137151

# On nonsymmetric rank one singular perturbations of selfadjoint operators

Mykola Dudkin  National Technical University of Ukraine ”Kyiv Polytechnic
Institute”, 37 Prospect Peremogy, Kyiv, 03056, Ukraine  dudkinimath.kiev.ua
Tetiana Vdovenko  National Technical University of Ukraine ”Kyiv Polytechnic
Institute”, 37 Prospect Peremogy, Kyiv, 03056, Ukraine<EMAIL_ADDRESS>
(Date: 22/09/2015; Revised 23/02/2016)

###### Abstract.

We consider nonsymmetric rank one singular perturbations of a selfadjoint
operator, i.e., an expression of the form
$\tilde{A}=A+\alpha\left\langle\cdot,\omega_{1}\right\rangle\omega_{2}$,
$\omega_{1}\not=\omega_{2}$, $\alpha\in{\mathbb{C}}$, in a general case
$\omega_{1},\omega_{2}\in{\mathcal{H}}_{-2}$.

Using a constructive description of the perturbed operator $\tilde{A}$, we
investigate some spectral and approximations properties of $\tilde{A}$. The
wave operators corresponding to the couple $A$, $\tilde{A}$ and a series of
examples are also presented.

###### Key words and phrases:

Singular perturbation, nonsymmetric perturbations, eigenvalue problem, M.
Krein’s formula.

###### 2000 Mathematics Subject Classification:

Primary: 47A10, 47A55, 47A75

##  1\. Introduction

The theory of rank one (symmetric) singular perturbations of a self adjoint
operator has obtained much attention of physicists and mathematicians. Several
papers and research monographs are devoted to this theory (see, f.e., [1, 3,
13] and references therein).

The aim of the current paper is to investigate a generalization of the
singular (symmetric) perturbation theory to the case of nonsymmetric
perturbations of the form
$\tilde{A}=A+\alpha\left\langle\cdot,\omega_{1}\right\rangle\omega_{2}$, where
$A=A^{*}$ is a given selfadjoint operator perturbed by
$\alpha\left\langle\cdot,\omega_{1}\right\rangle\omega_{2}$ with vectors
$\omega_{1}\not=\omega_{2}$ that belong to the negative space
${\mathcal{H}}_{-2}$ from the $A$-scale (the scale is generated by the
operator $A$) and $\alpha\in{\mathbb{C}}$.

If $\omega_{1}=\omega_{2}$ and $\alpha\in{\mathbb{R}}$, then we meet the
classical case, i.e., the well known theory of a (symmetric) singular
perturbation of selfadjoint operators [1, 3, 13].

In this article we continue our investigations started in [20], where we
considered only the case $\omega_{1},\omega_{2}\in{\mathcal{H}}_{-1}$. A
significant improvement of the previous studies is in the consideration of
perturbations by arbitrary vectors, i.e., vectors from ${\mathcal{H}}_{-2}$,
too.

Since the study of singularly perturbed operators is extended to perturbations
by nonsymmetric potentials, we will also expect that the spectral properties
(in particular the point spectrum of $\tilde{A}$) must be similar as in the
classical case (including the unexpected for the perception the so-called
associated pair of eigenvalues).

The next important topic of this article is to clarify approximation
properties of these operators. In particular, we investigate an approximation
of classical (symmetric) perturbations by nonsymmetric perturbations taking
into account $H_{-1}$\- and $H_{-2}$-perturbations.

When investigating wave operators for the couple $A$ and a nonsymmetrically
perturbed $\tilde{A}$, we solve a number of problems, including the following:
whether $\tilde{A}$ is a spectral type operator; whether $\tilde{A}$ is
additionally of a scalar type one; whether there exist wave operators for the
couple $A$ and $\tilde{A}$; and whether we can write explicit expressions for
these wave operators and, as a consequence, the scattering matrix.

In general, the idea and motivations for considering nonsymmetric
perturbations is not new. Closely related investigations are in [15, 16], the
ones carried out from the point of view of nonselfadjoint extensions in [19],
and a non-local interactions approach has been taken in [17, 18].

##  2\. The main definitions and properties

Let ${\mathcal{H}}$ be a separable Hilbert space with a scalar product
$(\cdot,\cdot)$ and the norm $\|\cdot\|=\sqrt{(\cdot,\cdot)}$. We consider a
selfadjoint semi-bounded operator $A=A^{*}$ defined on a domain ${\rm
Dom}A={\mathfrak{D}}(A)$ in ${\mathcal{H}}$. The sets $\sigma(\cdot)$,
$\sigma_{p}(\cdot)$, $\sigma_{ac}(\cdot)$, $\sigma_{c}(\cdot)$, $\rho(\cdot)$
denote the spectrum, the point spectrum, the absolutely continuous spectrum,
the continuous spectrum, and the regular points of a corresponding operator,
respectively.

An operator $A$ is associated with the $A$-scale of Hilbert spaces [3]. We
consider only a part of the $A$-scale, namely,

${\mathcal{H}}_{-2}\supset{\mathcal{H}}_{-1}\supset{\mathcal{H}}\equiv{\mathcal{H}}_{0}\supset{\mathcal{H}}_{+1}\supset{\mathcal{H}}_{+2}$
(1)

where ${\mathcal{H}}_{+1}:={\mathfrak{D}}(|A|^{1/2})$ and
${\mathcal{H}}_{+2}:={\mathfrak{D}}(A)$ are endowed with the norms
$\|\varphi\|_{k}=\|(|A|+I)^{k/2}\varphi\|$, $k=1,2$,
$\varphi\in{\mathcal{H}}_{k}(A)$, respectively ($I$ stands for the identity);
and ${\mathcal{H}}_{-k}:={\mathcal{H}}_{-k}(A)$ is the negative (dual) space,
i.e., the completion of ${\mathcal{H}}$ with respect to the norm
$\|f\|_{-k}=\|(|A|+I)^{-k/2}f\|$, $k=1,2$, $f\in{\mathcal{H}}$. Let
$\langle\cdot,\cdot\rangle$ denote the usual dual scalar product for the
spaces ${\mathcal{H}}_{k}$ and ${\mathcal{H}}_{-k}$. The inner product in
${\mathcal{H}}_{k}$ and ${\mathcal{H}}_{-k}$ is denoted by $(\cdot,\cdot)_{\pm
k}$, $k=1,2$.

The operator $A$ has an extension by continuity to ${\mathcal{H}}$
(${\mathcal{H}}_{1}$) and it is understood as a bounded operator from
${\mathcal{H}}$ (${\mathcal{H}}_{1}$) into ${\mathcal{H}}_{-2}$
(${\mathcal{H}}_{-1}$). We denote such an extension by ${\bf A}$ and ${\bf
R}_{z}=({\bf A}-z)^{-1}$, $z\in\rho({\bf A})$, is a corresponding resolvent.

In some cases, we can continue the usual dual scalar product
$\langle\cdot,\cdot\rangle$ to the case $\langle\omega,\phi\rangle$, where
$\omega,\phi\in{\mathcal{H}}_{-2}$ (of course $\omega\not=\phi$) in the
following way. For example, if we can decompose vectors
$\omega=\omega_{1}+\omega_{2}$ and $\phi=\phi_{1}+\phi_{2}$ so that ${\rm
spsupp}(\omega_{i})\subseteq\Pi_{i}$, ${\rm
spsupp}(\phi_{i})\subseteq\Pi_{i}$, $i=1,2$, $\Pi\cap\Pi=\varnothing$ and
$\omega_{1}\in{\mathcal{H}}_{+2}$, $\omega_{2}\in{\mathcal{H}}_{-2}$,
$\phi_{1}\in{\mathcal{H}}_{-2}$, $\phi_{2}\in{\mathcal{H}}_{+2}$, then we can
have
$\langle\omega,\phi\rangle=\langle\omega_{1},\phi_{1}\rangle+\langle\omega_{2},\phi_{2}\rangle<\infty$,
and $\langle\omega_{1},\phi_{2}\rangle=\langle\omega_{2},\phi_{1}\rangle=0$,
since ${\rm spsupp}(\omega_{1})\cap{\rm spsupp}(\phi_{2})=\varnothing$ and
${\rm spsupp}(\omega_{2})\cap{\rm spsupp}(\phi_{1})=\varnothing$. Here ${\rm
spsupp}(\cdot)$ denotes the spectral support of the corresponding vector in
the sense of the operator ${\bf A}$. By definition (cf. [9]), for
$\omega\in{\mathcal{H}}_{-2}$,

${\rm spsupp}(\omega):=\\{\lambda\in{\mathbb{R}}\ |\ \forall
O_{\lambda,\varepsilon}\exists\psi\in C_{0}({\mathbb{R}})\cap
L_{2}({\mathbb{R}},d\rho(\lambda))\ :\\\ {\rm supp}(\psi)\subset
O_{\lambda,\varepsilon}\ \text{and}\
\int\limits_{{\mathbb{R}}}\hat{\omega}(\lambda)\psi(\lambda)\,d\rho(\lambda)\not=0\\},$

where $O_{\lambda,\varepsilon}$ is an $\varepsilon$-neighborhood of a point
$\lambda$; $C_{0}({\mathbb{R}})$ is the set of continues functions with
compact supports on ${\mathbb{R}}$; $\hat{\omega}(\lambda)$ denotes the
Fourier image of the vector $\omega$. According to the central spectral
theorem [8], the Fourier transform between ${\mathcal{H}}_{-2}$ and
$L_{2}({\mathbb{R}},d\rho(\lambda))$ takes the operator ${\bf A}$ to the
multiplication operator by the independent variable $\lambda$ in the space
$L_{2}({\mathbb{R}},d\rho(\lambda))$ with the Borel measure $d\rho(\lambda)$.

Let us consider an operator $V$ in the $A$-scale, such that
${\mathfrak{D}}(V)\subseteq{\mathcal{H}}_{+k}$ and
${\mathfrak{R}}(V)\subseteq{\mathcal{H}}_{-k}$, $k=1,2$. In our case,
$V=V^{\omega_{1},\omega_{2}}=\langle\cdot,\omega_{1}\rangle\omega_{2}$,
$\omega_{1},\omega_{2}\in{\mathcal{H}}_{-k}$, $k=1,2$. Since the operator
${\bf A}$ is bounded and acts from ${\mathcal{H}}_{0}$ into the whole
${\mathcal{H}}_{-k}$, $k=1,2$, the expression ${\bf A}+V$ is a bounded linear
operator from ${\mathcal{H}}_{+k}$ into ${\mathcal{H}}_{-k}$. Let us remark
that due to [7] the adjoint operator $({\bf A}+V)^{+}$ is correctly defined
and acts also from ${\mathcal{H}}_{+k}$ into ${\mathcal{H}}_{-k}$, $k=1,2$.

Now, the formal expression $A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2}$
has a sense of an operator ${\bf A}+\langle\cdot,\omega_{1}\rangle\omega_{2}$
defined on ${\mathcal{H}}_{+k}$, acting from ${\mathcal{H}}_{+k}$ into
${\mathcal{H}}_{-k}$, $k=1,2$, and restricted to ${\mathcal{H}}$,

$A^{\omega_{1},\omega_{2}}=({\bf
A}+\langle\cdot,\omega_{1}\rangle\omega_{2})\restriction_{{\mathcal{H}}}.$ (2)

In what follows, we write usually $A$ instead of ${\bf A}$ and hence, $R_{z}$
will be used instead of ${\bf R}_{z}$.

Without loss of generality, supposing also that the operator $A$ is strongly
positive $A>0$, we give a constructive definition of the operator
$A^{\omega_{1},\omega_{2}}$.

###### Definition 1

Let $A$ be a positive self-adjoint operator on a separable Hilbert space
${\mathcal{H}}$, and consider $A^{\omega_{1},\omega_{2}}$ in (2) with
$\omega_{i}\in{\mathcal{H}}_{-2}\setminus{\mathcal{H}}$,
$\omega_{1}\not=\omega_{2}$. We put $\eta_{i}={\bf A}^{-1}\omega_{i}$,
$i=1,2$.

I. The operator $A^{\omega_{1},\omega_{2}}$ is called singularly rank-one
nonsymmetrically _uniquely_ perturbed with respect to the operator $A$ if
$|(A^{1/2}\eta_{2},A^{1/2}\eta_{1})|<\infty$. Moreover,

${\mathfrak{D}}(A^{\omega_{1},\omega_{2}})=\left\\{\psi=\varphi+b\eta_{2}\ |\
\varphi\in{\mathfrak{D}}(A),\
b=b(\varphi)=\frac{(A\varphi,\eta_{1})}{1+(A^{1/2}\eta_{2},A^{1/2}\eta_{1})}\right\\}$
(3)

in the case $(A^{1/2}\eta_{2},A^{1/2}\eta_{1})\not=-1$, and

${\mathfrak{D}}(A^{\omega_{1},\omega_{2}})={\mathfrak{D}}_{{\mathcal{H}}_{1}}\dot{+}\\{c\eta_{2}\\},\quad{\mathfrak{D}}_{{\mathcal{H}}_{1}}=\left\\{\varphi\in{\mathfrak{D}}(A)\
|\ (A\varphi,\eta_{1})=0\right\\}$ (4)

in the case $(A^{1/2}\eta_{2},A^{1/2}\eta_{1})=-1$, (this fact we denote by
$A^{\omega_{1},\omega_{2}}\in{\mathcal{P}}(A)$).

