal length
with an upper bound curvature for the forward and combined forward and
backward driving car respectively. We choose the Reeds-Shepp curves since in
parking maneuvers it is important that the car can move in both forward and
backward directions. These curves are computationally inexpensive to compute
as they are based on an analytic solution. As shown in Figure 5(a), this
heuristic takes into account the current heading as well as the turning
radius, which ensures that the vehicle approaches the goal with the
appropriate heading. This is especially important when the car gets closer to
the goal. Given that Reeds-Shepp curves are minimal, this heuristic is clearly
admissible.

(a) Non-holonomic without obstacles (Reeds-Shepp Curve).

(b) Holonomic with obstacles.

Figure 5: Paths based on relaxed models of heuristic functions.

Holonomic with Obstacles This heuristic function neglects the characteristics
of the vehicle and only accounts for obstacles. The estimate is based on the
shortest distance between the goal state and the state currently being
expanded. This distance is determined using the standard Dijkstra search in
two dimensions ($x$ and $y$ position). As the search is 2D and assumes the
object under control is holonomic, the path is not smooth. The search is
performed backward, it uses the initial state of the SMHA* as the goal state,
and the goal state of the SMHA* search as the start state to generate the
heuristic cost. The closed list of the Dijkstra search stores all the shortest
distances to the goal and guides the vehicle away from dead ends and around
obstacles. Since this heuristic function does not depend on any runtime sensor
information, it can be fully pre-computed offline and used as a lookup table
or simply translated and rotated to match the current goal instead of
initiating a new search while SMHA* progresses.

##  IV Simulation results

In order to benchmark the performance of the solution developed using SMHA*,
we chose the Hybrid A* as a reference. As discussed earlier, Hybrid A* gives a
comparable reference as it is also based on the orderly sampling approach and
also uses two heuristics to guide the search. The key difference is that in
the Hybrid A* approach, the maximum of both of the results of the heuristics
is considered to update the priority queue while in SMHA* the heuristics are
iteratively computed and both can update the priority queue.

The use cases chosen for the simulation depict common situations encountered
in a parking lot such as Entering Parking Lot and Exiting Parking Lot. The
simulations are performed by executing the Hybrid A* and SMHA* algorithm back-
to-back to compare the KPIs of the generated solution.

(a) .

(b)

Figure 6: Parking Lot with Start Position - Cyan and Goal Position - Green
(left) and cost-to-go map generated by 2D Dijkstra search (right).

In the rest of the section, an elaborate analysis of the simulation results
and solution paths that are generated for the use case Entering Parking Lot
using Bi-directional Search for parallel parking configuration is presented.
Figure 6(a) depicts the selected start and goal position (Parking Slot ID: 27)
on the parking layout.

First, the configuration space is explored using the 2D Dijkstra search to
generate the cost-to-go map as shown in Figure 6(b). The cost-to-go map shows
in a color scale distance from the goal pose considering obstacles but
neglecting non-holonomic constraints. The results of this step are stored in a
look-up and represents the Holonomic with Obstacles heuristic explained
earlier.

(a) State expansions.

(b) Path generated by Hybrid A*.

Figure 7: Entering Parking Lot with Bi-Directional - Hybrid A*.

Figure 7(a) depicts the state expansion pattern of the Hybrid A* algorithm.
The algorithm has searched the area around the start position with a bias
towards the goal position even though the solution path lies in another
direction. The heuristic strongly guides the search towards the shortest path
as far as possible within the obstacle-free area. As the search explores, it
expands all states with lower costs that can lead to the shortest path until
it reaches a point where the heuristic cost of expanding the points which do
not lead to the shortest path has a lower cost to reach the goal. As a result
of this poor pruning of the unwanted branches, the planner expands several
states around the start position before it realizes the optimal direction of
the path which leads to poor timing performance. The final path generated as
seen in Figure 7(b) has many orientation changes in the Forward Search step
and maneuvering step into the parking slot is determined by the backward
search. The solution path is smooth, but the direction of orientation is
reversed for the most part of the path which is not optimal.

(a) State expansions.

(b) Path generated.

Figure 8: Entering Parking Lot with Bi-Directional - SMHA*.

Figure 8(a) depicts the state expansion pattern of the SMHA* algorithm.
Similar to Hybrid A*, the algorithm has searched the area around the start
position with a bias towards the goal position even though the solution path
lies in another direction. But, the multi-heuristic approach, quickly balances
the bias towards finding the shortest path to finding a feasible path
considering the obstacles. In addition, due to the mutually informed
independent search by respective heuristics, the states that are expanded by
one heuristic function are not expanded by other heuristic functions. As a
result, the algorithm could prune the unwanted branches and realize the
optimal direction of the path much faster compared to Hybrid A*.

As seen in Figure 8(b), the path is smooth and the direction of orientation is
in the forward direction for most parts of the path, which is preferred.

KPI | Hybrid A* | SMHA*
---|---|---
Number of Expanded States | 2457 | 73
Execution Time (s) | 47.8 | 11.51
Path Length (m) | 90.3 | 90.58
Reverse Path Length (m) | 7.29 | 3.29
Direction Changes | 4 | 4
Number of Iterations | 23633 | 73
TABLE I: Entering parking lot with Bi-Directional Search - KPI Comparision.
Figure 9: Entering Parking Lot with Bi-Directional Search - KPI Comparision
(Expanded States (ES), Execution Time (ET), Forward Path Length (FPL), Reverse
Path Length (RPL), Direction Changes (DC), Iteration Count (IC)).

As seen from KPI values tabulated in Table I, the Hybrid A* algorithm expands
significantly more states and uses more time to generate the solution path
compared to SMHA* approach. Even though the heuristics used are the same, the
mutually informed independent search of SMHA* prunes the unwanted branches
much more significantly allowing faster convergence towards the solution and
improved execution time.

To give a comprehensive performance comparison of both the algorithms for the
use case Entering Parking Lot for parallel parking lot layout using Bi-
directional Search, a full simulation run through all the parking slots is
executed. In this simulation mode, each parking slot ID is selected as a goal
position sequentially and the back-to-back run of the Hybrid A* and SMHA*
algorithm is performed.

Figure 10: Entering Parking Lot with Bi-Directional Search - Mean Performance
Improvement.

As observed from Figure 10, SMHA* outperforms the Hybrid A* algorithm in terms
of both performance and solution path quality parameters. With the multi-
heuristic approach, the average execution time to generate the solution path
is reduced by 81%, which is a significant improvement demonstrating the
potential of multi heuristic approach to solve the given planning problem.

##  V Conclusion

The work focused on providing a Multi-Heuristic search-based approach to solve
the motion planning problem for autonomous parking. To benchmark the results
obtained, a state-of-the-art planning algorithm Hybrid A* was chosen as a
reference.

As the environment for the given use case involves only low-speed maneuvering,
a Single Track Bicycle Model which reflects the kinematics of the vehicle was
used as a motion model. The model emulates the non-holonomic nature of the
vehicle in all stages of the algorithm, motion primitives (node expansion),
and heuristic estimates. Thus, the paths generated are always driveable.

A collision check algorithm was implemented based on the Multi-Disk
Decomposition of the bounded volume. The algorithm was parameterized based on
the vehicle geometry making it a generic solution to fit with any vehicle type
and independent of the environment.

The path planning problem was solved using the Shared Multi-Heuristic A*
approach in which two heuristic functions were implemented with a round-robin
scheduling. The respective heuristic searches share the current path obtained
to a state. The heuristics were defined to capture the non-holonomic and
holonomic constraints of the vehicle. Two solution approaches were developed:
Forward Search and Bi-Directional Search .

The SMHA* algorithm solved the motion planning problem elegantly and
outperformed Hybrid A* with respect to the response time and path quality of
the generated solution path. The KPI comparison clearly indicates that SMHA*
is an ideal candidate for motion planning in slow-speed driving in
unstructured environments applications like autonomous valet parking.

##  VI Acknowledgments

The project leading to this study has received funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No 675999, ITEAM project.

VIRTUAL VEHICLE Research Center is funded within the COMET – Competence
Centers for Excellent Technologies – programme by the Austrian Federal
Ministry for Transport, Innovation and Technology (BMVIT), the Federal
Ministry of Science, Research and Economy (BMWFW), the Austrian Research
Promotion Agency (FFG), the province of Styria and the Styrian Business
Promotion Agency (SFG). The COMET programme is administrated by FFG.

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# On the domains of Bessel operators

Jan Dereziński
Department of Mathematical Methods in Physics,
Faculty of Physics, University of Warsaw,
Pasteura 5,
02-093 Warszawa, Poland,
email<EMAIL_ADDRESS>

Vladimir Georgescu
Laboratoire AGM, UMR 8088 CNRS,
CY Cergy Paris Université,
F-95000 Cergy, France,
email<EMAIL_ADDRESS>
###### Abstract

We consider the Schrödinger operator on the halfline with the potential
$(m^{2}-\frac{1}{4})\frac{1}{x^{2}}$, often called the Bessel operator. We
assume that $m$ is complex. We study the domains of various closed homogeneous
realizations of the Bessel operator. In particular, we prove that the domain
of its minimal realization for $|{\rm Re}(m)|<1$ and of its unique closed
realization for ${\rm Re}(m)>1$ coincide with the minimal second order Sobolev
space. On the other hand, if ${\rm Re}(m)=1$ the minimal second order Sobolev
space is a subspace of infinite codimension of the domain of the unique closed
Bessel operator. The properties of Bessel operators are compared with the
properties of the corresponding bilinear forms.

