\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 76, pp. 1--4.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/76\hfil Quasi-spectral decomposition]
{Quasi-spectral decomposition of the heat potential}

\author[T. Sh. Kal'menov, G. D. Arepova \hfil EJDE-2016/76\hfilneg]
{Tynysbek Sh. Kal'menov, Gaukhar D. Arepova}

\address{Tynysbek Sh. Kal'menov \newline
Institute of Mathematics and Mathematical Modeling, 125 Pushkin
str., 050010 Almaty, Kazakhstan} 
\email{kalmenov@math.kz}

\address{Gaukhar D. Arepova \newline
Institute of Mathematics and Mathematical Modeling, 125 Pushkin
str., 050010 Almaty, Kazakhstan and Al-Farabi Kazakh national
university, 71  Al-Farabi ave., 050040 Almaty, Kazakhstan}
\email{arepova@math.kz}

\thanks{Submitted January 25, 2016.  Published March 17, 2016.}
\subjclass[2010]{35K05, 47F05}
\keywords{Heat potential; quasi-spectral decomposition;
\hfill\break\indent  self-adjoint operator; unitary operator; 
the fundamental solution}

\begin{abstract}
 In this article, by multiplying of the unitary operator
 $$
 (Pf)(x,t)=f(x,T-t),\quad 0\leq t\leq T, 
 $$
 the heat potential turns into a self-adjoint operator.
 From the spectral decomposition of this completely continuous
 self-adjoint operator we obtain a quasi-spectral decomposition
 of the  heat potential operator.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In the works of Gohberg and  Krein \cite{GK}, 
it is proven that for any linear completely-continuous
operator $A$, in a Hilbert space  $H$,  has a triangular
representation  $A=U(A^{*}A)^{1/2}$, where $A^{*}$ is an
adjoint operator to $A$, and $U$ a unitary operator. 
When the operator $A$ is a completely-continuous Volterra operator
generated by a mixed solution of the Cauchy problem for parabolic
and hyperbolic equations proposes, it is of great interest. In this article
we give a new analogue of a triangular representation of
multi-dimensional heat potential and its quasi-spectral expansion.

\section{Main results}

Let $\Omega\subset R^{n}$ be a finite domain with a smooth boundary $\partial\Omega\in C^{1}$, and $D=\Omega\times(0,T)$.
In the domain $D$ we define the heat potential (see e.g. \cite{Biz1,Bla})
by the formula
\begin{equation}
u=\lozenge^{-1}f\equiv \int_{0}^{t}d\tau\int_{\Omega
}\varepsilon_{n}(x-\xi,t-\tau)f(\xi,\tau)d\xi \label{e2.1}
\end{equation}
where
\begin{equation}
\varepsilon_{n}(x,t)=\frac{\theta(t)}{(2\sqrt{\pi
t})^n}e^{-\frac{|x|^{2}}{4t}} \label{e2.2}
\end{equation}
is the fundamental solution of the heat equation
\begin{gather}
\lozenge \varepsilon_{n}(x,t)\equiv(\frac{\partial}{\partial
t}-\Delta_{x})\varepsilon_{n}(x,t)=\delta(x,t), \label{e2.3} 
\\
\varepsilon_{n}(x,t)|_{t=0}=0. \label{e2.4}
\end{gather}
For $f\in L_2(\Omega)$ it is easy to verify that
\begin{equation}
\lozenge u=\lozenge \lozenge^{-1}f
=\lozenge \int_{0}^{t}d\tau\int_{\Omega
}\varepsilon_{n}(x-\xi,t-\tau)f(\xi,\tau)d\xi=f(x,t), \quad
u|_{t=0}=0.\label{e2.5}
\end{equation}
In the work by Kalmenov, Tokmagambetov \cite{K5} (see
also \cite{K1,K2,K3,K4,ST}),  it
is shown that the heat potential $u=\lozenge^{-1}f$ at any 
$f\in L_2(\Omega)$ satisfies the following boundary conditions
\begin{equation}
\begin{aligned}
&\frac{u(x,t)}{2}-\int_{0}^{t}d\tau \int_{\partial \Omega}\Big(\frac{\partial\varepsilon_{n}}{\partial n_{\xi}}(x-\xi,t-\tau)u(\xi,\tau) \\
& -\varepsilon_{n}(x-\xi,\tau-t)\frac{\partial u}{\partial
n_{\xi}}(\xi,\tau)\Big)d\xi=0, \quad x\in\partial\Omega,\;
t\in[0,T].
\end{aligned} \label{e2.6}
\end{equation}
Conversely, for any $f\in L_2(D)$, solution of
\eqref{e2.5} defines the heat potential by formula \eqref{e2.1}.
Here,
 $\frac{\partial }{\partial n_{\xi}}$ is unit normal derivative at  $\partial\Omega$.