II. The operator $A^{\omega_{1},\omega_{2}}$ is called singularly rank-one
nonsymmetric _parametrically_ perturbed with respect to the operator $A$ if
$(A^{1/2}\eta_{2},A^{1/2}\eta_{1})$ does not exist. Moreover,

${\mathfrak{D}}(A^{\omega_{1},\omega_{2}})=\Big{\\{}\psi=\varphi+b\eta_{2}\ |\
\varphi\in{\mathfrak{D}}(A),\\\
\left.b=b(\varphi)=\frac{(A\varphi,\eta_{1})}{1+\tau+(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})}\right\\}$
(5)

in the case
$(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})\not=-\tau-1$,
where $\tau\in{\mathbb{C}}$ is a parameter, and

${\mathfrak{D}}(A^{\omega_{1},\omega_{2}})={\mathfrak{D}}_{{\mathcal{H}}_{1}}\dot{+}\\{c\eta_{2}\\},\quad{\mathfrak{D}}_{{\mathcal{H}}_{1}}=\left\\{\varphi\in{\mathfrak{D}}(A)\
|\ (A\varphi,\eta_{1})=0\right\\}$ (6)

in the case
$(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})=-\tau-1$,
(this fact we denote by $A^{\omega_{1},\omega_{2}}\in{\mathcal{P}}_{\tau}(A)$)

The action of the perturbed operator is given by the rule

$A^{\omega_{1},\omega_{2}}\psi=A\varphi$

in each case.

Remark 1. If $|(A^{1/2}\eta_{2},A^{1/2}\eta_{1})|<\infty$ in the second part
of Definition 1, then taking
$\tau=(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})$ we
obtain the first part of Definition 1, that was considered also in [20].

Remark 2. If $(A^{1/2}\eta_{2},A^{1/2}\eta_{1})$ does not exists i.e., we have
the second part of Definition 1, then still
$|(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})|<\infty$,
since $\omega_{1},\omega_{2}\in{\mathcal{H}}_{-2}$.

Remark 3. The defined in Definition 1 operator $A^{\omega_{1},\omega_{2}}$ can
be described in the following way. A linear closed operator
$A^{\omega_{1},\omega_{2}}\not=A$ densely defined on ${\mathcal{H}}$ is
nonsymmetric singularly perturbed with respect to the operator $A$ if both
sets,

$\displaystyle{\mathfrak{D}}$
$\displaystyle=\\{f\in{\mathfrak{D}}(A)\cap{\mathfrak{D}}(A^{\omega_{1},\omega_{2}})\
|\ Af=\tilde{A}f\\},$ (7) $\displaystyle{\mathfrak{D}}_{*}$
$\displaystyle=\\{f\in{\mathfrak{D}}(A)\cap{\mathfrak{D}}((A^{\omega_{1},\omega_{2}})^{*})\
|\ Af=\tilde{A}^{*}f\\},$ (8)

are dense in ${\mathcal{H}}$. In general
$\tilde{A}\in{\mathcal{P}}_{\tau}(A)$.

It is clear that for each operator
$A^{\omega_{1},\omega_{2}}\in{\mathcal{P}}_{\tau}(A)$, there exist densely
defined symmetric restrictions, i.e., operators
$\dot{A}:=A\restriction{\mathfrak{D}}$ and
$\dot{A}_{*}:=A\restriction{\mathfrak{D}}_{*}$ with nontrivial deficiency
indices

${\bf n}^{\pm}(\dot{A})={\rm dim\ ker}(\dot{A}\mp z)^{*}\not=0,\quad{\bf
n}^{\pm}(\dot{A}_{*})={\rm dim\ ker}(\dot{A}_{*}\mp z)^{*}\not=0,\quad
z\in\rho(A).$

(In this article we meet often the case where ${\bf n}^{\pm}(\dot{A})={\bf
n}^{\pm}(\dot{A}_{*})=1$.)

If ${\mathfrak{D}}={\mathfrak{D}}_{*}$ and $\tilde{A}=\tilde{A}^{*}$, then we
are in the case of the usual abstract definition of singularly perturbed
selfadjoint operators [3, 13] $\tilde{A}\in{\mathcal{P}}_{s}(A)$, that is, the
definition given above generalizes the known definition of a selfadjoint
singular perturbation to the case of a nonselfadjoint one.

The operator defined above, $A^{\omega_{1},\omega_{2}}$, has the following
general properties.

###### Proposition 1

For an arbitrary nonzero constant $a\in{\mathbb{C}}$ we have
$A^{a\omega_{1},\omega_{2}}=A^{\omega_{1},\bar{a}\omega_{2}}$.

###### Proof.

From Definition 1 in both cases (3), (4) and (5), (6) it follows that
${\mathfrak{D}}(A^{a\omega_{1},\omega_{2}})={\mathfrak{D}}(A^{\omega_{1},\bar{a}\omega_{2}})$
and
$A^{a\omega_{1},\omega_{2}}\psi=A^{\omega_{1},\bar{a}\omega_{2}}\psi=A\varphi$.
∎

###### Proposition 2

The adjoint operator $(A^{\omega_{1},\omega_{2}})^{*}$ satisfies the identity

$(A^{\omega_{1},\omega_{2}})^{*}=A^{\omega_{2},\omega_{1}}.$

###### Proof.

For the proof we use the second part of Definition 1 for
$A^{\omega_{1},\omega_{2}}$ and $A^{\omega_{2},\omega_{1}}$ and verify the
identity

$(A^{\omega_{1},\omega_{2}}f_{1},f_{2})=(f_{1},A^{\omega_{2},\omega_{1}}f_{2}),$
(9)

for $f_{1}\in{\mathfrak{D}}(A^{\omega_{1},\omega_{2}})$ and
$f_{2}\in{\mathfrak{D}}(A^{\omega_{2},\omega_{1}})$ of the form
$f_{1}=\varphi_{1}+b_{1}\eta_{2}$ and $f_{2}=\varphi_{2}+b_{2}\eta_{1}$,
correspondingly, $\varphi_{1},\varphi_{2}\in{\mathfrak{D}}(A)$. The left-hand
side of (9) is of the form

$(A^{\omega_{1},\omega_{2}}f_{1},f_{2})=(A^{\omega_{1},\omega_{2}}(\varphi_{1}+b_{1}\eta_{2}),(\varphi_{2}+b_{2}\eta_{1}))=(A\varphi_{1},\varphi_{2})+\bar{b}_{2}(A\varphi_{1},\eta_{1}),$
(10)

where

$b_{1}=b_{1}(\varphi_{1})=\frac{(A\varphi_{1},\eta_{1})}{1+\tau+(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})}.$

The right-hand side of (9) is of the form

$(f_{1},A^{\omega_{2},\omega_{1}}f_{2})=((\varphi_{1}+b_{1}\eta_{2}),A^{\omega_{2},\omega_{1}}(\varphi_{2}+b_{2}\eta_{1}))=(\varphi_{1},A\varphi_{2})+b_{1}(\eta_{2},A\varphi_{2}),$
(11)

where

$b_{2}=b_{2}(\varphi_{2})=\frac{(A\varphi_{2},\eta_{2})}{1+\tau+(A^{1/2}(A^{2}+1)^{-1/2}\eta_{1},A^{1/2}(A^{2}+1)^{-1/2}\eta_{2})}.$

The statement of the proposition follows from the obvious equality of last
terms from (10) and (11),

$\frac{\overline{(A\varphi_{2},\eta_{2})}}{1+\tau+\overline{(A^{1/2}(A^{2}+1)^{-1/2}\eta_{1},A^{1/2}(A^{2}+1)^{-1/2}\eta_{2})}}(A\varphi_{1},\eta_{1})\\\
=\frac{(A\varphi_{1},\eta_{1})}{1+\tau+(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})}(\eta_{2},A\varphi_{2}).$

The proof in case (5), i.e.,
$(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})\not=-\tau-1$
is completed. The case (6), i.e.,
$(A^{1/2}(A^{2}+1)^{-1/2}\eta_{2},A^{1/2}(A^{2}+1)^{-1/2}\eta_{1})=-\tau-1$ is
also valid.

The cases (3), (4) are particular with respect to (5), (6). ∎

##  3\. The description of a rank-one nonsymmetric singular perturbation by
resolvents

In this section we consider the perturbed operator with a parameter
$\alpha\in{\mathbb{C}}$ and denote it by
$\tilde{A}=A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2}$, where
$\omega_{i}\in{\mathcal{H}}_{-2}\setminus{\mathcal{H}}$ and
$\|\omega_{i}\|_{-1}=1$, $i=1,2$. The set of such operators is also denoted by
${\mathcal{P}}_{\tau}(A)$. At the beginning, let us briefly remark that if
$\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ then for the adjoint operator, we have
$\tilde{A}^{*}\in{\mathcal{P}}_{\tau}(A)$, which is also due to the
investigations in [7] and Proposition 2.

###### Theorem 1

For the resolvents $R_{z}=(A-z)^{-1}$ and $\tilde{R}_{z}=(\tilde{A}-z)^{-1}$
of operators $A=A^{*}>1$ and $\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ on a
separable Hilbert space ${\mathcal{H}}$, a formula similar to M. Krein’s
formula holds for $z,\xi,\zeta\in\rho(A)\cap\rho(\tilde{A})$,

$\tilde{R}_{z}=R_{z}+b_{z}(\cdot,n_{\bar{z}})m_{z},$ (12)

with

$n_{z}=(A-\xi)(A-z)^{-1}n_{\xi},\quad m_{z}=(A-\zeta)(A-z)^{-1}m_{\zeta},$
(13)

where $n_{z},m_{z}\in{\mathcal{H}}\setminus{\mathcal{H}}_{+2}$ and

$b_{z}^{-1}-b_{\xi}^{-1}=(\xi-z)(m_{\xi},n_{\bar{z}}).$ (14)

The vectors $n_{z},m_{z}$ and the number $b_{z}$ are connected with
$\omega_{1},\omega_{2}$ as follows:

$n_{z}=R_{z}\omega_{1},m_{z}=R_{z}\omega_{2},\quad-
b_{z}^{-1}={\alpha}^{-1}+\tau+\langle(A^{2}+1)^{-1}\omega_{2},(1+\bar{z}A)R_{\bar{z}}\omega_{1}\rangle,$
(15)

where $\alpha\not=0$.

The case $\alpha=0$ can be also included in the considerations, since in case
$\alpha=0$, we put $b_{z}\equiv 0$ and obtain $\tilde{R}_{z}\equiv R_{z}$.

###### Proof.

From the expression
$\tilde{A}=A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2}$ and for some
$z\in\rho(A)\cap\rho(\tilde{A})$ we have
$\tilde{A}-z=A-z+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2}$, and hence

$(\tilde{A}-z)^{-1}=(A-z)^{-1}-\alpha\langle\cdot,(A-\bar{z})^{-1}\omega_{1}\rangle(\tilde{A}-z)^{-1}\omega_{2},$
(16)

where $(A-z)^{-1}$ and $(\tilde{A}-z)^{-1}$ are considered as operators from
${\mathcal{H}}_{-2}$ into ${\mathcal{H}}$. In the case of the first part of
Definition 1 we can continue our proof as in [20]. In the case of the second
(general) part of Definition 1 we need to consider the next expression
following from (16):

$(\tilde{A}-z)^{-1}=(A-z)^{-1}-\alpha\\{\langle
A(A^{2}+1)^{-1/2}\cdot,A(A-\bar{z})^{-1}\omega_{1}\rangle\\\
+\langle(A^{2}+1)^{-1}\cdot,(1+\bar{z}A)(A-\bar{z})^{-1}\omega_{1}\rangle\\}(\tilde{A}-z)^{-1}\omega_{2}.$
(17)

In particular, for $\omega_{2}\in{\mathcal{H}}_{-2}$ we have

$(\tilde{A}-z)^{-1}\omega_{2}=(A-z)^{-1}\omega_{2}-\alpha\\{\tau+\langle(A^{2}+1)^{-1}\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}\omega_{1}\rangle\\}(\tilde{A}-z)^{-1}\omega_{2}.$

where instead of $\langle\omega_{2},A(A^{2}+1)^{-1}\omega_{1}\rangle$ we write
$\tau$. Substituting

$(\tilde{A}-z)^{-1}\omega_{2}=\frac{\alpha}{1+\alpha\\{\tau+\langle(A^{2}+1)^{-1}\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}\omega_{1}\rangle\\}}(A-z)^{-1}\omega_{2}.$

into (17) we get

$(\tilde{A}-z)^{-1}=(A-z)^{-1}-\frac{1}{\alpha^{-1}+\tau+\langle(A^{2}+1)^{-1}\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}\omega_{1}\rangle}\\\
\times(\cdot,(A-\bar{z})^{-1}\omega_{1})(A-z)^{-1}\omega_{2}.$ (18)

where $\alpha\not=0$. If we put

$n_{z}=(A-z)^{-1}\omega_{1},\quad m_{z}=(A-z)^{-1}\omega_{2},$ (19)

and

$b_{z}^{-1}=-({\alpha^{-1}+\tau+\langle(A^{2}+1)^{-1}\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}\omega_{1}\rangle}),$
(20)

then we obtain (15).