###### Acknowledgements

Jan Dereziński is grateful to F. Gesztesy for drawing attention to the
references [1, 2, 3] and for useful comments. His work was supported by
National Science Center (Poland) under the grant UMO-2019/35/B/ST1/01651. The
authors thank the referees for their comments.

Keywords: Schrödinger operatros, Bessel operators, unbounded operators,
Sobolev spaces.

MSC 2020: 47E99, 81Q80

##  1 Introduction

###  1.1 Overview of closed realizations of the Bessel operator

The Schrödinger operator on the half-line given by the expression

$L_{\alpha}:=-\frac{{\rm d}^{2}}{{\rm
d}x^{2}}+\Big{(}\alpha-\frac{1}{4}\Big{)}\frac{1}{x^{2}}$ (1.1)

is often called the Bessel operator. The name is justified by the fact that
its eigenfunctions and many other related objects can be expressed in terms of
Bessel-type functions.

There exists a large literature devoted to self-adjoint realizations of (1.1)
for real $\alpha$. The theory of closed realizations of (1.1) for complex
$\alpha$ is also interesting. Let us recall the basic elements of this theory,
following [7, 6].

For any complex $\alpha$ there exist two most obvious realizations of
$L_{\alpha}$: the minimal $L_{\alpha}^{\min}$, and the maximal
$L_{\alpha}^{\max}$. The complex plane is divided into two regions by the
parabola defined by

$\alpha=(1+{\rm i}\omega)^{2},\quad\omega\in{\mathbb{R}},$ (1.2)

(or, if we write $\alpha=\alpha_{\mathrm{R}}+{\rm i}\alpha_{\mathrm{I}}$, by
$\alpha_{\mathrm{R}}+\sqrt{\alpha_{\mathrm{R}}^{2}+\alpha_{\mathrm{I}}^{2}}=2$).
To the right of this parabola, that is, for $|{\rm Re}\sqrt{\alpha}|\geq 1$,
we have $L_{\alpha}^{\min}=L_{\alpha}^{\max}$. For $|{\rm
Re}\sqrt{\alpha}|<1$, that is to the left of (1.2), ${\cal
D}(L_{\alpha}^{\min})$ has codimension $2$ inside ${\cal
D}(L_{\alpha}^{\max})$. The operators ${\cal D}(L_{\alpha}^{\min})$ and ${\cal
D}(L_{\alpha}^{\max})$ are homogeneous of degree $-2$.

Let us note that in the region $|{\rm Re}\sqrt{\alpha}|<1$ the operators
$L_{\alpha}^{\min}$ and $L_{\alpha}^{\max}$ are not the most important
realizations of $L_{\alpha}$. Much more useful are closed realizations of
$L_{\alpha}$ situated between $L_{\alpha}^{\min}$ and $L_{\alpha}^{\max}$,
defined by boundary conditions near zero. (Among these realizations, the best
known are self-adjoint ones corresponding to real $\alpha$ and real boundary
conditions). All of this is described in [7].

Among these realizations for $\alpha\neq 0$ only two, and for $\alpha=0$ only
one, are homogeneous of degree $-2$. All of them are covered by the
holomorphic family of closed operators $H_{m}$, introduced in [6] and defined
for ${\rm Re}(m)>-1$ as the restriction of $L_{m^{2}}^{\max}$ to functions
that behave as $x^{\frac{1}{2}+m}$ near zero. Note that

$\displaystyle L_{m^{2}}^{\min}=$ $\displaystyle
H_{m}=L_{m^{2}}^{\max},\quad{\rm Re}(m)\geq 1;$ (1.3) $\displaystyle
L_{m^{2}}^{\min}\mathchar 13608\relax$ $\displaystyle H_{m}\mathchar
13608\relax L_{m^{2}}^{\max},\quad|{\rm Re}(m)|<1.$ (1.4)

###  1.2 Main results

Our new results give descriptions of the domains of various realizations of
$L_{\alpha}$ for $\alpha\in{\mathbb{C}}$. First of all, we prove that for
$|{\rm Re}\sqrt{\alpha}|<1$ the domain of $L_{\alpha}^{\min}$ does not depend
on $\alpha$ and coincides with the minimal 2nd order Sobolev space

${\cal H}_{0}^{2}({\mathbb{R}}_{+}):=\\{f\in{\cal H}^{2}({\mathbb{R}}_{+})\ |\
f(0)=f^{\prime}(0)=0\\},$ (1.5)

where

${\cal H}^{2}({\mathbb{R}}_{+}):=\\{f\in L^{2}({\mathbb{R}}_{+})\ |\
f^{\prime\prime}\in L^{2}({\mathbb{R}}_{+})\\}$ (1.6)

is the (full) 2nd order Sobolev space. We also show that

$\\{\alpha\ |\ |{\rm Re}\sqrt{\alpha}|<1\\}\ni\alpha\mapsto L_{\alpha}^{\min}$
(1.7)

is a holomorphic family of closed operators.

We find the constancy of the domain of the minimal operator quite surprising
and interesting. It contrasts with the fact that ${\cal D}(L_{\alpha}^{\max})$
for $|{\rm Re}\sqrt{\alpha}|<1$ depends on $\alpha$. Similarly, ${\cal
D}(H_{m})$ for $|{\rm Re}(m)|<1$ depends on $m$.

The holomorphic family $L_{\alpha}^{\min}$ for $|{\rm Re}\sqrt{\alpha}|<1$
consists of operators whose spectrum covers the whole complex plane.
Therefore, the usual approach to holomorphic families of closed operators
based on the study of the resolvent is not available.

We also study $H_{m}$ for ${\rm Re}(m)\geq 1$ (which by (1.3) coincides with
$L_{m^{2}}^{\min}$ and $L_{m^{2}}^{\max}$). We prove that for ${\rm Re}(m)>1$
its domain also coincides with ${\cal H}_{0}^{2}({\mathbb{R}}_{+})$. The most
unusual situation occurs in the case ${\rm Re}(m)=1$. In this case we show
that the domain of $H_{m}$ is always larger than
$\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$ and depends on $m$.

Specifying to real $\alpha$, the main result of our paper can be summarized as
follows: Let $L_{\alpha}^{\min}$ be the closure in $L^{2}({\mathbb{R}}_{+})$
of the operator $-\partial_{x}^{2}+\frac{\alpha-\frac{1}{4}}{x^{2}}$ with
domain $C_{\rm c}^{\infty}({\mathbb{R}}_{+})$.
1) If $\alpha<1$ then $L_{\alpha}^{\min}$ is Hermitian (symmetric) but not
self-adjoint and its domain is $\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$.
2) If $\alpha=1$ then $L_{\alpha}^{\min}$ is self-adjoint and
$\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$ is a dense subspace of infinite
codimension of its domain.
3) If $\alpha>1$ then $L_{\alpha}^{\min}$ is self-adjoint with domain
$\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$.

As a side remark, let us mention two open problems about Bessel operators.

###### Open Problem 1.1

  1. 1.

Can the holomorphic family $H_{m}$ be extended beyond ${\rm Re}(m)>-1$?
(Probably not).

  2. 2.

Can the holomorphic family $L_{\alpha}^{\min}$ (hence also
$L_{\alpha}^{\max}$) be extended beyond $|{\rm Re}\sqrt{\alpha}|<1$? (Probably
not).

Question 1 has already been mentioned in [6]. We hope that both questions can
be answered by methods of [10].

###  1.3 Bilinear Bessel forms

With every operator $T$ on a Hilbert space ${\cal H}$ one can associate the
sesquilinear form

$(f|Tg),\qquad f,g\in{\cal D}(T).$ (1.8)

One can try to extend (1.8) to a larger domain. If $T$ is self-adjoint, there
is a natural extension to the so-called form domain of $T$, ${\cal
Q}(T):={\cal D}(\sqrt{|T|})$. Interpreting $T$ as a bounded map from ${\cal
Q}(T)$ to its anti-dual, we obtain the sesquilinear form

$(f|Tg),\qquad f,g\in{\cal Q}(T),$ (1.9)

which extends (1.8).

We would like to have a similar construction for Bessel operators, including
non-self-adjoint ones. Before we proceed we should realize that identities
involving non-self-adjoint operators do not like complex conjugation.
Therefore, instead of sesquilinear forms it is more natural to use bilinear
forms.

Our analysis of bilinear Bessel forms is based on the pair of formal
factorizations of the Bessel operator

$\displaystyle-\partial_{x}^{2}+\Big{(}m^{2}-\frac{1}{4}\Big{)}\frac{1}{x^{2}}$
$\displaystyle=\Big{(}\partial_{x}+\frac{\frac{1}{2}+m}{x}\Big{)}\Big{(}-\partial_{x}+\frac{\frac{1}{2}+m}{x}\Big{)}$
(1.10)
$\displaystyle=\Big{(}\partial_{x}+\frac{\frac{1}{2}-m}{x}\Big{)}\Big{(}-\partial_{x}+\frac{\frac{1}{2}-m}{x}\Big{)}.$
(1.11)

In Theorems 8.2 and 8.3 for each ${\rm Re}(m)>-1$ we interpret (1.10) and
(1.11) as factorizations of the Bessel operator $H_{m}$ into two closed 1st
order operators. They define natural bilinear forms, which we call Bessel
forms. For each ${\rm Re}(m)>-1$ the corresponding Bessel form is unique,
except for ${\rm Re}(m)=0$, $m\neq 0$, when the two factorizations yield two
distinct Bessel forms.