Note that the operator $\lozenge^{-1}$ is completely-continuous on
$L_2$ for any $f\in L_2(\Omega)$, $u=\lozenge^{-1}f\in
W_2^{2,1}(D)$. The operator $\lozenge^{-1}$ is a Volterra
operator, i.e. it has no nontrivial eigenvectors.

Let us define the operator  $P$ by
\begin{equation}
(Pf)(x,t)=f(x,T-t),\quad  0\leq t\leq T. \label{e2.7}
\end{equation}
It is clear that $P$ is a bounded self-adjoint operator satisfying
\begin{equation}
P=P^{*},\quad P^{2}=I. \label{e2.8}
\end{equation}

\begin{lemma} \label{lem2.1}
The operator $P\lozenge^{-1}$ is a completely-continuous self-adjoint operator.
\end{lemma}

\begin{proof} 
Let us rewrite the operator $P\lozenge^{-1}$ in the  form 
\begin{equation}
\begin{aligned}
P\lozenge^{-1}f
&=P\Big(\int_{0}^{T}\theta(t-\tau)d\tau
\int_{\Omega}\varepsilon_{n}(x-\xi,t-\tau)f(\xi,\tau)d\xi\Big)\\
&=\int_{0}^{T}\theta(T-t-\tau)d\tau\int_{\Omega}\varepsilon_{n}(x-\xi,T-t-\tau)
f(\xi,\tau)d\xi.
\end{aligned} \label{e2.9}
\end{equation}
By using a direct computation for any $f,g\in L_2(D)$ it can be shown that
\begin{equation}
\begin{aligned}
&(P\lozenge^{-1}f,g)_{L_2(D)}\\
&=\int_{0}^{T}dt\int_{\Omega}(P\lozenge^{-1}f)(x,t)g(x,t)dx \\
&=\int_{0}^{T}dt\int_{\Omega}\int_{0}^{T}\theta(T-t-\tau)
 \int_{\Omega}\varepsilon_{n}(x-\xi,T-t-\tau)f(\xi,\tau)d\xi g(x,t)dx \\
&=\int_{0}^{T}\int_{\Omega}f(\xi,t)dx\int_{0}^{T}\theta(T-t-\tau)
 \int_{\Omega}\varepsilon_{n}(x-\xi,T-t-\tau)g(x,t)dxd\xi \\
&=\int_{0}^{T}d\tau\int_{\Omega}f(\xi,\tau)P
\Big(\int_{0}^{T}\theta(\tau-t)dt\int_{\Omega}\varepsilon_{n}(x-\xi,
 \tau-t)g(x,t)dx\Big)d\xi\\
&=(f,P\lozenge^{-1}g)_{L_2(D)}.
\end{aligned} \label{e2.10}
\end{equation}
On the other hand,
\begin{equation}
(P\lozenge^{-1}f,g)_{L_2(D)}=(f,(P\lozenge^{-1})^{*}g)_{L_2(D)}.\label{e2.11}
\end{equation}
Because of the arbitrariness of $f,g\in L_2(D)$ we obtain
$$
(P\lozenge^{-1})^{*}=P\lozenge^{-1}.
$$ 
This completes the proof.
\end{proof}

According to the theory of regular extensions of the linear
operator  (Otelbaev \cite{Otel} and  Vishik \cite{Bish})
self-adjoint differential operators are generated only by boundary
conditions.

\begin{lemma} \label{lem2.2}
For $f\in L_2(D)$ the function $u=P\lozenge^{-1}f \in
W_2^{1,2}(D)\cap W_2^1(\partial D)$ satisfies the equation
\begin{equation}
\lozenge Pu=f,\label{e2.12}
\end{equation}
the initial condition
\begin{equation}
u|_{t=T}=0,\label{e2.13}
\end{equation}
and the lateral boundary condition
\begin{equation}
\begin{aligned}
&-\frac{(Pu)(x,t)}{2}+\int_{0}^{t}d\tau \int_{\partial\Omega}(\frac{\partial\varepsilon_{n}}{\partial n_{\xi}}(x-\xi,\tau-t)Pu(\xi,\tau)d\xi) \\
&-\int_{0}^{t}d\tau \int_{\Omega}(\varepsilon_{n}(x-\xi,\tau-t)P\frac{\partial u}{\partial n_{\xi}}(\xi,\tau)d\tau)=0,\quad
 x\in\partial\Omega, t\in[0,T]. 
 \end{aligned} \label{e2.14}
\end{equation}
Conversely, if  $u\in W_2^{1,2}(D)\cap W_2^1(\partial D)$
satisfies  \eqref{e2.12}, the initial condition
\eqref{e2.13} and the lateral boundary condition \eqref{e2.14},
then $u=P\lozenge^{-1}f$.
\end{lemma}