Analogously, starting with
$\tilde{A}^{*}=A+\bar{\alpha}\langle\cdot,\omega_{2}\rangle\omega_{1}$, we
also go to (19) and (20) in the equivalent form

$\bar{b}_{z}^{-1}=-({\bar{\alpha}^{-1}+\bar{\tau}+\langle(A^{2}+1)^{-1}\omega_{1},(1+zA)(A-z)^{-1}\omega_{2}\rangle}),$

Let us remark that if
$\omega_{1},\omega_{2}\in{\mathcal{H}}_{-2}\setminus{\mathcal{H}}$ then
$n_{z},m_{z}\in{\mathcal{H}}\setminus{\mathcal{H}}_{2}$ (see, for example
[14]), and, more precisely, if
$\omega_{1},\omega_{2}\in{\mathcal{H}}_{-2}\setminus{\mathcal{H}}_{-1}$, then
$n_{z},m_{z}\in{\mathcal{H}}\setminus{\mathcal{H}}_{+1}$ and if
$\omega_{1},\omega_{2}\in{\mathcal{H}}_{-1}\setminus{\mathcal{H}}$, then
$n_{z},m_{z}\in{\mathcal{H}}_{+1}\setminus{\mathcal{H}}_{+2}$.

By using notations (19) in the form $n_{z}=(A-z)^{-1}\omega_{1}$ and
$n_{\xi}=(A-\xi)^{-1}\omega_{1}$ we obtain
$\omega_{1}=(A-z)n_{z}=(A-\xi)n_{\xi}$ and consequently the first expression
in (13),

$n_{z}=(A-\xi)(A-z)^{-1}n_{\xi}.$

This expression makes sense in ${\mathcal{H}}$ if we consider $A$ on
${\mathcal{H}}$ (but not ${\bf A}$). Analogously we obtain the second
expression in (13). By a use of the Hilbert identity with (13), we obtain
(14),

$\displaystyle b_{z}^{-1}-$ $\displaystyle b_{\xi}^{-1}$
$\displaystyle=-\langle(A^{2}+1)^{-1}\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}\omega_{1}\rangle+\langle(A^{2}+1)^{-1}\omega_{2},(1+\bar{\xi}A)(A-\bar{\xi})^{-1}\omega_{1}\rangle$
$\displaystyle=\langle(A^{2}+1)^{-1}\omega_{2},\left((1+\bar{z}A)(A-\bar{\xi})^{-1}-(1+\bar{\xi}A)(A-\bar{z})^{-1}\right)\omega_{1}\rangle$
$\displaystyle=(\xi-z)\langle(A-\xi)^{-1}\omega_{2},(A-\bar{z})^{-1}\omega_{1}\rangle$
$\displaystyle=(\xi-z)(m_{\xi},n_{\bar{z}}).$

This completes the proof. ∎

Let us remark that it is possible that $b_{z}=\infty$, and it is so iff
$z\in\sigma_{p}(\tilde{A})$, but (12) is also valid in such a case.

At the end of the article, we give an example that illustrates the following
corollary from Theorem 1.

###### Corollary 1

If the operators $A=A^{*}$ and $\tilde{A}\in{\mathcal{P}}(A)$ have inverses on
a separable Hilbert space ${\mathcal{H}}$, i.e.,
$0\in\rho(A)\cap\rho(\tilde{A})$, then (12), (13) and (14) have the form

$\tilde{A}^{-1}=A^{-1}+b_{0}(\cdot,n_{0})m_{0},$ (21)

where

$n_{0}=A^{-1}\omega_{1},\quad m_{0}=A^{-1}\omega_{2},\quad-
b_{0}^{-1}={\alpha}^{-1}+\langle\omega_{2},A^{-1}\omega_{1}\rangle.$ (22)

###### Proof.

The proof follows from Theorem 1 in two ways. The first one is to take into
account that $|\langle\omega_{2},A^{-1}\omega_{1}\rangle|<\infty$ and repeat
the proof of Theorem 1. The second one is to substitute $z=0$ into (12), (13)
and (14). ∎

##  4\. Spectral properties of rank one nonsymmetric singular perturbations

As a starting point, let us remark that the continuous spectrum
$\sigma_{c}(A)$ of the operator $A$ is unchanged by finite rank perturbations,
i.e., $\sigma_{c}(A)=\sigma_{c}(\tilde{A})$,
$\tilde{A}\in{\mathcal{P}}_{\tau}(A)$.

###### Theorem 2

Let the perturbed operator $\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ possess a new
eigenvalue $\lambda\in{\mathbb{C}}$ comparing to $A$, i.e., there exist
$\lambda\in\sigma_{p}(\tilde{A})$, $\lambda\not\in\sigma_{p}(A)$. Then for the
corresponding eigenvectors $\varphi$, $\psi$:
$\tilde{A}\varphi=\lambda\varphi$ and $\tilde{A}^{*}\psi=\bar{\lambda}\psi$,
the following relations hold true:

$(\lambda-z)b_{z}(\varphi,n_{\bar{z}})=1,\quad\varphi=(A-z)(A-\lambda)^{-1}m_{z};$
(23)
$(\bar{\lambda}-\bar{z})\bar{b}_{z}(\psi,m_{z})=1,\quad\psi=(A-\bar{z})(A-\bar{\lambda})^{-1}n_{\bar{z}}.$
(24)

###### Proof.

The proof follows from Theorem 1. Let $\tilde{A}\varphi=\lambda\varphi$, i.e.,

$\tilde{R}_{z}\varphi=R_{z}\varphi+b_{z}(\varphi,n_{\bar{z}})m_{z}=(\lambda-z)^{-1}\varphi.$

Hence,
$b_{z}(\varphi,n_{\bar{z}})m_{z}=[(\lambda-z)^{-1}-R_{z}]\varphi=(\lambda-z)^{-1}(A-\lambda)(A-z)^{-1}\varphi,$

$(\lambda-z)b_{z}(\varphi,n_{\bar{z}})(A-z)(A-\lambda)^{-1}m_{z}=\varphi.$
(25)

Multiplying the last expression by $n_{\bar{z}}$ we obtain that

$(\lambda-z)b_{z}(\varphi,n_{\bar{z}})\left((A-z)(A-\lambda)^{-1}m_{z},n_{\bar{z}}\right)=(\varphi,n_{\bar{z}}),$

and, hence,

$(\lambda-z)b_{z}(\varphi,n_{\bar{z}})=1.$ (26)

We remark that (25) and (26) gives $\varphi=(A-z)(A-\lambda)^{-1}m_{z}$. This
proves (23). Analogously, considering $\tilde{A}^{*}\psi=\bar{\lambda}\psi$ we
can prove (24). ∎

###### Corollary 2

Let us put $z=0$ in the case $\tilde{A}\in{\mathcal{P}}(A)$ under the
conditions of Theorem 2. Then (23) and (24) have the form

$\lambda
b_{0}(\varphi,n_{0})=1,\quad\varphi=A(A-\lambda)^{-1}m_{0};\quad\bar{\lambda}\bar{b}_{0}(\psi,m_{0})=1,\quad\psi=A(A-\bar{\lambda})^{-1}n_{0}.$

###### Proposition 3

Let the perturbed operator $\tilde{A}\in{\mathcal{P}}(A)$ possess a new
eigenvalue $\lambda\in{\mathbb{C}}$ in comparison with $A$ and let the
eigenvectors be $\varphi$ and $\psi$, i.e., $\tilde{A}\varphi=\lambda\varphi$
and $\tilde{A}^{*}\psi=\bar{\lambda}\psi$ then the relations (25) and (26) ,
in terms of $\omega_{1},\omega_{2}$, have a form

$\alpha\langle(A-\lambda)^{-1}\omega_{2},\omega_{1}\rangle=-1,\quad\varphi=(A-\lambda)^{-1}\omega_{2};$
(27)
$\bar{\alpha}\langle(A-\bar{\lambda})^{-1}\omega_{1},\omega_{2}\rangle=-1,\quad\psi=(A-\bar{\lambda})^{-1}\omega_{1}.$
(28)

###### Proof.

Taking $n_{z}$ and $m_{z}$ as in (15), we obtain (27) and (28). The second way
to prove is the following. Instead of (12) we take the representation
$\tilde{A}=A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2}$ and conduct
similar conversions with the corresponding calculations. ∎

##  5\. The inverse spectral problem for a rank one nonsymmetric singular
perturbation

If we regard the formulation of the Theorem 2 as a direct spectral problem,
then we can consider the following Theorem as a corresponding inverse problem.

###### Theorem 3

For a given positive selfadjoint operator $A=A^{*}$ on a separable Hilbert
space ${\mathcal{H}}$ and $\lambda\in{\mathbb{C}}$ and vectors
$\varphi,\psi\in{\mathcal{H}}\setminus{\mathcal{H}}_{+1}$
($\varphi,\psi\in{\mathcal{H}}_{+1}\setminus{\mathcal{H}}_{+2}$), there exist
a unique $\tilde{A}\in{\mathcal{P}}_{\tau}(A)$
($\tilde{A}\in{\mathcal{P}}(A)$) such that $\tilde{A}\varphi=\lambda\varphi$
and $\tilde{A}^{*}\psi=\bar{\lambda}\psi$. Moreover, the operator $\tilde{A}$
is defined by (12) as follows:

$\tilde{R}_{z}=R_{z}+b_{z}(\cdot,n_{\bar{z}})m_{z},$ (29)

with

$m_{z}=(A-\lambda)(A-z)^{-1}\varphi,\quad
n_{\bar{z}}=(A-\bar{\lambda})(A-\bar{z})^{-1}\psi$ (30)

and

$b_{z}^{-1}=(\lambda-z)(\varphi,n_{\bar{z}}),\quad\left(\bar{b}_{z}^{-1}=(\bar{\lambda}-\bar{z})(\psi,m_{z})\right).$
(31)

In general, the proof of Theorem 3 does not differ from the proof of the
similar Theorem in [20] and hence we give a sketch of the proof there.

###### Proof.

Let us remark that if
$\varphi,\psi\in{\mathcal{H}}\setminus{\mathcal{H}}_{+1}$, then there exists
$\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ (possibly
$\tilde{A}\in{\mathcal{P}}(A)$) and if
$\varphi,\psi\in{\mathcal{H}}_{+1}\setminus{\mathcal{H}}_{+2}$, then there
exists exactly $\tilde{A}\in{\mathcal{P}}(A)$.

The proof needs a supplementary proposition that has a general character. One
of them is known from [10].

###### Proposition 4

Let there be given a (half-bounded) positive selfadjoint operator $A$ with
domain ${\mathfrak{D}}(A)$ in a separable Hilbert space ${\mathcal{H}}$. Then
for an arbitrary vector $\eta\in{\mathcal{H}}\setminus{\mathfrak{D}}(A)$ and
an arbitrary number $z\in{\mathbb{C}}$, ${\rm Im}(z)\not=0$, there exists a
restriction $\dot{A}$ of the operator $A$ such that $\eta=n_{z}$ is its defect
vector namely $(\dot{A}-z)^{*}n_{z}=0$.

This proposition is used for $\dot{A}$ and $\dot{A}_{*}$.

Next we use the result which, in some sense, is inverse to the one pointed out
in Theorem 1, — that is a formula similar to M. Krein’s formula but from the
perturbation point of view, i.e., the perturbation of the resolvent of the
selfadjoint operator by a one-dimensional skew projection.

###### Proposition 5

Let there be given a positive selfadjoint operator $A$ on a separable Hilbert
space ${\mathcal{H}}$. The operator-valued function

$\tilde{R}_{z}:=(A-z)^{-1}+b_{z}(\cdot,n_{\bar{z}})m_{z}$ (32)

  * 1)

is a resolvent of a closed operator, if $n_{z}$ and $b_{z}$ satisfy conditions
(13) and (14) and $n_{\bar{z}}$, ($m_{\bar{z}}$) is not an eigenvector of
$A-z$, ($A-\bar{z}$);

  * 2)

is the resolvent of a rank one singularly perturbed operator iff, additionally
to the previous assumption, the inclusions
$n_{z},m_{z}\in{\mathcal{H}}\setminus{\mathcal{H}}_{+2}$ hold true.

Let us give also a sketch of the proof. The part 1) is verified due to Theorem
7.7.1 [11] as follows: $\tilde{R}_{z}$ is the resolvent of a closed operator
iff

  * a)

$\tilde{R}_{z}$ satisfies the Hilbert identity
$\tilde{R}_{z}-\tilde{R}_{\xi}=(z-\xi)\tilde{R}_{z}\tilde{R}_{\xi},\ {\rm
Im}z,\xi\not=0;$

  * b)

$\tilde{R}_{z}$ has the trivial kernel, $\ker(\tilde{R}_{z})=\\{0\\},\ {\rm
Im}z\not=0.$

The part a) is verified by substituting (32) into the Hilbert identity for the
resolvent. The condition b) is verified by directly checking (32) for the
vectors $f\perp n_{\bar{z}}$ and $n_{\bar{z}}$. The condition 2) follows from
the fact that $n_{z},m_{z}\in{\mathcal{H}}\setminus{\mathfrak{D}}(A)$.