Instead of ${\cal H}_{0}^{2}({\mathbb{R}}_{+})$, the major role is now played
by the minimal 1st order Sobolev space

$\displaystyle{\cal H}_{0}^{1}({\mathbb{R}}_{+}):=\\{f\in{\cal
H}^{1}({\mathbb{R}}_{+})\ |\ f(0)=0\\},$ (1.12)

subspace of the (full) 1st order Sobolev space

$\displaystyle{\cal H}^{1}({\mathbb{R}}_{+}):=\\{f\in L^{2}({\mathbb{R}}_{+})\
|\ f^{\prime}\in L^{2}({\mathbb{R}}_{+})\\}.$ (1.13)

Note that ${\cal H}_{0}^{1}({\mathbb{R}}_{+})$ is the domain of Bessel forms
for ${\rm Re}(m)>0$.

The analysis of Bessel forms and their factorizations shows a variety of
behaviors depending on the parameter $m$. In particular, there is a kind of a
phase transition at ${\rm Re}(m)=0$. Curiously, in the analysis of the domain
of Bessel operators the phase transition occurs elsewhere: at ${\rm Re}(m)=1$.

###  1.4 Comparison with literature

The fact that ${\cal D}(L_{\alpha}^{\min})$ does not depend on $\alpha$ for
real $\alpha\in[0,1[$ was first proven in [1], see also [2, 3]. Actually, the
arguments of [1] are enough to extend the result to complex $\alpha$ such that
$|\alpha-\frac{1}{4}|<\frac{3}{4}$. The proof is based on the bound
$\|Q\|=\frac{3}{4}$ of the operator $Q$ on $L^{2}({\mathbb{R}}_{+})$ given by
the integral kernel

$Q(x,y)=\frac{1}{x^{2}}(x-y)\theta(x-y),$ (1.14)

where $\theta$ is the Heaviside function. Our proof is quite similar. Instead
of (1.14) we consider for $|{\rm Re}(m)|<1$ the operator $Q_{m^{2}}$ with the
kernel

$Q_{m^{2}}(x,y)=\frac{1}{2mx^{2}}(x^{\frac{1}{2}+m}y^{\frac{1}{2}-m}-x^{\frac{1}{2}-m}y^{\frac{1}{2}+m})\theta(x-y).$
(1.15)

Note that $Q_{\frac{1}{4}}$ coincides with (1.14). We prove that the norm of
$Q_{m^{2}}$ is the inverse of the distance of $m^{2}$ to the parabola (1.2). A
simple generalization of the Kato-Rellich Theorem to closed operators implies
then our result about ${\cal D}(L_{\alpha}^{\min})$.

In the paper [6] on page 567 it is written “If $m\neq 1/2$ then ${\cal
D}(L^{\min}_{m})\neq\mathcal{H}_{0}^{2}$.” (In that paper $L_{m^{2}}^{\min}$
was denoted $L_{m}^{\min}$). This sentence was not formulated as a
proposition, and no proof was provided. Anyway, in view of the results of [2]
and of this paper, this sentence was wrong.

The analysis of Bessel forms in the self-adjoint case, that is for real
$m>-1$, is well known–it is essentially equivalent to the famous Hardy
inequality. This subject is discussed e.g. in the monograph [4] and in a
recent interesting paper [11] about a refinement of the 1-dimensional Hardy’s
inequality. The latter paper contains in particular many references about
factorizations of Bessel operators in the self-adjoint case.

Results about Bessel forms and their factorizations for complex parameters are
borrowed to a large extent from [6]. We include them in this paper, because
they provide an interesting complement to the analysis of domains of Bessel
operators.

##  2 Basic closed realizations of the Bessel operator

The main topic of this preliminary section are closed homogeneous realizations
of $L_{\alpha}$. We recall their definitions following [6, 7].

We will denote by ${\mathbb{R}}_{+}$ the open positive half-line, that is
$]0,\infty[$. We will use $L^{2}({\mathbb{R}}_{+})$ as our basic Hilbert
space. We define $L_{\alpha}^{\max}$ to be the operator given by the
expression $L_{\alpha}$ with the domain

${\cal D}(L_{\alpha}^{\max})=\\{f\in L^{2}({\mathbb{R}}_{+})\mid
L_{\alpha}f\in L^{2}({\mathbb{R}}_{+})\\}.$

We also set $L_{\alpha}^{\min}$ to be the closure of the restriction of
$L_{\alpha}^{\max}$ to $C_{\rm c}^{\infty}({\mathbb{R}}_{+})$.

We will often write $m$ for one of the square roots of $\alpha$, that is,
$\alpha=m^{2}$. It is easy to see that the space of solutions of the
differential equation

$L_{\alpha}f=0$ (2.1)

is spanned for $\alpha\neq 0$ by $x^{\frac{1}{2}+m}$, $x^{\frac{1}{2}-m}$, and
for $\alpha=0$ by $x^{\frac{1}{2}}$, $x^{\frac{1}{2}}\log x$. Note that both
solutions are square integrable near $0$ iff $|{\rm Re}(m)|<1$. This is used
in [6] to show that we have

$\displaystyle{\cal D}(L_{\alpha}^{\max})$ $\displaystyle={\cal
D}(L_{\alpha}^{\min})+{\mathbb{C}}x^{\frac{1}{2}+m}\xi+{\mathbb{C}}x^{\frac{1}{2}-m}\xi,$
$\displaystyle|{\rm Re}\sqrt{\alpha}|<1,\ \alpha\neq 0;$ (2.2)
$\displaystyle{\cal D}(L_{0}^{\max})$ $\displaystyle={\cal
D}(L_{0}^{\min})+{\mathbb{C}}x^{\frac{1}{2}}\xi+{\mathbb{C}}x^{\frac{1}{2}}\log(x)\xi,$
$\displaystyle\alpha=0;$ (2.3) $\displaystyle{\cal D}(L_{\alpha}^{\max})$
$\displaystyle={\cal D}(L_{\alpha}^{\min}),$ $\displaystyle|{\rm
Re}\sqrt{\alpha}|\geq 1.$ (2.4)

Above (and throughout the paper) $\xi$ is any
$C_{\mathrm{c}}^{\infty}[0,\infty[$ function such that $\xi=1$ near $0$.

Following [6], for ${\rm Re}(m)>-1$ we also introduce another family of closed
realizations of Bessel operators: the operators $H_{m}$ defined as the
restrictions of $L_{m^{2}}^{\max}$ to

${\cal D}(H_{m}):={\cal
D}(L_{m^{2}}^{\min})+{\mathbb{C}}x^{\frac{1}{2}+m}\xi.$ (2.5)

We will use various basic concepts and facts about 1-dimensional Schrödinger
operators with complex potentials. We will use [9] as the main reference, but
clearly most of them belong to the well-known folklore. In particular, we will
use two kinds of Green’s operators. Let us recall this concept, following [9].
Let $L_{\mathrm{c}}^{1}({\mathbb{R}}_{+})$ be the set of integrable functions
of compact support in ${\mathbb{R}}_{+}$. We will say that an operator
$G:L_{\mathrm{c}}^{1}({\mathbb{R}}_{+})\to AC^{1}({\mathbb{R}}_{+})$ is a
Green’s operator of $L_{\alpha}$ if for any $g\in
L_{\mathrm{c}}^{1}({\mathbb{R}}_{+})$

$L_{\alpha}Gg=g.$ (2.6)

##  3 The forward Green’s operator

Let us introduce the operator $G_{\alpha}^{\to}$ defined by the kernel

$\displaystyle G_{\alpha}^{\to}(x,y)$
$\displaystyle:=\frac{1}{2m}\big{(}x^{\frac{1}{2}+m}y^{\frac{1}{2}-m}-x^{\frac{1}{2}-m}y^{\frac{1}{2}+m}\big{)}\theta(x-y),$
$\displaystyle\alpha\neq 0;$ (3.1) $\displaystyle G_{0}^{\to}(x,y)$
$\displaystyle:=x^{\frac{1}{2}}y^{\frac{1}{2}}\log\Big{(}\frac{x}{y}\Big{)}\theta(x-y),$
$\displaystyle\alpha=0.$ (3.2)

Note that $G_{\alpha}^{\to}$ is a Green’s operator in the sense of (2.6).
Besides,

$\displaystyle{\rm supp}G_{\alpha}^{\to}g\subset{\rm supp}g+{\mathbb{R}}_{+},$
(3.3)

which is why it is sometimes called the forward Green’s operator.

Unfortunately, the operator $G_{\alpha}^{\to}$ is unbounded on
$L^{2}({\mathbb{R}}_{+})$. To make it bounded, for any $a>0$ we can compress
it to the finite interval $[0,a]$, by introducing the operator
$G_{\alpha}^{a\to}$ with the kernel

$G_{\alpha}^{a\to}(x,y):={\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{[0,a]}(x)G_{\alpha}^{\to}(x,y){\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{[0,a]}(y).$ (3.4)

It is also convenient to consider the operator $L_{\alpha}$ restricted to
$[0,a]$. One of its closed realizations, is defined by the zero boundary
condition at $0$ and no boundary conditions at $a$ (see [9] Def. 4.14). It
will be denoted $L_{\alpha,0}^{a}$. By Prop. 7.3 of [9] we have
$G_{\alpha}^{a\to}=(L_{\alpha,0}^{a})^{-1}$, and hence

${\cal D}(L_{\alpha,0}^{a})=G_{\alpha}^{a\to}L^{2}[0,a].$ (3.5)

Now we can describe the domain of $L_{\alpha}^{\min}$ with the help of the
forward Green’s operator.

###### Proposition 3.1

Suppose that $f\in{\cal D}(L_{\alpha}^{\max})$. Then the following statements
are equivalent:

  1. 1.