\begin{proof}
In view of  $\lozenge Pu=f$, where $u\in W_2^{1,2}(D)\cap W_2^1(\partial  D)$ satisfies the initial condition \eqref{e2.13}
and the lateral boundary condition \eqref{e2.14}, it is easy to
prove (see \cite{K5}) that $v=Pu=\diamondsuit^-1f$, where
\begin{equation}
v=\lozenge^{-1}\lozenge\vartheta=\int_{0}^{t}d\tau\int_{\Omega }\varepsilon_{n}(x-\xi,\tau-t)(\frac{\partial}{\partial\tau}-
\Delta_{\xi})\vartheta(\xi,\tau)d\xi.\label{e2.16}
\end{equation}
It is easy to check as in \cite{K5} that
\begin{gather}
\begin{aligned}
&-\frac{\vartheta(x,t)}{2}+\int_{0}^{t}d\tau \int_{\partial \Omega}(\frac{\partial\varepsilon_{n}}{\partial n_{\xi}}(x-\xi,t-\tau)\vartheta(\xi,\tau)\\
&-\varepsilon_{n}(x-\xi,\tau-t)\frac{\partial u}{\partial n_{\xi}}(\xi,\tau))d\xi=0, \quad x\in\partial\Omega, \; t\in[0,T].
\end{aligned} \label{e2.17}
\\
v\big{|}_{t=0}=0 \label{e2.17*}
\end{gather}
By taking into account $v=Pu$ we will rewrite
\eqref{e2.17}--\eqref{e2.17*} in the  form
\begin{gather}
\begin{aligned}
&-\frac{(Pu)(x,t)}{2}+\int_{0}^{t}d\tau \int_{\partial \Omega}(\frac{\partial\varepsilon_{n}}{\partial n_{\xi}}(x-\xi,t-\tau)(Pu)(\xi,\tau)\\
&-\varepsilon_{n}(x-\xi,\tau-t)\frac{\partial Pu}{\partial n_{\xi}}(\xi,\tau))d\xi=0, \quad x\in\partial\Omega,\; t\in[0,T].
\end{aligned} \label{e2.19} \\
u\big{|}_{t=T}=0 \label{e2.19*}
\end{gather}
This completes the proof. 
\end{proof}

Since the operator $P\lozenge^{-1}$ is completely-continuous and
self-adjoint throughout $L_2(\Omega)$, then it has a complete
orthonormal system of eigenvectors $e_{k}(x,t)$ associated with  real
eigenvalues  $\lambda_{k}$,
\begin{equation}
\lambda_{k}(P\lozenge^{-1})e_{k}=e_{k}. \label{e2.20}
\end{equation}
Then
\begin{equation}
\begin{aligned}
P\lozenge^{-1}f
&=\sum_{k}(P\lozenge^{-1}f,e_{k})_{0}e_{k}
 =\sum_{k}(f,(P\lozenge^{-1})e_{k})_{0}e_{k}\\
&= \sum_{k}(f,\frac{e_{k}}{\lambda_{k}})e_{k}
 =\sum_{k}\frac{1}{\lambda_{k}}(f,e_{k})e_{k}.
 \end{aligned} \label{e2.21}
\end{equation}
Applying the operator  $P$ to both sides of  \eqref{e2.21}, we obtain
\begin{equation}
\lozenge^{-1}f=\sum_{k}\frac{1}{\lambda_{k}}(f,e_{k})Pe_{k}.\label{e2.22}
\end{equation}
The decomposition of  $\lozenge^{-1}f$ through orthonormal system
$Pe_{k}$ is called a quasi-spectral expansion of the heat
potential $\lozenge^{-1}$.
This proves the following theorem.

\begin{theorem} \label{thm2.1}
Let $e_{k}$ be a complete orthonormal system of eigenvectors of the self-adjoint operator $\lambda_{k}(P\lozenge^{-1})e_{k}=e_{k}$. Then, for any $f\in L_2(D)$, $\lozenge^{-1}f$ has quasi-spectral expansion in the  form
\begin{equation}
\lozenge^{-1}f=\sum_{k}\frac{1}{\lambda_{k}}(f,e_{k})Pe_{k}.\label{e2.23}
\end{equation}
\end{theorem}

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\end{thebibliography}

\end{document}