We now return to the sketch of the proof of Theorem 3. Taking $n_{z},m_{z}$ in
the form (30) and (31) we will check identity (14). Since the vectors $n_{z}$
and $m_{z}$ belong to ${\mathcal{H}}\setminus{\mathcal{H}}_{+2}$, by
Proposition 5, i.e., by part 2), the operator $\tilde{A}$ is singularly
perturbed with respect of $A$. The identity $\tilde{A}\varphi=\lambda\varphi$
is checked by direct calculation. The uniqueness is proved by contradiction. ∎

###### Proposition 6

For a given selfadjoint operator $A$ on a separable Hilbert space
${\mathcal{H}}$, and $\lambda\in{\mathbb{C}}$, and vectors
$\varphi,\psi\in{\mathcal{H}}_{+1}\setminus{\mathcal{H}}_{+2}$, there exist a
unique $\tilde{A}\in{\mathcal{P}}(A)$ such that
$\tilde{A}\varphi=\lambda\varphi$, $\tilde{A}^{*}\psi=\bar{\lambda}\psi$.
Moreover, the operator $\tilde{A}$ is defined by the expression
$\tilde{A}=A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2},$ where
$\omega_{1}=(A-\bar{\lambda})\psi,\quad\omega_{2}=(A-\lambda)\varphi,$ and
$\alpha^{-1}=-\langle(A-\lambda)\omega_{2},\omega_{1}\rangle$, or
$\alpha^{-1}=-\langle(A-\bar{\lambda})\omega_{1},\omega_{2}\rangle.$

###### Proof.

The corresponding proof is an implication of Theorems 1 and 3. ∎

##  6\. A dual pair of eigenvalues

Since $\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ is a non-self-adjoint operator,
the definition of a dual pair of eigenvalues is different from [2].

###### Definition 2

A couple of numbers $\lambda,\mu\in{\mathbb{C}}$ is called a dual pair of
eigenvalues of a singularly perturbed operator
$\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ iff

$\displaystyle\tilde{A}\varphi_{\lambda}=\lambda\varphi_{\lambda},\quad\tilde{A}\varphi_{\mu}=\mu\varphi_{\mu},$
(33)
$\displaystyle\tilde{A}^{*}\psi_{\bar{\lambda}}=\bar{\lambda}\psi_{\bar{\lambda}},\quad\tilde{A}^{*}\psi_{\bar{\mu}}=\bar{\mu}\psi_{\bar{\mu}},$
(34)
$\displaystyle(\bar{\lambda}-\bar{\mu})((A-\mu)^{-1}\varphi_{\lambda},\psi_{\bar{\lambda}})=(\varphi_{\lambda},\psi_{\bar{\lambda}}).$
(35)

The next theorem describes a method how to construct an operator with a dual
pair.

###### Theorem 4

Let $A=A^{*}\geq c$ be a semibounded selfadjoint operator defined on
${\mathfrak{D}}(A)$ in a separable Hilbert space ${\mathcal{H}}$. For an
arbitrary $\mu\in\rho(A)$ and vectors
$\varphi_{\lambda},\psi_{\bar{\lambda}}\in{\mathcal{H}}\setminus{\mathcal{H}}_{+2}$
there exists a unique nonsymmetric singularly perturbed operator
$\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ such that $(\mu,\lambda)$ is a dual
pair, where
$\bar{\lambda}:=\bar{\mu}+\frac{(\varphi_{\lambda},\psi_{\bar{\lambda}})}{((A-\mu)^{-1}\varphi_{\lambda},\psi_{\bar{\lambda}})}$
is an eigenvalue with the eigenvector
$\varphi_{\mu}=(A-\lambda)(A-\mu)^{-1}\varphi_{\lambda}$. The adjoint operator
$\tilde{A}^{*}\in{\mathcal{P}}_{\tau}(A)$ has also eigenvectors
$\psi_{\bar{\lambda}}$ and
$\psi_{\bar{\mu}}=(A-\bar{\lambda})(A-\bar{\mu})\psi_{\bar{\lambda}}$.

Remark 4. If $\tilde{A}\in{\mathcal{P}}(A)$, then in the form
$\tilde{A}=A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2}$, we can calculate
the coupling constant $\alpha=-\frac{1}{(\varphi_{\lambda},\omega_{1})}$ (or
$\bar{\alpha}=-\frac{1}{(\psi_{\bar{\lambda}},\omega_{2})}$) and the
corresponding vectors

$\omega_{2}=(A-\mu)\varphi_{\lambda}-\frac{(\psi_{\bar{\lambda}},\varphi_{\lambda})}{((A-\mu)\psi_{\bar{\lambda}},\varphi_{\lambda})}\varphi_{\lambda},\quad\omega_{1}=(A-\bar{\mu})\psi_{\bar{\lambda}}-\frac{(\varphi_{\lambda},\psi_{\bar{\lambda}})}{((A-\mu)\varphi_{\lambda},\psi_{\bar{\lambda}})}\psi_{\bar{\lambda}}.$
(36)

###### Proof.

The proof is given by a direct verification of only (33) and (34, since (35 is
the condition of Theorem. ∎

For a real dual pair $\lambda,\mu\in{\mathbb{R}}$ we have the following
corollary that follows from Theorem 4.

###### Corollary 3

Let $A=A^{*}\geq 0$ be a positive selfadjoint operator defined on
${\mathfrak{D}}(A)$ in a separable Hilbert space ${\mathcal{H}}$, so that
$\sigma(A)=\sigma_{c}(A)=[0,\infty)$. For an arbitrary number $\mu<0$ and
vectors
$\varphi_{\lambda},\psi_{\bar{\lambda}}\in{\mathcal{H}}_{+1}\setminus{\mathcal{H}}_{+2}$
there exists a unique nonsymmetric singular perturbation of rank one,
$\tilde{A}\in{\mathcal{P}}(A)$, such that it has the dual pair
$(\mu,\lambda)$, where
$\lambda=\mu+\frac{(\varphi_{\lambda},\psi{\lambda})}{((A-\mu)^{-1}\varphi_{\lambda},\psi_{\lambda})}$,
as its eigenvalues with the eigenvectors $\varphi_{\lambda}$ and
$\varphi_{\mu}=(A-\lambda)(A-\mu)^{-1}\varphi_{\lambda}$. Moreover, the
operator $\tilde{A}^{*}\in{\mathcal{P}}(A)$ has the same eigenvalues but with
different eigenvectors, $\psi_{\lambda}$ and
$\psi_{\mu}=(A-\lambda)(A-\mu)\psi_{\lambda}$.

##  7\. Approximations properties of perturbed operators

Different approximations of singularly perturbed selfadjoint operators are
presented in [4]. We have some generalization to the case of a nonsymmetric
rank one perturbation.

###### Theorem 5

Let $A$ be a semibounded self-adjoint operator defined on ${\mathfrak{D}}(A)$
in a separable Hilbert space ${\mathcal{H}}$, and two vectors
$\omega_{1},\omega_{2}\in{\mathcal{H}}_{-2}$ such that
$\langle\omega_{2},(A-z)^{-1}\omega_{1}\rangle$ does not exist. Then there
exist two sequences $\omega_{1,n},\omega_{2,n}\in{\mathcal{H}}_{-2}$
converging to $\omega_{i}$, $i=1,2$, correspondingly, such that the sequence
of operators
$\tilde{A}_{n}=A+\alpha\langle\cdot,\omega_{1,n}\rangle\omega_{2,n}\in{\mathcal{P}}(A)$
converge to the operator
$\tilde{A}=A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2}\in{\mathcal{P}}_{\tau}(A)$
in the norm resolvent sense if

$\lim\limits_{n\to\infty}\langle\omega_{2,n},A(A^{2}+1)^{-1}\omega_{1,n}\rangle=\tau.$

###### Proof.

Without loss of generality let us assume that $A=A^{*}\geq 0$. Resolvents of
the corresponding operators have the following forms:

$(\tilde{A}_{n}-z)^{-1}=(A-z)^{-1}\\\
-\frac{1}{\alpha^{-1}+\langle\omega_{2,n},(A-\bar{z})^{-1}\omega_{1,n}\rangle}\langle\cdot,(A-\bar{z})^{-1}\omega_{1,n}\rangle(A-z)^{-1}\omega_{2,n};$
(37) $(\tilde{A}-z)^{-1}=(A-z)^{-1}\\\
-\frac{1}{\alpha^{-1}+\langle\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}(A^{2}+1)^{-1}\omega_{1}\rangle}\langle\cdot,(A-\bar{z})^{-1}\omega_{1}\rangle(A-z)^{-1}\omega_{2}.$
(38)

The difference of the resolvents has the form

$\begin{split}(\tilde{A}_{n}-&z)^{-1}-(\tilde{A}-z)^{-1}=\\\
&-\frac{1}{\alpha^{-1}+\langle\omega_{2,n},(A-\bar{z})^{-1}\omega_{1,n}\rangle}\langle\cdot,(A-\bar{z})^{-1}\omega_{1,n}\rangle(A-z)^{-1}\omega_{2,n}\\\
&+\frac{1}{\alpha^{-1}+\langle\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}(A^{2}+1)^{-1}\omega_{1}\rangle}\langle\cdot,(A-\bar{z})^{-1}\omega_{1}\rangle(A-z)^{-1}\omega_{2}\\\
&=\left[-\frac{1}{\alpha^{-1}+\langle\omega_{2,n},(A-\bar{z})^{-1}\omega_{1,n}\rangle}\right.\\\
&+\left.\frac{1}{\alpha^{-1}+\langle\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}(A^{2}+1)^{-1}\omega_{1}\rangle}\right]\\\
&\times\langle\cdot,(A-\bar{z})^{-1}\omega_{1,n}\rangle(A-z)^{-1}\omega_{2,n}+\\\
&+\frac{1}{\alpha^{-1}+\langle\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}(A^{2}+1)^{-1}\omega_{1}\rangle}\\\
&\times\langle\cdot,\left((A-\bar{z})^{-1}\omega_{1,n}-(A-\bar{z})^{-1}\omega_{1}\right)\rangle(A-z)^{-1}\omega_{2}\\\
&+\frac{1}{\alpha^{-1}+\langle\omega_{2},(1+\bar{z}A)(A-\bar{z})^{-1}(A^{2}+1)^{-1}\omega_{1}\rangle}\end{split}$
$\times\langle\cdot,(A-\bar{z})^{-1}\omega_{1}\rangle\left((A-z)^{-1}\omega_{2,n}-(A-z)^{-1}\omega_{2}\right).\qquad\qquad\qquad\qquad$

To prove that the resolvents of $\tilde{A}_{n}$ converge to the resolvent
$\tilde{A}$ with respect to the operator norm it is enough to show that

$\|(A-z)^{-1}\omega_{i,n}-(A-z)^{-1}\omega_{i}\|\longrightarrow 0,\quad
i=1,2,$ (39)

and

$\langle\omega_{2,n},(A-\bar{z})^{-1}\omega_{1,n}\rangle\longrightarrow\langle\omega_{2,n},A(A^{2}+1)^{-1}\omega_{1,n}\rangle\\\
+\tau+\langle\omega_{2,n},(1+\bar{z}A)(A-\bar{z})^{-1}(A^{2}+1)^{-1}\omega_{1,n}\rangle,$
(40)

as $n\longrightarrow\infty$. Indeed, if
$\omega_{i,n}\longrightarrow\omega_{i}$, in ${\mathcal{H}}_{-2}$, $i=1,2$,
then (39) is valid.