$f\in{\cal D}(L_{\alpha}^{\min})$.

  2. 2.

For some $a>0$ and $g^{a}\in L^{2}[0,a]$ we have
$f\Big{|}_{[0,a]}=G_{\alpha}^{\to}g^{a}\Big{|}_{[0,a]}$.

  3. 3.

For all $a>0$ there exists $g^{a}\in L^{2}[0,a]$ such that
$f\Big{|}_{[0,a]}=G_{\alpha}^{\to}g^{a}\Big{|}_{[0,a]}$.

Proof. The boundary space ([9] Def. 5.2) of $L_{\alpha}$ is trivial at
$\infty$ (see [9] Prop. 5.15). Therefore, for any $a>0$ we have

$f\in{\cal D}(L_{\alpha}^{\min})\ \Leftrightarrow\ f\Big{|}_{[0,a]}\in{\cal
D}(L_{\alpha,0}^{a}).$ (3.6)

Hence it is enough to apply (3.5). $\Box$

Define the operator $Q_{\alpha}:=\frac{1}{x^{2}}G_{\alpha}^{\to}.$ Its
integral kernel is

$\displaystyle Q_{\alpha}(x,y)$
$\displaystyle=\frac{1}{2m}(x^{-\frac{3}{2}+m}y^{\frac{1}{2}-m}-x^{-\frac{3}{2}-m}y^{\frac{1}{2}+m})\theta(x-y),$
$\displaystyle\alpha\neq 0;$ (3.7) $\displaystyle Q_{0}(x,y)$
$\displaystyle:=x^{-\frac{3}{2}}y^{\frac{1}{2}}\log\Big{(}\frac{x}{y}\Big{)}\theta(x-y),$
$\displaystyle\alpha=0.$ (3.8)

###### Proposition 3.2

Assume that $|{\rm Re}\sqrt{\alpha}|<1$. Then the operator $Q_{\alpha}$ is
bounded on $L^{2}({\mathbb{R}}_{+})$, and

$\|Q_{\alpha}\|=\frac{1}{\mathrm{dist}\big{(}\alpha,(1+{\rm
i}{\mathbb{R}})^{2}\big{)}}$ (3.9)

Proof. Introduce the unitary operator $U:L^{2}({\mathbb{R}}_{+})\to
L^{2}({\mathbb{R}})$ given by

$(Uf)(t):={\rm e}^{\frac{t}{2}}f({\rm e}^{t}).$ (3.10)

Note that if an operator $K$ has the kernel $K(x,y)$, then $UKU^{-1}$, has the
kernel ${\rm e}^{\frac{t}{2}}K({\rm e}^{t},{\rm e}^{s}){\rm e}^{\frac{s}{2}}$.
Therefore, for any $\alpha$ the operator $UQ_{\alpha}U^{-1}$ has the kernel

$\displaystyle\frac{1}{2m}({\rm e}^{-(t-s)(1-m)}-{\rm
e}^{-(t-s)(1+m)})\theta(t-s),$ $\displaystyle\alpha\neq 0;$ (3.11)
$\displaystyle{\rm e}^{-(t-s)}(t-s)\theta(t-s),$ $\displaystyle\alpha=0.$
(3.12)

Thus, it is the convolution by the function

$\displaystyle t\to$ $\displaystyle\frac{1}{2m}({\rm e}^{-t(1-m)}-{\rm
e}^{-t(1+m)})\theta(t),$ $\displaystyle\alpha\neq 0;$ (3.13) $\displaystyle
t\to$ $\displaystyle{\rm e}^{-t}t\theta(t),$ $\displaystyle\alpha=0.$ (3.14)

Assume now that $|{\rm Re}\sqrt{\alpha}|<1$. Then the function (3.13) is
integrable and we can apply the Fourier transformation defined by $({\cal
F}u)(\omega)=(2\pi)^{-1/2}\int{\rm e}^{-i\omega t}u(t)\,{\rm d}t$. After this
transformation the operator $UQ_{\alpha}U^{-1}$ becomes the multiplication wrt
the Fourier transform of (3.13) or (3.14), that is

$\omega\mapsto\frac{1}{(1+{\rm i}\omega)^{2}-m^{2}}.$ (3.15)

Thus the norm of $UQ_{\alpha}U^{-1}$, and hence also of $Q_{\alpha}$, is the
supremum of the absolute value of (3.15). $\Box$

###### Remark 3.3

The operator $Q_{\alpha}$ belongs to the class of operators analyzed in [17]
on p. 271, which goes back to Hardy-Littlewood-Polya [12] p. 229.

Proposition (3.2) for $\alpha=\frac{1}{4}$ is especially important and simple.
This case was noted in cf. [6, p. 566] and [1, Lemma 2.2]. It can be written
as

$\displaystyle g(x):=x^{-2}\int_{0}^{x}(x-y)f(y){\rm d}y\ \Rightarrow\
\|g\|\leq\frac{4}{3}\|f\|.$ (3.16)

One can remark that (3.16) is essentially equivalent to the one-dimensional
version of the classical Rellich’s inequality, see e.g. [4, (6.1.1)]:

$\int_{0}^{\infty}\frac{|u|^{2}}{x^{4}}{\rm
d}x\leq\frac{16}{9}\int_{0}^{\infty}|u^{\prime\prime}|^{2}{\rm d}x,\quad u\in
C_{\mathrm{c}}^{\infty}({\mathbb{R}}_{+}),$ (3.17)

where we identify $f=u^{\prime\prime}$ and $g=\frac{u}{x^{2}}$.

The proof of the following proposition uses only the simple estimate (3.16).

###### Proposition 3.4

${\cal D}(L_{\alpha}^{\max})\cap{\cal
D}(L_{\beta}^{\max})=\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$ if
$\alpha\neq\beta$.

Proof. We have $f\in{\cal D}(L_{\alpha}^{\max})$ if and only if $f\in
L^{2}({\mathbb{R}}_{+})$ and $-f^{\prime\prime}+(\alpha-1/4)x^{-2}f\in
L^{2}({\mathbb{R}}_{+})$ hence if we also have $f\in{\cal
D}(L_{\beta}^{\max})$ then $(\alpha-\beta)x^{-2}f\in L^{2}({\mathbb{R}}_{+})$
and since $\alpha\neq\beta$ we get $x^{-2}f\in L^{2}({\mathbb{R}}_{+})$ hence
$f^{\prime\prime}\in L^{2}({\mathbb{R}}_{+})$. Recall that
$f,f^{\prime\prime}\in L^{2}({\mathbb{R}}_{+})$ implies
$f\in\mathcal{H}^{1}({\mathbb{R}}_{+})$ and
$\|f^{\prime}\|^{2}_{L^{2}({\mathbb{R}}_{+})}\leq\|f\|^{2}_{L^{2}({\mathbb{R}}_{+})}\|f^{\prime\prime}\|^{2}_{L^{2}({\mathbb{R}}_{+})}$.
It follows that $f$ is absolutely continuous and
$f(x)=a+\int_{0}^{x}f^{\prime}(y){\rm d}y$ for some constant $a$ and
$f^{\prime}$ is absolutely continuous and
$f^{\prime}(x)=b+\int_{0}^{x}f^{\prime\prime}(y){\rm d}y$ for some constant
$b$, thus

$f(x)=a+bx+\int_{0}^{x}\int_{0}^{y}f^{\prime\prime}(z){\rm d}z{\rm
d}y=a+bx+x^{2}g(x),\qquad g(x):=x^{-2}\int_{0}^{x}(x-y)f^{\prime\prime}(y){\rm
d}y.$

Then, by (3.16)

$\displaystyle\|g\|_{L^{2}({\mathbb{R}}_{+})}\leq\frac{4}{3}\|f^{\prime\prime}\|_{L^{2}({\mathbb{R}}_{+})}.$
(3.18)

Thus $x^{-2}f(x)=ax^{-2}+bx^{-1}+g(x)$ where $g\in L^{2}({\mathbb{R}}_{+})$,
so $\int_{0}^{1}|x^{-2}f(x)|^{2}{\rm d}x<\infty$ if and only if $a=b=0$, so
that $f(x)=\int_{0}^{x}(x-y)f^{\prime\prime}(y){\rm d}y$ and
$f^{\prime}(x)=\int_{0}^{x}f^{\prime\prime}(y){\rm d}y$, hence
$f\in\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$.

Reciprocally, if $f\in\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$ then $x^{-2}f\in
L^{2}({\mathbb{R}}_{+})$ with
$\|x^{-2}f\|_{L^{2}({\mathbb{R}}_{+})}\leq\frac{4}{3}\|f^{\prime\prime}\|_{L^{2}({\mathbb{R}}_{+})}$
by (3.16), hence $f\in{\cal D}(L_{\alpha}^{\max})$ for all $\alpha$. $\Box$

##  4 Domain of Bessel operators for $|{\rm Re}(m)|<1$

Below we state the first main result of our paper (which is an extension of a
result of [1]).

###### Theorem 4.1

If $|{\rm Re}\sqrt{\alpha}|<1$, then ${\cal
D}(L_{\alpha}^{\min})=\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$. Moreover,

$\left\\{\alpha\in{\mathbb{C}}\ |\ |{\rm
Re}\sqrt{\alpha}|<1\right\\}\ni\alpha\mapsto L_{\alpha}^{\min}$ (4.1)

is a holomorphic family of closed operators.

The proof of this theorem is based on the following lemma.