Hence to prove the theorem it is sufficient to show that there exist sequences
of vectors $\omega_{i,n}$, $i=1,2$, with the properties

$\omega_{i,n}\longrightarrow\omega_{1}\ \text{in}\ {\mathcal{H}}_{-2},\
\text{and}\
\lim\limits_{n\to\infty}\langle\omega_{2,n},A(A^{2}+1)^{-1}\omega_{1,n}\rangle=\tau.$

Since the case $\omega_{i,n}\in{\mathcal{H}}_{-1}$ would be trivial, we
consider the general case $\omega_{i,n}\in{\mathcal{H}}_{-2}$. Let us
introduce the real sequence $a_{n}=\langle
E_{[0,n]}\omega_{2},A(A^{2}+1)^{-1}\omega_{1}\rangle,$ where $E$ is the
spectral measure of $A$. Since $\omega_{i}\not\in{\mathcal{H}}_{-1}$, $i=1,2$,
there exists an interval $[c_{n},d_{n}]$ inside $[0,n]$, so that
$b_{n}=\langle E_{[c_{n},d_{n}]}\omega_{2},A(A^{2}+1)^{-1}\omega_{1}\rangle$
could be such that $|b_{n}|>|\tau-a_{n}|$. We choose the sequence
$\omega_{i,n}=E_{[0,n]}\omega_{i}+\varepsilon_{i,n}E_{[c_{n},d_{n}]}\omega_{i},\
i=1,2.$ where $\varepsilon_{i,n}$ are taken to be

$\varepsilon_{1,n}=\frac{\sqrt{|\tau-
a_{n}|}}{\sqrt{b_{n}}},\quad\varepsilon_{2,n}=({\rm sgn}(\tau-a_{n}){\rm
sgn}b_{n})\varepsilon_{1,n}.$

It is obvious that $|\varepsilon_{i,n}|\leq 1$ and
$[\langle\omega_{2,n},A(A^{2}-1)^{-1}\omega_{1,n}\rangle=a_{n}+\varepsilon_{1,n}\varepsilon_{2,n}b_{n}=\tau.$
And we also have that
$\|\omega_{i,n}-\omega_{i}\|_{-2}\leq\|E_{[0,\infty]}\omega_{i}\|_{-2}\longrightarrow
0,$ as $n\longrightarrow\infty$. Hence $\tilde{A}_{n}$ converges to
$\tilde{A}$ in the norm resolvent sense. ∎

##  8\. Wave operators

Assuming that $i\not\in\sigma(\tilde{A})$ we define the wave operators for $A$
and $\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ in a usual way [3],

$W_{\pm}(A,\tilde{A})=s-\lim\limits_{t\rightarrow\pm\infty}U_{t}=s-\lim\limits_{t\rightarrow\pm\infty}e^{i\tilde{A}t}e^{-iAt}P^{ac},$
(41)

where

$P^{ac}f=\int_{-\infty}^{\infty}\hat{f}^{ac}(\lambda)\,dE^{ac}(\lambda)(A+i)^{-1}\omega_{1},\quad
f^{ac}\in{\mathcal{H}}^{ac},$

$P^{ac}$ and $\tilde{P}^{ac}$ denote the spectral projections onto the
absolutely continuous parts $\sigma_{ac}(A)$ and $\sigma_{ac}(\tilde{A})$ of
the spectrum of the operators $A$ and $\tilde{A}$; ${\mathcal{H}}^{ac}$ and
$\tilde{\mathcal{H}}^{ac}$ are the corresponding subspaces,

${\mathcal{H}}^{ac}=P^{ac}{\mathcal{H}},\
\tilde{\mathcal{H}}^{ac}=\tilde{P}^{ac}{\mathcal{H}};$
$f=\int_{-\infty}^{\infty}\hat{f}(\lambda)\,dE(\lambda)g=\int_{-\infty}^{\infty}\hat{\tilde{f}}(\lambda)\,d\tilde{E}(\lambda)\tilde{g},\quad
g=(A+i)^{-1}\omega_{1},\ \tilde{g}=(\tilde{A}+i)^{-1}\omega_{2}.$

This definition of wave operators is correct due to Theorem 1.5 from [11],
namely, $\tilde{A}$ is the spectral type operator and what is more, with this
connection $\tilde{A}$ is the scalar type operator
$\sigma_{ac}(A)=\sigma_{ac}(\tilde{A})$, and there exist wave operators for
the couple $A$ and $\tilde{A}$.

###### Theorem 6

Let, on a separable Hilbert space ${\mathcal{H}}$, there be given a self-
adjoint operator $A$ and its nonsymmetric singular perturbation of the form

$\tilde{A}=A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2},\quad\omega_{1},\omega_{2}\in{\mathcal{H}}_{-2}\setminus{\mathcal{H}},\
\alpha\in{\mathbb{C}}.$

Then there exist the defined in (41) wave operators $W_{\pm}$ in the form:

$W_{\pm}f=\int_{-\infty}^{\infty}(1+\tau+\alpha F(\lambda\pm
i0))\hat{f}^{ac}(\lambda)\,d\tilde{E}(\lambda)(\tilde{A}+i)^{-1}\omega_{2},$
(42)

where

$P^{ac}f=\int_{-\infty}^{\infty}\hat{f}^{ac}(\lambda)\,dE(\lambda)(A+i)^{-1}\omega_{1},\quad
F(z)=\langle(A-z)^{-1}\omega_{2},(1+\bar{z}A)(A^{2}+I)^{-1}\omega_{1}\rangle.$

###### Proof.

The proof can be carried out by a direct calculation as in [3] or by
substituting the corresponding expressions in the formulas obtained in [20]
and taking into account that $\tilde{A}\in{\mathcal{P}}_{\tau}(A)$ instead of
$\tilde{A}\in{\mathcal{P}}(A)$. ∎

The adjoint operator has the form

$W_{+}^{*}g=\int_{-\infty}^{\infty}\frac{1}{1+\tau+\alpha
F(\lambda+i0)}\hat{\tilde{g}}^{ac}(\lambda)\,dE(A+i)^{-1}\omega_{1},$

where

$\tilde{P}^{ac}g=\int_{-\infty}^{\infty}\hat{\tilde{g}}^{ac}\,d\tilde{E}(\lambda)(\tilde{A}+i)^{-1}\omega_{2},\quad\tilde{g}=(\tilde{A}+i)^{-1}\omega_{2}=\frac{1}{1+\tau+F(-i)}g.$

Then $S(\tilde{A},A)=W_{+}^{*}W_{-}$ is a scattering operator and

$S(\tilde{A},A,\lambda,\tau)=\frac{1+\tau+\alpha F(\lambda-i0)}{1+\tau+\alpha
F(\lambda+i0)}.$ (43)

is a scattering matrix consisting of one (complex) number.

##  9\. Examples

Example 1. Let us illustrate Theorems 1, 2, and 3. Let
${\mathcal{H}}=L_{2}([2,\infty),dx)=L_{2}$ and $A$ be the operator of
multiplication by the independent variable $x^{2}$, namely

$Af(x)=x^{2}f(x),\quad{\mathfrak{D}}(A)=\\{f(x)\in L_{2}\ |\ x^{2}f(x)\in
L_{2}\\}.$

It is obvious that the operator $A\geq{2}$ and $A$ has purely absolutely
continuous spectrum, i.e., $\sigma(A)=\sigma_{ac}(A)=[{2},\infty)$.

Let us put ${\mathcal{H}}_{1}=L_{2}([{2},\infty),x^{2}dx)$ and
${\mathcal{H}}_{-1}=L_{2}([{2},\infty),x^{-2}dx)$ be the dual spaces. In such
a case, ${\mathfrak{D}}(A)={\mathcal{H}}_{2}=L_{2}([{2},\infty),x^{4}dx)$, and
${\mathcal{H}}_{-2}=L_{2}([{2},\infty),x^{-4}dx)$.

Let us take $\omega_{1}=\frac{1}{{x-1}}$ and $\omega_{2}=\frac{1}{{x+1}}$. It
is obvious that
$\omega_{1},\omega_{2}\in{\mathcal{H}}_{-1}\setminus{\mathcal{H}}$. So we can
illustrate the operator $\tilde{A}\in{\mathcal{P}}(A)$ in the form (12).

If we suppose additionally that $\tilde{A}$ possess a new point of spectrum
$\lambda=0$, i.e., $0\in\sigma_{p}(\tilde{A})$, then due to Proposition 3 we
calculate $\alpha=-\frac{2}{1-\ln 3}$ by the formula (27), since

$\langle
A^{-1}\omega_{2},\omega_{1}\rangle=\int_{2}^{\infty}\frac{dx}{x^{2}(x^{2}-1)}=\frac{1-\ln
3}{2}.$

Hence,

$\tilde{A}f(x)=xf(x)-\frac{2}{1-\ln
3}\frac{1}{{x+1}}\int_{2}^{\infty}\frac{f(x)}{{x-1}}\,dx.$

Thus

$\varphi=\frac{1}{x^{2}(x+1)},\quad\psi=\frac{1}{x^{2}(x-1)}.$

To illustrate Theorem 1, formulas (12) and Theorem 3, formulas (30), (31) (for
the case where $\lambda=0$) we must put

$n_{z}=\frac{1}{(x^{2}-z)(x-1)},\quad m_{z}=\frac{1}{(x^{2}-z)(x+1)},\quad
b_{z}^{-1}=\frac{2}{1-\ln
3}-\int_{{2}}^{\infty}\frac{dx}{(x^{2}-z)(x^{2}-1)}.$

It is obvious that $\tilde{A}\in{\mathcal{P}}(A)$.

Example 2. This example illustrates Theorem 4. Let
${\mathcal{H}}=L_{2}({\mathbb{R}}^{1},dx)$ and $A$ be a Laplace operator,
namely $Af(x)=-f^{\prime\prime}(x)$,
${\mathfrak{D}}(A)=W_{2}^{2}({\mathbb{R}}^{1})$ is the Sobolev space. The
operator $A\geq 0$ has purely absolutely continuous spectrum, namely
$\sigma(A)=\sigma_{c}(A)=[0,\infty)$.

We put $\varphi_{\lambda}=e^{-|x-1|}$ and $\psi_{\lambda}=e^{-|x+1|}$,
$\mu=-1$ (we consider the case $\lambda,\mu\in{\mathbb{R}}$ and
$\lambda,\mu\in\rho(A)$). To calculate $\lambda$, we need

$(\varphi_{\lambda},\psi_{\lambda})=\int_{{\mathbb{R}}}e^{-|x-1|}e^{-|x+1|}\,dx=3e^{-2}.$

By using [1], we calculate

$(A+1)^{-1}\varphi_{\lambda}=\int_{{\mathbb{R}}}\frac{1}{2}e^{-|x-\tau|}e^{-|\tau-1|}\,d\tau=\left\\{\begin{array}[]{l}\frac{x}{2}e^{-x+1},\
x>1,\\\ \frac{2-x}{2}e^{x-1},\ x<1,\end{array}\right.$ (44)

and $((A+1)^{-1}\varphi_{\lambda},\psi_{\lambda})=\frac{13}{4}e^{-2}.$ Also,
$\lambda=-\frac{1}{13}<0$, $\alpha=-\frac{4}{13}e^{2}$. And from (36) we have
that

$\omega_{1}=\delta_{-1}(x)-\frac{12}{13}e^{-|x+1|},\quad\omega_{2}=\delta_{+1}(x)-\frac{12}{13}e^{-|x-1|}.$

The operator
$\tilde{A}=A+\alpha\langle\cdot,\omega_{1}\rangle\omega_{2}\in{\mathcal{P}}(A)$
is such that $\tilde{A}\varphi_{\lambda}=\lambda\varphi_{\lambda}$,
$\tilde{A}^{*}\psi_{\lambda}=\lambda\psi_{\lambda}$,
$\tilde{A}\varphi_{\mu}=\mu\varphi_{\mu}$,
$\tilde{A}^{*}\psi_{\mu}=\mu\psi_{\mu}$, and
$\varphi_{\mu}=(A-\lambda)(A-\mu)^{-1}\varphi_{\lambda}=e^{-|x+1|}$,
$\psi_{\mu}=e^{-|x-1|}$.

Example 3. We illustrate once more Theorem 1. Let
${\mathcal{H}}=L_{2}({\mathbb{R}}^{3})$ and $A$ play the role of the Laplace
operator, namely $Af(x)=-\Delta f(x)$,
${\mathfrak{D}}(A)=W_{2}^{2}({\mathbb{R}}^{3})$ is the Sobolev space. The
operator is positive $A\geq 0$ and has purely absolutely continuous spectrum,
i.e., $\sigma(A)=\sigma_{c}(A)=[0,\infty)$. Let us consider the expression (4)
that describes the $\delta$-interaction with retardation, i.e., the formal
expression $\tilde{A}=-\Delta+\alpha\langle\cdot,\delta_{0}\rangle\delta_{1}$,
where $\delta_{0}$ is a $\delta$-function at the point
$0=(0,0,0)\in{\mathbb{R}}^{3}$ and $\delta_{1}$ is a $\delta$-function at the
point ${\bf 1}=(1,0,0)\in{\mathbb{R}}^{3}$.

Using [1] we can write the resolvent of such operators at the regular point
$i\in{\mathbb{C}}$ (corresponding to one parameter family) i.e its integral
kernel,

$(-\tilde{\Delta}-i)^{-1}(x,p)=\frac{e^{-|x-p|}}{4\pi|x-p|}+\frac{1}{\alpha^{-1}+(4\pi
e)^{-1}}\frac{e^{-(|x|+|{\bf 1}-p|)}}{(4\pi)^{2}|x||{\bf 1}-p|}$

since $|\langle\delta_{1},(A-\bar{z})^{-1}\delta_{0}\rangle|=\frac{1}{4\pi
e}$, $z=\sqrt{i}$, we take ($\Im\sqrt{i}>0$). It is obvious that
$-\tilde{\Delta}\in{\mathcal{P}}(-\Delta)$.

Example 4. Let us again illustrate Theorem 4. Let
${\mathcal{H}}=L_{2}([1,\infty),dx)$ and $A$ be the multiplication operator on
$x^{2}$, namely $Af(x)=x^{2}f(x)$, $f\in{\mathfrak{D}}(A)$, where
${\mathfrak{D}}(A):=\\{f(x)\in L_{2}\ |\ x^{2}f(x)\in L_{2}\\}$. It is obvious
that $A\geq 1$ and ${\mathfrak{D}}(A)=\sigma(A)=[1,\infty)$. We put
$0=\mu\notin\sigma(A)$ and $\varphi=\varphi_{\lambda}=x^{-1\frac{1}{3}}$,
$\psi=\psi_{\lambda}=x^{-1\frac{2}{3}}$, $\varphi,\psi\in{\mathcal{H}}$ but
$\varphi,\psi\not\in{\mathcal{H}}_{+1}$. In particular
${\mathcal{H}}_{+2}=L_{2}([1,\infty),x^{4}dx)$. Then

$(\varphi,\psi)=\int_{1}^{\infty}\frac{dx}{x^{3}}=\frac{1}{2},\quad((A-\mu)^{-1}\varphi,\psi)=\int_{1}^{\infty}\frac{dx}{x^{5}}=\frac{1}{4}.$

Hence, $\lambda={2}\in\sigma(A)$, and

$\varphi_{\mu}=(A-{2})A^{-1}\varphi_{\lambda}=\frac{x^{2}-{2}}{x^{10/3}},\quad\psi_{\mu}=(A-{2})A^{-1}\psi_{\lambda}=\frac{x^{2}-{2}}{x^{11/3}}.$

Also from (30) we have that

$m_{z}=(A-\lambda)(A-z)^{-1}\varphi_{\lambda}=\frac{x^{2}-2}{x^{2}-z}\frac{1}{x^{4/3}},\quad
n_{z}=(A-\bar{\lambda})(A-\bar{z})^{-1}\psi_{\lambda}=\frac{x^{2}-2}{x^{2}-z}\frac{1}{x^{5/3}},$

and from (31) we have
$b_{z}=(\lambda-z)^{-1}(\varphi_{\lambda},n_{\bar{z}})^{-1}=({2}-z)^{-1}(\varphi_{\lambda},n_{\bar{z}})^{-1}$,
where

$(\varphi_{\lambda},n_{\bar{z}})=\int_{1}^{\infty}\frac{1}{x^{4/3}}\frac{x^{2}-2}{x^{2}-z}\frac{1}{x^{5/3}}\,dx=\left(\frac{1}{z^{2}}-\frac{1}{2z}\right)\ln\sqrt{1-z}+\frac{1}{z}.$

Hence (29) has the form

$(\tilde{A}-z)^{-1}=\frac{1}{x^{2}-z}+b_{z}\left(\cdot,\frac{x^{2}-2}{x^{2}-\bar{z}}\frac{1}{x^{5/3}}\right)\frac{x^{2}-2}{x^{2}-z}\frac{1}{x^{4/3}}.$

Moreover,

$\omega_{1}=\frac{x^{2}}{x^{5/3}}-\frac{1/2}{1/4}\frac{1}{x^{5/3}}=\frac{x^{2}-2}{x^{5/3}},\quad\omega_{2}=\frac{x^{2}}{x^{4/3}}-\frac{1/2}{1/4}\frac{1}{x^{4/3}}=\frac{x^{2}-2}{x^{4/3}}.$

Since $\omega_{1},\omega_{2}\in{\mathcal{H}}_{-2}\setminus{\mathcal{H}}_{-1}$,
$\tilde{A}\in{\mathcal{P}}_{\tau}(A)$.

Example 5. This example is a modification of the previous one and illustrates
Corollary 3. Let ${\mathcal{H}}=L_{2}([1,\infty),dx)$ and, as above, $A$ be an
operator of multiplication by $x^{2}$, namely $Af(x)=x^{2}f(x)$,
$f\in{\mathfrak{D}}(A)$, where ${\mathfrak{D}}(A):=\\{f(x)\in L_{2}\ |\
x^{2}f(x)\in L_{2}\\}$. For simplicity we also put $0=\mu\notin\sigma(A)$, but
$\varphi=\varphi_{\lambda}=x^{-2\frac{1}{3}}$,
$\psi=\psi_{\lambda}=x^{-2\frac{2}{3}}$,
$\varphi,\psi\in{\mathcal{H}}_{+1}=L_{2}([1,\infty),x^{2}dx)$. In particular
${\mathcal{H}}_{+2}=L_{2}([1,\infty),x^{4}dx)$ and
$\varphi,\psi\notin{\mathcal{H}}_{+2}$, Then

$(\varphi,\psi)=\int_{1}^{\infty}\frac{dx}{x^{5}}=\frac{1}{4},\quad((A-\mu)^{-1}\varphi,\psi)=\int_{1}^{\infty}\frac{dx}{x^{7}}=\frac{1}{6}.$

Hence, $\lambda=3/2\in\sigma(A)$; and

$\varphi_{\mu}=(A-3/2)A^{-1}\varphi_{\lambda}=\frac{x^{2}-3/2}{x^{13/3}},\quad\psi_{\mu}=(A-3/2)A^{-1}\psi_{\lambda}=\frac{x^{2}-3/2}{x^{14/3}}.$

And also from (30) we have that

$m_{z}=(A-\lambda)(A-z)^{-1}\varphi_{\lambda}=\frac{x^{2}-3/2}{x^{2}-z}\frac{1}{x^{7/3}},\quad
n_{z}=(A-\bar{\lambda})(A-\bar{z})^{-1}\psi_{\lambda}=\frac{x^{2}-3/2}{x^{2}-z}\frac{1}{x^{8/3}},$

and from (31) it follows that
$b_{z}=(\lambda-z)^{-1}(\varphi_{\lambda},n_{\bar{z}})^{-1}=(3/2-z)^{-1}(\varphi_{\lambda},n_{\bar{z}})^{-1}$,
where

$(\varphi_{\lambda},n_{\bar{z}})=\int_{1}^{\infty}\frac{1}{x^{7/3}}\frac{x^{2}-3/2}{x^{2}-z}\frac{1}{x^{8/3}}\,dx=\left(\frac{3}{2z^{3}}-\frac{1}{z^{2}}\right)\ln\sqrt{1-z}+\left(\frac{3}{4z^{2}}-\frac{1}{2z}\right)+\frac{3}{8z}.$

Hence (29) has the form

$(\tilde{A}-z)^{-1}=\frac{1}{x^{2}-z}+b_{z}\left(\cdot,\frac{x^{2}-3/2}{x^{2}-\bar{z}}\frac{1}{x^{8/3}}\right)\frac{x^{2}-3/2}{x^{2}-z}\frac{1}{x^{7/3}}.$

Moreover,

$\omega_{1}=\frac{x^{2}}{x^{8/3}}-\frac{1/4}{1/6}\frac{1}{x^{8/3}}=\frac{x^{2}-3/2}{x^{8/3}},\quad\omega_{2}=\frac{x^{2}}{x^{7/3}}-\frac{1/4}{1/6}\frac{1}{x^{7/3}}=\frac{x^{2}-3/2}{x^{7/3}}.$

Since $\omega_{1},\omega_{2}\in{\mathcal{H}}_{-2}\setminus{\mathcal{H}}_{-1}$,
$\tilde{A}\in{\mathcal{P}}_{\tau}(A)$, and we can exactly calculate the
coupling constant.

$\langle\varphi_{\lambda},\omega_{1}\rangle=\int_{1}^{\infty}\frac{1}{x^{7/3}}\frac{x^{2}-3/2}{x^{8/3}}\,dx=\langle\psi_{\lambda},\omega_{2}\rangle=\int_{1}^{\infty}\frac{1}{x^{8/3}}\frac{x^{2}-3/2}{x^{7/3}}\,dx=1/8.$

Due to (27), (28) we have that
$\alpha=-\frac{1}{\langle\varphi_{\lambda},\omega_{1}\rangle}=-8$.

Example 6. This example once more illustrates Theorems 1, 2, and 3. Let
${\mathcal{H}}=L_{2}({\mathbb{R}}^{1})$ and $A$ be also the Laplace operator,
namely $Af(x)=-f^{\prime\prime}(x)$,
${\mathfrak{D}}(A)=W_{2}^{2}({\mathbb{R}}^{1})$ is the Sobolev space. The
operator $A\geq 0$ is positive and has purely absolutely continuous spectrum,
i.e., $\sigma(A)=\sigma_{c}(A)=[0,\infty)$. Let us consider the expression (4)
that describes the $\delta$-interaction with retardation on the real line.

Using [1] we can write the resolvent of such operators (corresponding to one
parameter family) at a regular point $k^{2}$, ($\Im k^{2}>0$), i.e its
integral kernel,

$(-\tilde{\Delta}-k^{2})^{-1}(x,\xi)=(i/2k)e^{ik|x-\xi|}+\alpha(2k)^{-1}(i\alpha+2k)^{-1}e^{ik[|x-x_{2}|+|x_{1}-\xi|]},$

where

${\rm Im}k>0,\quad\alpha\in{\mathbb{C}},\quad
x,\xi,x_{1},x_{2}\in{\mathbb{R}}^{1},\quad x_{1}<x_{2}.$

It is not hard to understand that the essential spectrum is
$\sigma_{ess}(-\tilde{\Delta})=\sigma_{ac}(-\Delta)=[0,\infty)$, and, for the
singularly continuous spectrum, we have
$\sigma_{sc}(-\tilde{\Delta})=\emptyset$. Moreover, if ${\Re}\alpha<0$, then
the operators $-\tilde{\Delta}$ (and $-\tilde{\Delta}^{*}$) possess precisely
one negative, simple eigenvalue $\\{-\alpha^{2}/4\\}$ (and
$\\{-\bar{\alpha}^{2}/4\\}$) with the corresponding normalized eigenfunction

$\varphi=(-\alpha/2)^{1/2}e^{\alpha|x-y_{1}|/2},\quad\psi=(-\bar{\alpha}/2)^{1/2}e^{\bar{\alpha}|x-y_{2}|/2}.$

It is obvious that $-\tilde{\Delta}\in{\mathcal{P}}(-\Delta)$.

Acknowledgments. The authors are grateful to Professor L. P. Nizhnik for
stimulating discussions and providing important suggestions about an approach
of viewing nonsingular perturbations.

## References

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11institutetext: Department of Computer Languages and Computer Science.
University of Málaga, Bulevar Louis Pasteur, 35, Málaga, Spain, 29071
22institutetext: Biomedical Research Institute of Málaga (IBIMA), C/ Doctor
Miguel Díaz Recio, 28, Málaga, Spain, 29010

# Improved detection of small objects in road network sequences using CNN and
super resolution

Iván García-Aguilar 11 Rafael Marcos Luque-Baena 1122 Ezequiel López-Rubio
1122

###### Abstract

The vast number of existing IP cameras in current road networks is an
opportunity to take advantage of the captured data and analyze the video and
detect any significant events. For this purpose, it is necessary to detect
moving vehicles, a task that was carried out using classical artificial vision
techniques until a few years ago. Nowadays, significant improvements have been
obtained by deep learning networks. Still, object detection is considered one
of the leading open issues within computer vision.

The current scenario is constantly evolving, and new models and techniques are
appearing trying to improve this field. In particular, new problems and
drawbacks appear regarding detecting small objects, which correspond mainly to
the vehicles that appear in the road scenes. All this means that new solutions
that try to improve the low detection rate of small elements are essential.
Among the different emerging research lines, this work focuses on the
detection of small objects. In particular, our proposal aims to vehicle
detection from images captured by video surveillance cameras.

In this work, we propose a new procedure for detecting small-scale objects by
applying super-resolution processes based on detections performed by
convolutional neural networks _(CNN)_. The neural network is integrated with
processes that are in charge of increasing the resolution of the images to
improve the object detection performance. This solution has been tested for a
set of traffic images containing elements of different scales to test the
efficiency according to the detections obtained by the model, thus
demonstrating that our proposal achieves good results in a wide range of
situations.

###### Keywords:

Object detection Small scale Super-resolution Convolutional neural networks.

##  1 Introduction

Currently, object detection is one of the most popular computer vision
applications of deep learning. The proliferation of this type of application
is caused by the increase of video data obtained from different sources, as
well as the improvement in the computational power of the hardware, thereby
facilitating the successful accomplishment of this task. The area of road
network management can be considered as one of the potential application
domains of this type of technology because there are many objects to detect
(vehicles), the scenarios are very heterogeneous due to the position of the
camera, angle, orientation, and distance to the road, and there is a huge
number of already installed traffic cameras with the possibility of capturing
relevant information.

The proposed solution has as its primary objective the detection of reduced
size elements (objects) using convolutional neural networks. Small elements
are those which occupy a small region within the entire image. At present,
some pre-trained models have significant intrinsic problems that need to be
addressed. Object detection models determine the number of features by
aggregating information from the raw pixels across the layers of the
convolutional network. Most of them reduce the resolution of the images in
intermediate layers. This fact causes the loss of the features of small
objects, which disappear during the processing carried out by the network,
thus avoiding their detection. The low detection rate of this type of element
is also caused by the generic background clutter in the images to be inferred,
thus making the task of detecting new elements more complex due to many
potential object locations. The small objects to be detected have simple
shapes that cannot decompose into smaller parts or features. Another point to
consider is the quality of the image since there are video surveillance
systems with low image quality. Poor quality images negatively affect the
performance of object detection methods, because these types of sequences,
especially in the case of traffic videos, are dense, with small vehicles and
partially occluded. On the other hand, some objects are similar in feature,
shape, color, or pattern. Therefore, existing pre-trained high-quality models
cannot distinguish among them accurately.

The small object detection problem is relevant in many areas. Direct
applications include the following: the enhancement in the classification and
detection of elements through images captured by satellites, the improvement
in video surveillance through the treatment of images provided by security
cameras established at high points, and the increase in the detection of
pedestrians or traffic. This article focuses on the last of these direct
applications. Thanks to this new solution, it will be possible to detect a
larger number of elements and improve the confidence of each detection without
the need to retrain the model so that this solution may be advantageous for
automated traffic control systems.