###### Lemma 4.2

Let $|{\rm Re}\sqrt{\alpha}|<1$ and $f\in{\cal D}(L_{\alpha}^{\min})$. Then

$\|x^{-2}f\|\leq\frac{1}{\mathrm{dist}\big{(}\alpha,(1+{\rm
i}{\mathbb{R}})^{2}\big{)}}\|L_{\alpha}^{\min}f\|.$ (4.2)

Proof. Let $a>0$. Set $g:=L_{\alpha}^{\min}f$, $f^{a}:=f\Big{|}_{[0,a]}$,
$g^{a}:=g\Big{|}_{[0,a]}$. Let $G_{\alpha}^{a\to}$ be as in (3.4). As in the
proof of Prop. 3.1,

$f^{a}=G_{\alpha}^{a\to}g^{a}.$ (4.3)

So

$\displaystyle\|x^{-2}f\|$ $\displaystyle=\lim_{a\to\infty}\|x^{-2}f^{a}\|$
(4.4) $\displaystyle=\lim_{a\to\infty}\|x^{-2}G_{\alpha}^{a\to}g^{a}\|$
$\displaystyle=\|Q_{\alpha}g\|\leq\frac{1}{\mathrm{dist}\big{(}\alpha,(1+{\rm
i}{\mathbb{R}})^{2}\big{)}}\|g\|.\quad\Box$ (4.5)

Proof of Theorem 4.1. We can cover the region on the lhs of (4.1) by disks
touching the boundary of this region, that is, (1.2). Inside each disk we
apply Thm A.1 and Lemma 4.2. We obtain in particular, that if $|{\rm
Re}\sqrt{\alpha}_{i}|<1$, $i=1,2$, then ${\cal D}(L_{\alpha_{1}}^{\min})={\cal
D}(L_{\alpha_{2}}^{\min})$. But clearly ${\cal
D}(L_{\frac{1}{4}}^{\min})=\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$. $\Box$

###### Theorem 4.3

We have

$\displaystyle{\cal D}(L_{\alpha}^{\max})$ $\displaystyle={\cal
H}_{0}^{2}+{\mathbb{C}}x^{\frac{1}{2}+m}\xi+{\mathbb{C}}x^{\frac{1}{2}-m}\xi,$
$\displaystyle|{\rm Re}\sqrt{\alpha}|<1,\ \alpha\neq 0;$ (4.6)
$\displaystyle{\cal D}(L_{\alpha}^{\max})$ $\displaystyle={\cal
H}_{0}^{2}+{\mathbb{C}}x^{\frac{1}{2}}\xi+{\mathbb{C}}x^{\frac{1}{2}}\log(x)\xi,$
$\displaystyle\alpha=0.$ (4.7)

Besides,

${\cal D}(L_{\alpha_{1}}^{\max})\neq{\cal
D}(L_{\alpha_{2}}^{\max}),\qquad\alpha_{1}\neq\alpha_{2},\quad|{\rm
Re}\sqrt{\alpha}_{i}|<1,\quad i=1,2.$ (4.8)

Furthermore,

$\left\\{\alpha\in{\mathbb{C}}\ |\ |{\rm
Re}\sqrt{\alpha}|<1\right\\}\ni\alpha\mapsto L_{\alpha}^{\max}$ (4.9)

is a holomorphic family of closed operators.

Proof. Using ${\cal D}(L_{\alpha}^{\min})={\cal H}_{0}^{2}$, (2.2) and (2.3)
can be now rewritten as (4.6) and (4.7).

Clearly, $x^{\frac{1}{2}+m}\xi$ and $x^{\frac{1}{2}}\log(x)\xi$ do not belong
to ${\cal H}_{0}^{2}({\mathbb{R}}_{+})$ (because their second derivatives are
not square integrable). Therefore, ${\cal D}(L_{\alpha}^{\max})\neq{\cal
H}_{0}^{2}({\mathbb{R}}_{+})$. This together with Proposition 3.4 implies
(4.8).

We have $(L_{\alpha}^{\min})^{*}=L_{\overline{\alpha}}^{\max}$. Therefore, to
obtain the holomorphy we can use Proposition A.2. $\Box$

The most important holomorphic family of Bessel operators is

$\\{m\in{\mathbb{C}}\ |\ {\rm Re}(m)>-1\\}\ni m\mapsto H_{m}.$ (4.10)

Its holomorphy has been proven in [6]. Using arguments similar to those in the
proof of Theorem 4.3 we obtain a closer description of this family in the
region $|{\rm Re}(m)|<1$.

###### Theorem 4.4

We have

$\displaystyle{\cal D}(H_{m})$ $\displaystyle={\cal
H}_{0}^{2}+{\mathbb{C}}x^{\frac{1}{2}+m}\xi,$ $\displaystyle|{\rm Re}(m)|<1.$
(4.11)

Besides, if $m_{1}\neq m_{2}$ and $|{\rm Re}(m_{i})|<1$, $i=1,2$, then ${\cal
D}(H_{m_{1}})\neq{\cal D}(H_{m_{2}})$.

##  5 Two-sided Green’s operator

For any $m\in{\mathbb{C}}$, $m\neq 0$, let us introduce the operator $G_{m}$
with the kernel

$\displaystyle G_{m}(x,y)$
$\displaystyle:=\frac{1}{2m}\left(x^{\frac{1}{2}+m}y^{\frac{1}{2}-m}\theta(y-x)+x^{\frac{1}{2}-m}y^{\frac{1}{2}+m}\theta(x-y)\right).$
(5.1)

Recall that $\theta$ is the Heaviside function. (5.1) is one of Green’s
operators of $L_{m^{2}}$ in the sense of (2.6), Following [9], we will call it
the two-sided Green’s operator.

The operator $G_{m}$ is not bounded on $L^{2}({\mathbb{R}}_{+})$ for any
$m\in{\mathbb{C}}$. However, at least for ${\rm Re}(m)>-1$, it is useful in
the $L^{2}$ setting.

###### Proposition 5.1

Let ${\rm Re}(m)>-1$, $m\neq 0$ and $a>0$.

  1. 1.

If $g\in L^{2}[0,a]$, then

$f(x)=G_{m}g(x)=\int_{0}^{\infty}G_{m}(x,y)g(y){\rm d}y$ (5.2)

is well defined, belongs to $\in AC^{1}]0,\infty[$ and $L_{\alpha}f=g$.

  2. 2.

Conversely, if $f\in AC^{1}]0,\infty[$, $L_{\alpha}f=g\in L^{2}[0,a]$, then
there exist $c_{+},c_{-}$ such that

$\displaystyle f(x)$
$\displaystyle=c_{+}x^{\frac{1}{2}+m}+c_{-}x^{\frac{1}{2}-m}+G_{m}g(x),\quad
m\neq 0.$ (5.3)

Proof. Note first that ${\rm Re}(m)>1$ implies $x^{\frac{1}{2}\pm m}$ is
locally in $L^{2}$. Using this, the proof of the first part of the proposition
is a straightforward computation done, in a more general setting, in [9], see
§2.7 and Definition 2.10 there. For the second part, note that
$L_{\alpha}(f-G_{m}g)=0$ by the first part of the proposition, and that the
two functions $x^{\frac{1}{2}\pm m}$ give a basis of the nullspace of
$L_{\alpha}$. $\Box$

Let us introduce the operator $Z_{m}:=\frac{1}{x^{2}}G_{m}$ with the kernel

$\displaystyle Z_{m}(x,y)$
$\displaystyle=\frac{1}{2m}\left(x^{-\frac{3}{2}+m}y^{\frac{1}{2}-m}\theta(y-x)+x^{-\frac{3}{2}-m}y^{\frac{1}{2}+m}\theta(x-y)\right).$
(5.4)

###### Proposition 5.2

Let ${\rm Re}(m)>1$. Then $Z_{m}$ is bounded and

$\|Z_{m}\|=\frac{1}{\mathrm{dist}\big{(}m^{2},(1+{\rm
i}{\mathbb{R}})^{2}\big{)}}$ (5.5)

Proof. If $U$ is given by (3.10), then $UZ_{m}U^{-1}$ has the kernel

$\displaystyle\frac{1}{2m}\left({\rm e}^{-(m-1)(s-t)}\theta(s-t)+{\rm
e}^{-(m+1)(s-t)}\theta(t-s)\right).$ (5.6)

If ${\rm Re}(m)>1$, after the Fourier transformation (defined as in the proof
of Proposition 3.2) it becomes the multiplication by the function

$\displaystyle\omega\mapsto\frac{1}{2m}\Big{(}\frac{1}{(m-1-{\rm
i}\omega)}+\frac{1}{1+m+{\rm i}\omega)}\Big{)}=\frac{1}{m^{2}-(1+{\rm
i}\omega)^{2}},$ (5.7)

whose supremum is the right hand side of (5.5). $\Box$

##  6 Domain of Bessel operators for ${\rm Re}(m)>1$

For ${\rm Re}(m)\geq 1$ there is a unique closed Bessel operator. We will see
in the following theorem that its domain is again the minimal 2nd order
Sobolev space, except at the boundary ${\rm Re}(m)=1$, cf. Section 7.

###### Theorem 6.1

Let ${\rm Re}(m)>1$. Then ${\cal
D}(H_{m})=\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$.

Proof. We know that $\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})\subset{\cal
D}(L_{m^{2}}^{\max})$ for any $m$. But for ${\rm Re}(m)>1$ we have
$L_{m^{2}}^{\max}=H_{m}$. This proves the inclusion
$\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})\subset{\cal D}(H_{m})$.