Currently, there are some models aimed at the detection of these elements
through two workflows. The first of them follows the usual flow in which a
series of candidate regions are generated to perform the classification of
each proposed area subsequently. The second method establishes the detection
of objects as a regression or classification problem to adopt a framework to
achieve final results. Among the methods based on regional proposals, we find
mainly R-CNN, Faster R-CNN, and Mask R-CNN, while in regression methods, we
can mention SSD and YOLO [1]. As an example, we find the model called
_EfficientDet_ , which has an average detection rate of 51% for medium-sized
elements. In contrast, smaller elements are detected in only 12% of the cases.
It should also be noted that there are no datasets devoted explicitly to the
detection of small objects, which is why most of the attained detections
correspond to larger elements. When applying existing models such as _Faster-
RCNN_ it is found that it misses several small objects in part due to the size
of the anchor boxes of the elements detected in the image.

The proposal put forward in this article is based on the design,
implementation, and subsequent testing of a technique based on convolutional
neural networks, capable of detecting small-scale elements and improving the
class inference, without the need to retrain the pre-trained model. To achieve
this goal, the hyper-parameters of the network have been modified to reduce
false negatives. Subsequently, super-resolution processes have been applied to
the input image to generate a series of images centered on each of the
detected elements to increase its resolution. Then the model yields an
enhanced inference about the object class. Using this method, implemented in
conjunction with image preprocessing through a denoising filter, small object
detection and class inference are significantly improved.

Additionally, it has been necessary to create a new dataset to perform the
relevant tests to evaluate the qualitative improvement attained by the newly
developed technique. Most existing datasets are not adequate, such as _KITTI_
[2], which is composed of a series of images collected by cameras located on
mobile vehicles. This set of images contains several vehicles. However, the
size of these images is large, and it should be noted that not all images
include vehicles, and in other cases, the number of vehicles is small. It
should be noted that nowadays, traffic-oriented video surveillance systems
produce sharp and clear images. With this in mind, it can therefore be
established that there are no appropriate public datasets on which to perform
tests, so a new dataset of vehicle detection data has been developed from a
series of traffic videos captured by cameras and surveillance systems intended
for that purpose. Three test sets filmed in different locations and with a
series of challenges such as image quality, light interference, motion blur,
and light interference, among others, have been compiled. This dataset
consists of 476 images with a total of 14557 detections. These images contain
a set of challenges to be solved, as shown in Figure 1.

Figure 1: Main challenges to be solved

Under the premises described above, it can be concluded that there are
significant shortcomings in the state of the art methods for the detection of
small-scale elements, given the low detection rate of current models and the
few performance-enhancing procedures.

The rest of this article is organized as follows. Section 2 sets out the
related work. Throughout Section 3, the improvements developed are detailed,
explaining in depth the implemented workflow. Section 4 includes the performed
tests along with their respective results. Finally, in Section 5, the
conclusions and the future works to be developed according to the proposed
solution are outlined.

##  2 Related work

Given the context of our proposal, computationally intensive image processing
is required. Papers such as ([3]; [4]) have identified new methods that
significantly accelerate the detection of visual elements, thus allowing high-
speed image inference.

Currently, there are pre-trained models such as, for example, _YOLO_ [5]. This
model divides the image into a grid to subsequently predict the class labels
for each bounding box. Another frequently used model is _CenterNet_. This
model detects each of the objects as a triplet instead of a pair, thus
improving the accuracy in detecting and inferring each element’s class.

According to previous studies, it has been established that methods based on
regional proposals obtain a higher hit rate when detecting elements in an
image. On the other hand, it should be noted that there are some detectors
based on _RCNNs_ such as ([6]; [7]). However, the detection of small-scale
elements is referenced without going into great detail. Other works such as
([8]; [9]) study the modification of the model architecture to detect small
size elements such as faces and pedestrians, respectively. Precisely in
detecting small elements such as pedestrians, there are some advances and
interesting developments. For example, [10] proposes a pedestrian detector
that operates with multilayer neural representations to predict pedestrian
locations accurately. Also, [11] presents a new knowledge distillation
framework to learn a pedestrian detector that reduces the computational cost
associated with detections while maintaining high accuracy. In the field of
vehicle detection, we can highlight works such as the one proposed by [12]
that recognizes the direction of the vehicle in the input image. This method
may be suitable to be able to classify the vehicle model based on vehicle
appearance. Nevertheless, as the distance from the vehicle gets larger, the
image quality deteriorates, thereby resulting in a worse accuracy in
recognizing small elements.

On the other hand, image super-resolution processes establish a series of
techniques to increase the resolution of an input image. There are several
advances and developments for super-resolution processes applied to a single
image. For example, [13] proposed a network based on super-resolution
processes known as _SRCNN_. This network establishes an end-to-end mapping
between images given as input with low resolution and those processed by the
model. Furthermore, [14] proposes a network known as _VDSR_ which stacks a
series of convolutional layers with residual character learning. Also, [15]
proposed a network denoted as _EDSR_ which achieves high performance for _SR_
processes by eliminating batch normalization layers, resulting in the model
known as _SRResNet_.

Although these processes remain at a primitive stage and are complex, there
are significant advances, mainly focused on the creation of algorithms that
perform these functions effectively. One of the main approaches in this field
is presented by [16], as they proposed super-resolution algorithms based on
deep learning. One of the most notable applications in the area of super-
resolution corresponds to the processing of satellite images, wherein [17]
establishes a quantitative analysis to determine the success of improvements
in the resolution of these images through the modification of layers in the
detection model. Finally, it is worth mentioning articles that are related to
our work. First of all, [18] uses high-resolution aerial imagery to improve
vehicle detection using a simple linear model. In that article, a significant
improvement of the results is obtained thanks to super-resolution processes.
Works such as [19] advocate the generation of preprocessed low-resolution
images to increase the coverage in the detections inferred by the model.

Furthermore, [20] proposes a method based on super-resolution using deep
neural networks based on _GANs_ to reconstruct a low-resolution image and
obtain a new image that improves license plate recognition. Throughout this
paper, a super-resolution model is developed which makes use of deep residual
blocks. They use adversarial loss allowing the discriminator to distinguish
between the input image and the image processed with the super-resolution
model. On the other hand, the perceptual loss is used to help the generator
better reconstruct the high-resolution image. The license plate recognition
improves from 38.45% to 73.65% by the application of this method.

In addition to this, [21] proposes super-resolution processes based on a
series of regions detected by an _RPN_. These are applied to improve the
detection and classification of visual elements.

Although these approaches are related to the one presented in this paper, they
fail to propose a scheme based on current convolutional neural network models.
Thanks to the super-resolution processes as well as the pre-processing stage,
our proposal finds out new visual elements. In this paper, we hypothesize an
improvement in the detection of small-scale objects using super-resolution
(_SR_) techniques for the enhancement in the inference provided by models such
as _CenterNet HourGlass_ , specifically for sequences obtained by video
surveillance cameras set at high points.

##  3 Methodology

Next, our small object detection system is presented. Figure 2 shows the
global workflow of our proposed framework.

Figure 2: Workflow of the proposed technique

In what follows, the methodology of our proposal is explained in detail. As
set out in point 1 of the workflow, let us consider a deep learning neural
network for object detection which takes an image $\mathbf{X}$ as input and
returns a set of detections $S$ as output:

$S=\mathcal{F}\left(\mathbf{X}\right)$ (1)

$S=\left\\{\left(a_{i},b_{i},c_{i},d_{i},q_{i},r_{i}\right)\;|\;i\in\left\\{1,...,N\right\\}\right\\}$
(2)

where $N$ is the number of detections,
$\left(a_{i},b_{i}\right)\in\mathbb{\mathbb{R}}^{2}$ are the coordinates of
the upper left corner of the i-th detection within the image $\mathbf{X}$,
$\left(c_{i},d_{i}\right)\in\mathbb{\mathbb{\mathbb{R}}}^{2}$ are the
coordinates of the lower right corner of the $i$-th detection within
$\mathbf{X}$, $q_{i}$ is the class label of the detection, and
$r_{i}\in\mathbb{R}$ is the class score of the detection. The higher $r_{i}$,
the more confidence that an object of class $q_{i}$ is actually there. It is
assumed that the origin of the coordinate system of all images is at the
center of the image. Next, step two is performed based on the selection of
each of the initially established detections. Given a low-resolution input
image $\mathbf{X}_{LR}$ as shown, for example, in figure 3, our first step is
to process it with the object detection network to yield a set of tentative
detections $S_{LR}$:

$S_{LR}=\mathcal{F}\left(\mathbf{X}_{LR}\right)$ (3)

Subsequently, as detailed in the third step of the workflow, a new image is
generated by applying super-resolution and denoising processes to the initial
image given as input.

Figure 3: Frame processing

A super-resolution deep network is used to obtain a high-resolution version
$\mathbf{X}_{HR}$ with zoom factor $Z$ of the low-resolution input image
$\mathbf{X}_{LR}$ as shown in point 2 of the figure 3. To implement this
super-resolution process, the OpenCV library has been used. There are some
pre-trained models for the execution of processes that increase the initial
resolution of an image. The considered models are the following:

  * •

Fsrcnn: Fast Super-Resolution Convolutional Neural Network [22].

  * •

Edsr: Enhanced Deep Residual Networks Single Image Super-Resolution [15].

  * •

Espcn: Efficient Sub-Pixel Convolutional Neural Network [23].

  * •

Lapsrn: The Laplacian Pyramid Super-Resolution Network [24].

According to the final image quality obtained by each of these models as well
as the time required to carry out the process, the _Fsrcnn_ model has been
selected because it is one of the models that process the images with the best
speed and quality.

Subsequently, to improve the detections by the network, noise elimination is
performed. This process is performed through the use of the non-local
denoising
algorithm111http://www.ipol.im/pub/algo/bcm_non_local_means_denoising/ making
use of various computational optimizations [25]. This method is applied when
Gaussian white noise is expected (Point 3 of the Figure 3). After an empirical
study, it was concluded that this type of processing improves the number of
detected elements.

Then, for each detection in $S_{LR}$, a subimage $\mathbf{X}_{i}$ with the
same size as $\mathbf{X}_{LR}$ is extracted from $\mathbf{X}_{HR}$. The
subimage $\mathbf{X}_{i}$ is centered at the center of the detection:

$\mathbf{y}_{i}=\left(\frac{a_{i}+c_{i}}{2},\frac{b_{i}+d_{i}}{2}\right)$ (4)

$\hat{\mathbf{y}}_{i}=Z\mathbf{y}_{i}$ (5)

where $\mathbf{y}_{i}$ is the center of $\mathbf{X}_{i}$ expressed in
coordinates of $\mathbf{X}_{LR}$, while $\hat{\mathbf{y}}_{i}$ is the center
of $\mathbf{X}_{i}$ expressed in coordinates of $\mathbf{X}_{HR}$. The object
detection network is applied to $\mathbf{X}_{i}$ in order to yield a new list
of detections:

$S_{i}=\mathcal{F}\left(\mathbf{X}_{i}\right)$ (6)

As indicated in step 5 of the workflow, each of the generated images is again
given to the object detection model to improve the inference or detect
elements not initially spotted.

The detections of $S_{i}$ are expressed in coordinates of $\mathbf{X}_{i}$, so
that they must be translated to coordinates of $\mathbf{X}_{LR}$. The equation
to translate a point $\tilde{\mathbf{h}}$ in coordinates of $\mathbf{X}_{i}$
to coordinates $\mathbf{h}$ of $\mathbf{X}_{LR}$ is:

$\mathbf{h}=\mathbf{y}_{i}+\frac{1}{Z}\tilde{\mathbf{h}}$ (7)

As shown in Figure 4, for the same object there will be multiple possible
detections before performing the intersection over union (IOU) operation.

Figure 4: Video frame before filtering matching detections

After that, all the detections coming from the detection set $S_{i}$ are
processed to estimate whether they correspond to new objects, or they match
some detection already present in $S_{LR}$. To this end, the intersection over
union measure is computed for each pair of detections $D_{j}$ and $D_{k}$:

$IOU=\frac{Area\left(D_{j}\cap D_{k}\right)}{Area\left(D_{j}\cup
D_{k}\right)}$ (8)

The detections $D_{j}$ and $D_{k}$ are judged to correspond to the same object
whenever $IOU>\theta$, where $\theta$ is a tunable parameter. The final set of
detections is comprised of those detections that persist after filtering the
matching detections. This point finally corresponds to step 6 of the workflow.
At the end of this process, an image will be obtained with a higher number of
detections and with an improvement in the class inference of each element, see
Figure 5.

Figure 5: Final result after filtering matching detections

##  4 Experimental results

In order to carry out the tests on the data set designed for the experiments,
it is necessary to establish a series of parameters such as the maximum number
of detected objects, the minimum inference to take into account an object as
well as the threshold $\theta$ for the intersection over union index to
eliminate simultaneous detections for the same object. These parameters are
established in Table 1 and selected according to the best results obtained
after an empirical study.

Parameter | Value
---|---
Maximum number of detections | 100
Minimum percentage of inference | 0.3
IOU threshold $\theta$ | 0.1
Table 1: Selected values of the hyperparameters.

Regarding the dataset used for the tests, as stated in the Introduction
(Section 1), three videos from video surveillance systems were manually
labeled, all of which pointed to roads with a large number of vehicles, video
one is named as _sb-camera1-0820am-0835am_ , video two as _sb-
camera3-0820am-0835am_ and video three as _sb-camera4-0820am-0835am_. As can
be seen in the name of each of the videos, they are formed by the number of
the camera referred to as well as the time that the images were acquired. This
dataset consists of 476 images with a total of 14557 detections. Four specific
categories related to the detection of vehicles on roads have been selected.
Namely, the labels _car_ , _truck_ , _motorcycle_ , and _bus_ have been chosen
to analyze the results. The dataset is composed of a series of images with
specific interesting characteristics for the study of the obtained results,
mainly highlighting the following aspects:

  * •

The vehicles set in each frame occupy a small area of the image, thus
qualifying them as small size elements. In datasets such as _KITTI_ , we have
a large number of vehicles. However, the size of the vehicles is much larger
compared to the dataset manually developed for this article.

  * •

There is an imbalance of classes. There are more abundant categories of
elements such as cars versus motorcycles which appear in a limited number of
frames.

Next, a series of examples are shown in Figures 6,7,8,9, 10 and 11 in which
the qualitative results obtained with the proposed technique are shown for
some instances of the considered dataset.

To check the results obtained by the proposed technique, in addition to the
images coming to the selected videos, the technique has been tested on a
series of previously available ground truth frames from other benchmark videos
as can be seen in Figures 9, 10 and 11. To obtain quantitative performance
results, the evaluation process developed by _COCO_
222https://github.com/cocodataset/cocoapi/blob/master/PythonAPI/pycocotools/cocoeval.py
has been used. The standard measure for object detection tasks is the mean
average precision (mAP), which takes into account the bounding box prediction
and the class inference for each object. Thus, the bounding box of a
prediction is considered a correct detection depending on the overlap with the
ground truth bounding box, as measured by the intersection over the union
known as _IoU_. In addition, the class inference must be correct for that
element. As output, several mAPs measures are obtained, divided by the object
size and the quality of the inference class. According to the tests carried
out, our technique solves a series of problems depending on the context of the
image. Firstly, it detects many elements that were not detected a priori by
the model in the first pass. Secondly, in the initial inference, we can
observe in certain areas that the class of the elements is not correct, and
after applying our technique, this problem is solved.

Figure 6: Frame 1 of the first video denoted as sb-camera1-0820am-0835am. The
left side shows the results obtained by the unmodified model while the right
side shows the detections after applying our proposal.

Figure 7: Frame 6 of the second video denoted as sb-camera3-0820am-0835am. The
left side shows the results obtained by the unmodified model while the right
side shows the detections after applying our proposal using
CenterNetHourGlass104Keypoints.

Figure 8: Frame 136 of the third video denoted as sb-camera4-0820am-0835am.
The left side shows the results obtained by the unmodified model while the
right side shows the detections after applying our proposal using
CenterNetHourGlass104Keypoints.

Figure 9: Frame 5837 of the M-30-HD dataset [26]. The left side shows the
results obtained by the unmodified model while the right side shows the
detections after applying our proposal using CenterNetHourGlass104Keypoints.

Figure 10: Frame 2 of the UA-DETRAC MVI_39311 dataset [27, 28, 29]. The left
side shows the results obtained by the unmodified model while the right side
shows the detections after applying our proposal using
CenterNetHourGlass104Keypoints.

Figure 11: Frame 1 of the UA-DETRAC MVI_40903 dataset [27, 28, 29]. The left
side shows the results obtained by the unmodified model while the right side
shows the detections after applying our proposal using
CenterNetHourGlass104Keypoints.

However, it should be noted that our solution is not infallible, and its
performance depends on the context in which it is applied. For example, in
Figure 12 we can see how the number of elements is increased. However, we also
obtain false positives such as the one in the center of the image or the
incorrect inference of the motorcycle located at the center-left side. This
problem can be alleviated by increasing the minimum percentage of inference to
take into account a detection.

Figure 12: Frame 5 of the UA-DETRAC MVI_40852 dataset [27, 28, 29]. The left
side shows the results obtained by the unmodified model while the right side
shows the detections after applying our proposal using
CenterNetHourGlass104Keypoints.

To carry out the tests, several variants of either the _EfficientNet_ or
_Centernet_ models have been used. For _EfficientNet_ , some variants have
been defined, which range from zero to seven, each representing variants for
efficiency and accuracy according to a given scale. Thanks to this scaling
heuristic, it is possible for the base model denoted as _D0_ to outperform the
models at each scale, also avoiding an extensive search in the hyperparameter
space. At first, one may get the impression that _EfficientNet_ constitutes a
continuous family of models defined by an arbitrary choice of scale; however,
other aspects such as resolution, width, and depth are contemplated. According
to resolution, models with low scale such as _D0_ or _D1_ apply zero-padding
near the boundaries of specific layers, thereby wasting computational
resources. As for depth and width, the size of the channels must be a multiple
of eight. For _Centernet_ , three variants have been used, each of them with a
defined architecture, featuring as main differences the size of the image
given as input as well as the way of processing each of the detections. It has
been tested with variants based on keypoints as well as traditional models.
These pre-trained models have been extracted from Tensorflow Model
Zoo333https://github.com/tensorflow/models/blob/master/research/object_detection/g3doc/tf2_detection_zoo.md.
It is important to note that no training process using traffic sequences is
performed on these pre-trained models.

The performance of these initial models and the proposed methodology is
quantitatively compared. Since the _car_ class is majority, the performance in
the detection of cars is considered as the measure to compare. For this
purpose, only class number 3 has been considered, corresponding in _COCO_ to
the _car_ element, thus obtaining Tables 4, 4 and 4. In each of these tables,
the mean average precision (mAP) is determined for both the initial detections
as well as the results obtained after applying our proposal. Columns second to
fourth refer to detections of objects of any size. The second column
corresponds to the mAP of the elements detected with an _IoU_ between 50 and
90 percent. The third column reports detections above 50 percent of _IoU_ ,
while the fourth column corresponds to the detections above 75 percent of this
metric. Additionally, results are presented for small-sized objects with an
_IoU_ between 50 and 95 percent and medium-sized items with an _IoU_ above 50
percent, in the fifth and sixth columns, respectively.

Metrics obtained for large-sized objects have been omitted since there are no
samples in the established dataset. As shown in each of the tables obtained
for the images that make up the defined dataset, there is a clear improvement
in the mAP measure.

To better illustrate the increase in the number of detections, the _CenterNet
HourGlass104 Keypoints_ model has been selected for this test as it is one of
the best performing models compared to the rest of the models used for this
experiment. For each of the sets of images that make up the videos from which
the dataset has been made, the improvement obtained in the number of elements
by the described technique compared to the results initially provided by the
model has been graphically represented in Figures 13, 14 and 15.

Video 1 - sb-camera1-0820am-0835am
---
Model | IoU=0.50:0.95—area=all | IoU¿0.50—area=all | IoU¿0.75—area=all | IoU=0.50:0.95—area=Small | IoU¿0.50—area=Medium
CenterNet HourGlass104 Keypoints | 0.244 / 0.361 | 0.420 / 0.628 | 0.263 / 0.381 | 0.258 / 0.375 | 0.193 / 0.283
CenterNet MobileNetV2 FPN Keypoints | 0.083 / 0.161 | 0.191 / 0.358 | 0.053 / 0.121 | 0.085 / 0.163 | 0.081 / 0.122
CenterNet Resnet101 V1 FPN | 0.273 / 0.350 | 0.512 / 0.656 | 0.248 / 0.318 | 0.274 / 0.355 | 0.244 / 0.273
EfficientDet D3 | 0.159 / 0.336 | 0.276 / 0.585 | 0.169 / 0.364 | 0.159 / 0.343 | 0.154 / 0.247
EfficientDet D4 | 0.236 / 0.442 | 0.424 / 0.790 | 0.237 / 0.459 | 0.243 / 0.451 | 0.168 / 0.298
EfficientDet D5 | 0.245 / 0.378 | 0.433 / 0.672 | 0.252 / 0.389 | 0.251 / 0.395 | 0.158 / 0.164

Table 2: Results obtained for the first video. On the left is the mAP obtained
by the base model and on the right of each box is the mAP obtained by our
proposal (higher is better). The best results are marked in bold.

Video 2 - sb-camera3-0820am-0835am
---
Model | IoU=0.50:0.95—area=all | IoU¿0.50—area=all | IoU¿0.75—area=all | IoU=0.50:0.95—area=Small | IoU¿0.50—area=Medium
CenterNet HourGlass104 Keypoints | 0.090 / 0.186 | 0.176 / 0.363 | 0.074 / 0.154 | 0.092 / 0.191 | 0.043 / 0.112
CenterNet MobileNetV2 FPN Keypoints | 0.117 / 0.158 | 0.309 / 0.390 | 0.047 / 0.083 | 0.119 / 0.159 | 0.039 / 0.085
CenterNet Resnet101 V1 FPN | 0.111 / 0.188 | 0.242 / 0.403 | 0.072 / 0.124 | 0.111 / 0.191 |  0.096 / 0.093
EfficientDet D3 | 0.092 / 0.294 | 0.177 / 0.590 | 0.073 / 0.224 | 0.091 / 0.297 | 0.105 / 0.171
EfficientDet D4 | 0.236 / 0.345 | 0.480 / 0.689 | 0.171 / 0.261 | 0.237 / 0.350 | 0.232 / 0.244
EfficientDet D5 | 0.178 / 0.292 | 0.345 / 0.588 | 0.145 / 0.219 | 0.178 / 0.295 | 0.207 / 0.221

Table 3: Results obtained for the second video. On the left is the mAP
obtained by the base model and on the right of each box is the mAP obtained by
our proposal (higher is better). The best results are marked in bold.

Video 3 - sb-camera4-0820am-0835am
---
Model | IoU=0.50:0.95—area=all | IoU¿0.50—area=all | IoU¿0.75—area=all | IoU=0.50:0.95—area=Small | IoU¿0.50—area=Medium
CenterNet HourGlass104 Keypoints | 0.074 / 0.219 | 0.137 / 0.394 | 0.072 / 0.217 | 0.075 / 0.227 | 0.054 / 0.060
CenterNet MobileNetV2 FPN Keypoints | 0.060 / 0.102 | 0.159 / 0.237 | 0.026 / 0.064 | 0.066 / 0.108 | 0.009 / 0.014
CenterNet Resnet101 V1 FPN | 0.039 / 0.082 | 0.075 / 0.160 | 0.028 / 0.065 | 0.040 / 0.084 |  0.016 / 0.015
EfficientDet D3 | 0.029 / 0.163 | 0.050 / 0.292 | 0.022 / 0.164 | 0.029 / 0.168 | 0.012 / 0.040
EfficientDet D4 | 0.129 / 0.280 | 0.236 / 0.518 | 0.118 / 0.264 | 0.130 / 0.287 | 0.068 / 0.151
EfficientDet D5 | 0.056 / 0.204 | 0.099 / 0.384 | 0.051 / 0.184 | 0.058 / 0.207 | 0.006 / 0.082

Table 4: Results obtained for the third video. On the left is the mAP obtained
by the base model and on the right of each box is the mAP obtained by
performing our proposal (higher is better). The best results are marked in
bold.

Figure 13: Comparison of the number of detections for the first sequence.
Figure 14: Comparison of the number of detections for the second sequence.

Figure 15: Comparison of the number of detections for the third sequence.

According to Tables 4, 4, 4 , and Figures 13, 14 and 15, it can be stated that
the application of the technique described in this work not only improves the
accuracy of the elements initially detected by the model, but also detects
objects that were not identified a priori. Thus, we can corroborate that in
each and every one of the frames that make up the test videos, we have
obtained a larger number of elements, denoted in the graphs by the orange
dots, compared to the initial detections established by the gray dots.

##  5 Conclusions

Throughout this paper, a new technique has been developed which employs deep
convolutional neural networks for super-resolution to detect small-scale
visual elements and improve the reliability of the estimation of their
classes. The method makes a first pass of the object detection deep neural
network. Then it super-resolves the video frame. After that, a window is
defined for each detected object, which has the detected object at its center,
and a object detection is carried out for that window. This object detection
results in more accurate estimations of the characteristics of the previously
detected objects, as well as the detection of additional objects that were
missed by the object detection network in the first pass.

A quantitative comparison of our approach has been carried out. The results
obtained show that the application of super-resolution processes does improve
the detections made on an image since, based on the metrics obtained using the
_COCO_ evaluator, the quantitative object detection results are enhanced.
Qualitative results further confirm the advantages of our proposal since the
improvement of the reliability of the object detections and the discovery of
additional objects can be observed.

Let us remember that, in all cases, the images that make up the model on which
the results have been validated correspond to video surveillance systems
placed in high points. Regarding future lines to extend the research reported
in this article, it should be noted that a novel approach is envisaged on
which work is already underway. It consists of reducing the number of windows
that need to be processed during the object detection. This enhancement might
result in a reduction of the computational load associated with the execution
of our method.

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