Let us prove the converse inclusion. Let $f\in{\cal D}(H_{m})$. It is enough
to assume that $f\in L^{2}[0,1]$. Let $g:=H_{m}f$. Then $g\in L^{2}[0,1]$. By
Prop. 5.1, we can write

$f(x)=c_{+}x^{\frac{1}{2}+m}+c_{-}x^{\frac{1}{2}-m}+\frac{x^{\frac{1}{2}+m}}{2m}\int_{x}^{1}y^{\frac{1}{2}-m}g(y){\rm
d}y+\frac{x^{\frac{1}{2}-m}}{2m}\int_{0}^{x}y^{\frac{1}{2}+m}g(y){\rm d}y.$
(6.1)

For $x>1$ we have

$f(x)=c_{+}x^{\frac{1}{2}+m}+x^{\frac{1}{2}-m}\left(c_{-}+\frac{1}{2m}\int_{0}^{1}y^{\frac{1}{2}+m}g(y){\rm
d}y\right),$ (6.2)

hence $c_{+}=0$. We have, for $x\to 0$,

$\displaystyle\left|x^{\frac{1}{2}+m}\int_{x}^{1}y^{\frac{1}{2}-m}g(y){\rm
d}y\right|$ $\displaystyle\leq x\int_{0}^{1}|g(x)|{\rm d}y\to 0;$ (6.3)
$\displaystyle\left|x^{\frac{1}{2}-m}\int_{0}^{x}y^{\frac{1}{2}+m}g(y)\right|{\rm
d}y$ $\displaystyle\leq x\int_{0}^{x}|g(y)|{\rm d}y\to 0.$ (6.4)

$x^{\frac{1}{2}-m}$ is not square integrable near zero. Hence $c_{-}=0$. Thus

$f(x)=\frac{x^{\frac{1}{2}+m}}{2m}\int_{x}^{1}y^{\frac{1}{2}-m}g(y){\rm
d}y+\frac{x^{\frac{1}{2}-m}}{2m}\int_{0}^{x}y^{\frac{1}{2}+m}g(y){\rm d}y.$
(6.5)

By (6.3) and (6.4), $\lim\limits_{x\to 0}f(x)=0$. Now

$\displaystyle f^{\prime}(x)=\frac{(\frac{1}{2}+m)x^{-\frac{1}{2}+m}}{2m}$
$\displaystyle\int_{x}^{1}y^{\frac{1}{2}-m}g(y){\rm
d}y+\frac{(\frac{1}{2}-m)x^{-\frac{1}{2}-m}}{2m}\int_{0}^{x}y^{\frac{1}{2}+m}g(y){\rm
d}y,$ (6.6)
$\displaystyle\left|x^{-\frac{1}{2}-m}\int_{0}^{x}y^{\frac{1}{2}+m}g(y){\rm
d}y\right|$ $\displaystyle\leq\int_{0}^{x}|g(y)|{\rm d}y\to 0,$ (6.7)
$\displaystyle\left|x^{-\frac{1}{2}+m}\int_{x}^{1}y^{\frac{1}{2}-m}g(y){\rm
d}y\right|$ $\displaystyle\leq\int_{0}^{\epsilon}|g(y)|{\rm
d}y+x^{-\frac{1}{2}+{\rm Re}(m)}\int_{\epsilon}^{1}y^{\frac{1}{2}-{\rm
Re}(m)}|g(y)|{\rm d}y.$ (6.8)

For any $\epsilon>0$, the second term on the right of (6.8) goes to zero. The
first, by making $\epsilon$ small, can be made arbitrarily small. Therefore
(6.8) goes to zero. Thus $\lim\limits_{x\to 0}f^{\prime}(0)=0$.

Finally

$\displaystyle f^{\prime\prime}(x)+g(x)=$
$\displaystyle\frac{(m^{2}-\frac{1}{4})x^{-\frac{3}{2}+m}}{2m}\int_{x}^{1}y^{\frac{1}{2}-m}g(y){\rm
d}y+\frac{(m^{2}-\frac{1}{4})x^{-\frac{3}{2}-m}}{2m}\int_{0}^{x}y^{\frac{1}{2}+m}g(y){\rm
d}y$ (6.9) $\displaystyle=$
$\displaystyle\Big{(}m^{2}-\frac{1}{4}\Big{)}Z_{m}g(x).$ (6.10)

By Proposition 5.2 $Z_{m}$ is bounded. Hence $f^{\prime\prime}\in
L^{2}({\mathbb{R}}_{+})$. Therefore,
$f\in\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$. $\Box$

##  7 Domain of Bessel operators for ${\rm Re}(m)=1$

In this section we treat the most complicated situation concerning the domain
of $H_{m}$, namely the case ${\rm Re}(m)=1$. By (1.3) we then have
$H_{m}=L_{m^{2}}^{\min}=L_{m^{2}}^{\max}$. We prove the following theorem.

###### Theorem 7.1

Let ${\rm Re}(m)=1$.

  1. 1.

$\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$ is a dense subspace of ${\cal
D}(H_{m})$ of infinite codimension.

  2. 2.

If $\xi$ is a $C_{\mathrm{c}}^{2}[0,\infty[$ function equal $1$ near zero,
then $x^{\frac{1}{2}+m}\xi\in{\cal D}(H_{m})$ but
$x^{\frac{1}{2}+m}\xi\not\in\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$.

  3. 3.

If ${\rm Re}(m^{\prime})=1$ and $m\neq m^{\prime}$, then ${\cal
D}(H_{m})\cap{\cal D}(H_{m^{\prime}})=\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$.

By (1.3), it is clear that $\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})\subset{\cal
D}(H_{m})$ and $x^{\frac{1}{2}+m}\xi\in{\cal D}(H_{m})$. The density of
$\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$ in ${\cal D}(H_{m})$ is a consequence
of $H_{m}=L_{m^{2}}^{\min}$. The last assertion of the theorem is a special
case of Proposition 3.4. In the rest of this section we construct an infinite
dimensional vector subspace ${\cal V}$ of ${\cal D}(H_{m})$ such that ${\cal
V}\cap\big{(}\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})+{\mathbb{C}}x^{\frac{1}{2}+m}\xi\big{)}=\\{0\\}$,
which will finish the proof of the theorem.

Let us study the behaviour at zero of the functions in ${\cal D}(H_{m})$. For
functions in the subspace
$\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})+{\mathbb{C}}x^{\frac{1}{2}+m}\xi$ this
is easy, cf. the next lemma, but this is not so trivial for the other
functions.

###### Lemma 7.2

If
$f=f_{0}+cx^{\frac{1}{2}+m}\xi\in\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})+{\mathbb{C}}x^{\frac{1}{2}+m}\xi$
then

$c=\lim_{x\to 0}x^{-\frac{1}{2}-m}f(x).$ (7.1)

Proof. If $f_{0}\in\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})$ then
$f_{0}(x)=\int\limits_{0}^{x}(x-y)f_{0}^{\prime\prime}(y){\rm d}y$. Therefore,
$\sqrt{3}|f_{0}(x)|\leq x^{\frac{3}{2}}\|f_{0}^{\prime\prime}\|_{L^{2}[0,x]}$
and since ${\rm Re}(m+\frac{1}{2})=\frac{3}{2}$ we get $\lim\limits_{x\to
0}x^{-m-\frac{1}{2}}f_{0}(x)=0$, which implies (7.1). $\Box$

Let $a>0$. Let $G_{m}^{a}$ be the operator $G_{m}$ compressed to the interval
$[0,a]$. Its kernel is

$G_{m}^{a}(x,y)={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{[0,a]}(x)G_{m}(x,y){\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{[0,a]}(y).$ (7.2)

We will write $L_{\alpha}^{a,\max}$ for the maximal realization of operator
$L_{\alpha}$ on $L^{2}[0,a]$.

###### Lemma 7.3

Let ${\rm Re}(m)>-1$. Then $G_{m}^{a}$ is a bounded operator on $L^{2}[0,a]$.
If $g\in L^{2}[0,a]$, then $G_{m}^{a}g\in{\cal D}(L_{m^{2}}^{a,\max})$ and
$L_{m^{2}}^{a,\max}G_{m}^{a}g=g$. Consequently, $G_{m}^{a}$ is injective.

Proof. We check that (7.2) belongs to $L^{2}\big{(}[0,a]\times[0,a]\big{)}$.
This proves that $G_{m}^{a}$ is Hilbert Schmidt, hence bounded. $G_{m}^{a}$ is
a right inverse of $L_{m^{2}}^{a,\max}$, because $G_{m}$ is a right inverse of
$L_{m^{2}}$ (see Proposition 5.1). $\Box$

###### Lemma 7.4

Let ${\rm Re}(m)=1$. Let $g\in L^{2}[0,a]$ and $f=G_{m}^{a}g$. Then

$\lim_{x\to
0}\left(2mx^{-\frac{1}{2}-m}f(x)-\int_{x}^{a}y^{\frac{1}{2}-m}g(y){\rm
d}y\right)=0.$ (7.3)

Therefore, if

$\lim_{x\to 0}\int_{x}^{a}y^{\frac{1}{2}-m}g(y){\rm d}y$ (7.4)

does not exist, then
$f=G_{m}^{a}g\not\in\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})+{\mathbb{C}}x^{\frac{1}{2}+m}\xi$.

Proof. We have

$2mx^{-\frac{1}{2}-m}f(x)=\int_{x}^{a}y^{-\frac{1}{2}-m}g(y){\rm
d}y+x^{-2m}\int_{0}^{x}y^{\frac{1}{2}+m}g(y){\rm d}y.$ (7.5)

Since ${\rm Re}(m)=1$ the absolute value of the second term on the right hand
side is less than

$x^{-\frac{1}{2}}\int_{0}^{x}(y/x)^{\frac{3}{2}}|g(y)|{\rm d}y\leq
x^{-\frac{1}{2}}\int_{0}^{x}|g(y)|{\rm d}y\leq\|g\|_{L^{2}[0,x]}$

This proves (7.3). If
$f=G_{m}^{a}g\in\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})+{\mathbb{C}}x^{\frac{1}{2}+m}\xi$,
then by (7.3) and (7.1) there exists (7.4). This proves the second statement
of the lemma. $\Box$

###### Lemma 7.5

Let ${\rm Re}(m)=1$. There exists an infinite dimensional subspace ${\cal
V}\subset{\cal D}(H_{m})$ such that

${\cal
V}\cap\big{(}\mathcal{H}_{0}^{2}({\mathbb{R}}_{+})+{\mathbb{C}}x^{\frac{1}{2}+m}\xi\big{)}=\\{0\\}.$
(7.6)

Proof. For each $\tau\in]\frac{1}{2},1[$ let $g_{\tau}\in C^{2}(]0,1])$, for
$0<x<\frac{1}{2}$ given by

$g_{\tau}(x)=x^{-\frac{3}{2}+m}\big{(}\ln(1/x)\big{)}^{-\tau}$

and arbitrary on $[\frac{1}{2},1]$. Then for $x<\frac{1}{2}$ we have

$|g_{\tau}(x)|^{2}=x^{-1}\big{(}\ln(1/x)\big{)}^{-2\tau}=(2\tau-1)^{-1}\frac{{\rm
d}}{{\rm d}x}\big{(}\ln(1/x)\big{)}^{1-2\tau}.$

Hence

$\int_{0}^{\frac{1}{2}}|g_{\tau}(x)|^{2}{\rm d}x=(2\tau-1)^{-1}(\ln
2)^{1-2\tau},$

and $g_{\tau}\in L^{2}[0,1]$. Moreover, if $x<\frac{1}{2}$ then

$x^{\frac{1}{2}-m}g_{\tau}(x)=x^{-1}\big{(}\ln(1/x)\big{)}^{-\tau}=(\tau-1)^{-1}\frac{{\rm
d}}{{\rm d}x}\big{(}\ln(1/x)\big{)}^{1-\tau}.$

Hence

$\int_{x}^{\frac{1}{2}}y^{-\frac{1}{2}}g_{\tau}(y){\rm d}y=(\tau-1)^{-1}(\ln
2)^{1-\tau}+(1-\tau)^{-1}\big{(}\ln(1/x)\big{)}^{1-\tau}\to\infty\quad\text{as
}x\to 0.$

Let ${\cal G}$ be the vector subspace of $L^{2}[0,1]$ generated by the
functions $g_{\tau}$ with $\frac{1}{2}<\tau<1$. Note that each finite set
$\\{g_{\tau}\mid\tau\in A\\}$ with $A\subset]\frac{1}{2},1[$ finite is
linearly independent. Indeed, if $\sum\limits_{\tau\in A}c_{\tau}g_{\tau}=0$
and $\sigma=\min A$ and $\tau\neq\sigma$ then
$\frac{g_{\tau}(x)}{g_{\sigma}(x)}=\big{(}\ln(1/x)\big{)}^{\sigma-\tau}\to 0$
as $x\to 0$ so we get $c_{\sigma}=0$, etc. Moreover, for each not zero
$g=\sum\limits_{\tau\in A}c_{\tau}g_{\tau}\in{\cal G}$ (with $c_{\tau}\neq 0$)
we have $\lim\limits_{x\to 0}\left|\int_{x}^{1}y^{-\tfrac{1}{2}}g(y){\rm
d}y\right|=\infty$. Indeed, we may assume $c_{\sigma}=1$ and then,

$\displaystyle\int_{x}^{\frac{1}{2}}y^{-\frac{1}{2}}g(y){\rm d}y=$
$\displaystyle(1-\sigma)^{-1}\big{(}\ln(1/x)\big{)}^{1-\sigma}$
$\displaystyle+\sum\limits_{\tau\in A}c_{\tau}(\tau-1)^{-1}(\ln
2)^{1-\tau}+\sum_{\tau\neq\sigma}c_{\tau}(1-\tau)^{-1}\big{(}\ln(1/x)\big{)}^{1-\tau},$

and the first term on the right hand side tends to $+\infty$ more rapidly than
all the other, hence

$\left|\int_{x}^{\frac{1}{2}}y^{-\frac{1}{2}}g(y){\rm
d}y\right|\geq\frac{1}{2(1-\sigma)}\big{(}\ln(1/x)\big{)}^{1-\sigma}$

if $x$ is small enough.

Finally, let $\varphi\in C_{\mathrm{c}}^{\infty}[0,\infty[$ equal $1$ on
$[0,1]$. Let us define ${\cal V}$ as the space of functions on
${\mathbb{R}}_{+}$ of the form $f=\varphi G_{m}g$ with $g\in{\cal G}$. By
Lemma 7.3, $G_{m}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{[0,1]}(x)$ is injective. Hence ${\cal
V}$ is infinite dimensional and it satisfies (7.6) by Lemma 7.4. $\Box$

##  8 Bilinear forms associated with Bessel operators

As noted in the introduction, in this section we will avoid complex
conjugation. Thus in the place of the usual sesquilinear scalar product

$(f|g):=\int_{0}^{\infty}\overline{f(x)}g(x){\rm d}x,$ (8.7)

we will prefer to use the bilinear product

$\langle f|g\rangle:=\int_{0}^{\infty}f(x)g(x){\rm d}x,$ (8.8)

Clearly, (8.8) is well defined for $f,g\in L^{2}({\mathbb{R}}_{+})$. Instead
of the usual adjoint $T^{*}$ we will use the transpose $T^{\\#}$, defined with
respect to (8.8), see [9].

An important role will be played by the 1st order operators given by the
formal expression

$A_{\rho}:=\partial_{x}-\frac{\rho}{x}.$ (8.9)

A detailed analysis of (8.9) has been done in [6], where the notation was
slightly different: $A_{\rho}:=-{\rm i}(\partial_{x}-\frac{\rho}{x})$. Let us
recall the main points of that analysis.

In the usual way we define two closed realizations of $A_{\rho}$: the minimal
and the maximal one, denoted $A_{\rho}^{\min}$, resp. $A_{\rho}^{\max}$. The
following theorem was (mostly) proven in Section 3 of [6]. For the proof of
the infinite codimensionality assertion in 6 see the proof of Lemma 3.9 there
(where $\gamma$ is arbitrary $>\frac{1}{2}$).

###### Theorem 8.1

  1. 1.

$A_{\rho}^{\min}\subset A_{\rho}^{\max}$.

  2. 2.

$A_{\rho}^{\min\\#}=-A_{-\rho}^{\max}$,
$A_{\rho}^{\max\\#}=-A_{-\rho}^{\min}$.

  3. 3.

$A_{\rho}^{\min}$ and $A_{\rho}^{\max}$ are homogeneous of degree $-1$.

  4. 4.

$A_{\rho}^{\min}=A_{\rho}^{\max}$ iff $|{\rm Re}(\rho)|\geq\frac{1}{2}$. If
this is the case, we will often denote them simply by $A_{\rho}$

  5. 5.

If ${\rm Re}(\rho)\neq\frac{1}{2}$, then ${\cal D}(A_{\rho}^{\min})={\cal
H}_{0}^{1}$.

  6. 6.

If ${\rm Re}(\rho)=\frac{1}{2}$, then ${\cal
H}_{0}^{1}+{\mathbb{C}}x^{\rho}\xi$ is a dense subspace of ${\cal
D}(A_{\rho})$ of infinite codimension.

  7. 7.

If $|{\rm Re}(\rho)|<\frac{1}{2}$, then ${\cal D}(A_{\rho}^{\max})={\cal
H}_{0}^{1}+{\mathbb{C}}x^{\rho}\xi\neq{\cal H}_{0}^{1}$.

  8. 8.

If ${\rm Re}(\rho),{\rm Re}(\rho^{\prime})\in]-\frac{1}{2},\frac{1}{2}]$ and
$\rho\neq\rho^{\prime}$ then ${\cal D}(A_{\rho}^{\max})\neq{\cal
D}(A_{\rho^{\prime}}^{\max})$.

Now let us describe possible factorizations of $H_{m}$ into operators of the
form $A_{\rho}^{\min}$ and $A_{\rho}^{\max}$. On the formal level they
correspond to one of the factorizations (1.10) and (1.11).

###### Theorem 8.2

  1. 1.

For ${\rm Re}(m)>-1$ we have

$\langle f|H_{m}g\rangle=\langle
A_{\frac{1}{2}+m}^{\max}f|A_{\frac{1}{2}+m}^{\max}g\rangle,\quad f\in{\cal
D}(A_{\frac{1}{2}+m}^{\max}),\quad g\in{\cal
D}(A_{\frac{1}{2}+m}^{\max})\cap{\cal D}(H_{m}).$ (8.10)

Moreover,

$\displaystyle{\cal D}(H_{m})$ $\displaystyle=\left\\{f\in{\cal
D}(A_{\frac{1}{2}+m}^{\max})\ |\ A_{\frac{1}{2}+m}^{\max}f\in{\cal
D}(A_{-\frac{1}{2}-m}^{\min})\right\\},$ (8.11) $\displaystyle H_{m}f$
$\displaystyle=-A_{-\frac{1}{2}-m}^{\min}A_{\frac{1}{2}+m}^{\max}f,\qquad
f\in{\cal D}(H_{m}).$ (8.12)

  2. 2.

For ${\rm Re}(m)>0$ we have

$\langle f|H_{m}g\rangle=\langle
A_{\frac{1}{2}-m}^{\min}f|A_{\frac{1}{2}-m}^{\min}g\rangle,\quad f\in{\cal
D}(A_{\frac{1}{2}-m}^{\min}),\quad g\in{\cal
D}(A_{\frac{1}{2}-m}^{\min})\cap{\cal D}(H_{m}).$ (8.13)

Moreover,

$\displaystyle{\cal D}(H_{m})$ $\displaystyle=\left\\{f\in{\cal
D}(A_{\frac{1}{2}-m}^{\min})\ |\ A_{\frac{1}{2}-m}^{\min}f\in{\cal
D}(A_{-\frac{1}{2}+m}^{\max})\right\\},$ (8.14) $\displaystyle H_{m}f$
$\displaystyle=-A_{-\frac{1}{2}+m}^{\max}A_{\frac{1}{2}-m}^{\min}f,\qquad
f\in{\cal D}(H_{m}).$ (8.15)

The factorizations described in Theorem 8.2 can be used to define bilinear
forms corresponding to $H_{m}$. For details of the proof, we refer again to
[6], especially pages 571–574 and 577.

###### Theorem 8.3

The following bilinear forms are extensions of

$\langle f|H_{m}g\rangle=\langle H_{m}f|g\rangle,\qquad f,g\in{\cal
D}(H_{m}),$ (8.16)

to larger domains:

  1. 1.

For $1\leq{\rm Re}(m)$,

$\displaystyle\langle A_{\frac{1}{2}+m}f|A_{\frac{1}{2}+m}g\rangle$
$\displaystyle=\langle A_{\frac{1}{2}-m}f|A_{\frac{1}{2}-m}g\rangle,$
$\displaystyle\quad f,g\in{\cal H}_{0}^{1}.$ (8.17)

  2. 2.

For $0<{\rm Re}(m)<1$,

$\displaystyle\langle A_{\frac{1}{2}+m}f|A_{\frac{1}{2}+m}g\rangle$
$\displaystyle=\langle
A_{\frac{1}{2}-m}^{\min}f|A_{\frac{1}{2}-m}^{\min}g\rangle,$
$\displaystyle\quad f,g\in{\cal H}_{0}^{1}.$ (8.18)

  3. 3.

For ${\rm Re}(m)=0$,

$\displaystyle\langle A_{\frac{1}{2}+m}f|A_{\frac{1}{2}+m}g\rangle,$
$\displaystyle\quad f,g\in{\cal D}(A_{\frac{1}{2}+m})\supset{\cal
H}_{0}^{1}+{\mathbb{C}}x^{\frac{1}{2}+m}\xi,$ (8.19) $\displaystyle\langle
A_{\frac{1}{2}-m}f|A_{\frac{1}{2}-m}g\rangle,$ $\displaystyle\quad f,g\in{\cal
D}(A_{\frac{1}{2}-m})\supset{\cal
H}_{0}^{1}+{\mathbb{C}}x^{\frac{1}{2}-m}\xi.$ (8.20)

  4. 4.

For $-1<{\rm Re}(m)<0$,

$\displaystyle\langle
A_{\frac{1}{2}+m}^{\max}f|A_{\frac{1}{2}+m}^{\max}g\rangle,$
$\displaystyle\quad f,g\in{\cal H}_{0}^{1}+{\mathbb{C}}x^{\frac{1}{2}+m}\xi.$
(8.21)

Note that for ${\rm Re}(m)>0$ both factorizations yield the same quadratic
form. This is no longer true for ${\rm Re}(m)=0$, $m\neq 0$, when there are
two distinct quadratic forms with distinct domain corresponding to $H_{m}$.
Finally, for $-1<m<0$, and also for $m=0$, we have a unique factorization.

Let us finally specialize Theorem 8.3 to real $m$. The following theorem is
essentially identical with Thm 4.22 of [6].

###### Theorem 8.4

For real $-1<m$ the operators $H_{m}$ are positive and self-adjoint. The
corresponding sesquilinear form can be factorized as follows:

  1. 1.

For $1\leq m$,

$\displaystyle(\sqrt{H_{m}}f|\sqrt{H_{m}}g)=(A_{\frac{1}{2}+m}f|A_{\frac{1}{2}+m}g)$
$\displaystyle=(A_{\frac{1}{2}-m}f|A_{\frac{1}{2}-m}g),$ $\displaystyle\quad
f,g\in{\cal Q}(H_{m})={\cal H}_{0}^{1}.$ (8.22)

$H_{m}$ is essentially self-adjoint on
$C_{\mathrm{c}}^{\infty}({\mathbb{R}}_{+})$.

  2. 2.

For $0<m<1$,

$\displaystyle(\sqrt{H_{m}}f|\sqrt{H_{m}}g)=(A_{\frac{1}{2}+m}f|A_{\frac{1}{2}+m}g)$
$\displaystyle=(A_{\frac{1}{2}-m}^{\min}f|A_{\frac{1}{2}-m}^{\min}g),$
$\displaystyle\quad f,g\in{\cal Q}(H_{m})={\cal H}_{0}^{1}.$ (8.23)

$H_{m}$ is the Friedrichs extension of $L_{m^{2}}$ restricted to
$C_{\mathrm{c}}^{\infty}({\mathbb{R}}_{+})$.

  3. 3.

For $m=0$,

$\displaystyle(\sqrt{H_{0}}f|\sqrt{H_{0}}g)=(A_{\frac{1}{2}}f|A_{\frac{1}{2}}g),$
$\displaystyle\quad f,g\in{\cal Q}(H_{0})={\cal D}(A_{\frac{1}{2}})\mathchar
13609\relax{\cal H}_{0}^{1}+{\mathbb{C}}x^{\frac{1}{2}}\xi.$ (8.24)

$H_{0}$ is both the Friedrichs and Krein extension of $L_{0}$ restricted to
$C_{\mathrm{c}}^{\infty}({\mathbb{R}}_{+})$.

  4. 4.

For $-1<m<0$,

$\displaystyle(\sqrt{H_{m}}f|\sqrt{H_{m}}g)=(A_{\frac{1}{2}+m}^{\max}f|A_{\frac{1}{2}+m}^{\max}g),$
$\displaystyle\quad f,g\in{\cal Q}(H_{m})={\cal
H}_{0}^{1}+{\mathbb{C}}x^{\frac{1}{2}+m}\xi.$ (8.25)

$H_{m}$ is the Krein extension of $L_{m^{2}}$ restricted to
$C_{\mathrm{c}}^{\infty}({\mathbb{R}}_{+})$.

##  Appendix A Holomorphic families of closed operators and the Kato-Rellich
Theorem

In this appendix we describe a few general concepts and facts from the
operator theory, which we use in our paper.

The definition (or actually a number of equivalent definitions) of a
holomorphic family of bounded operators is quite obvious and does not need to
be recalled. In the case of unbounded operators the situation is more subtle.

Suppose that $\Theta$ is an open subset of ${\mathbb{C}}$, ${\cal H}$ is a
Banach space, and $\Theta\ni z\mapsto H(z)$ is a function whose values are
closed operators on ${\cal H}$. We say that this is a holomorphic family of
closed operators if for each $z_{0}\in\Theta$ there exists a neighborhood
$\Theta_{0}$ of $z_{0}$, a Banach space ${\cal K}$ and a holomorphic family of
bounded operators $\Theta_{0}\ni z\mapsto A(z)\in B({\cal K},{\cal H})$ such
that ${\rm Ran}A(z)={\cal D}(H(z))$ and

$\Theta_{0}\ni z\mapsto H(z)A(z)\in B({\cal K},{\cal H})$

is a holomorphic family of bounded operators.

The following theorem is essentially a version of the well-known Kato-Rellich
Theorem generalized from self-adjoint to closed operators:

###### Theorem A.1

Suppose that $A$ is a closed operator on a Hilbert space ${\cal H}$. Let $B$
be an operator ${\cal D}(A)\to{\cal H}$ such that

$\|Bf\|\leq c\|Af\|,\quad f\in{\cal D}(A).$ (A.26)

Then for $|z|<\frac{1}{c}$ the operator $A+zB$ is closed on ${\cal D}(A)$ and

$\left\\{z\in{\mathbb{C}}\ |\ |z|<c^{-1}\right\\}\ni z\mapsto A+zB$ (A.27)

is a holomorphic family of closed operators.

Proof. We easily check that the norms $\sqrt{\|f\|^{2}+\|Af\|^{2}}$ and
$\sqrt{\|f\|^{2}+\|(A+zB)f\|^{2}}$ are equivalent for $|z|<\frac{1}{c}$. Let
${\cal H}_{0}$ be the closure of ${\cal D}(A)$ in ${\cal H}$. The restriction
of $A$ to ${\cal H}_{0}$ is densely defined, so that we can define $A^{*}$.
The operator $(A^{*}A+{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})^{-\frac{1}{2}}$ is unitary from ${\cal
H}_{0}$ to ${\cal D}(A)$. Clearly, it is bounded in the sense of ${\cal
H}_{0}$. Now

${\mathbb{C}}\ni z\mapsto(A+zB)(A^{*}A+{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})^{-\frac{1}{2}}$
(A.28)

is obviously a polynomial of degree 1 with values in bounded operators (hence
obviously a holomorphic family). $\Box$

Let us also quote the following fact proven by Bruk [5], see also [10]:

###### Proposition A.2

If $z\mapsto A(z)$ is a holomorphic family of closed operators, then so is
$z\mapsto A(\overline{z})^{*}$.

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