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\AtBeginDocument{{\noindent\small
2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.\newline
{\em Electronic Journal of Differential Equations},
Conference 13, 2005, pp. 101--112.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
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\begin{document}
\title[\hfilneg EJDE/Conf/13 \hfil The division method for subspectra]
{The division method for subspectra of self-adjoint differential
vector-operators}
\author[M. S. Sokolov \hfil EJDE/Conf/13 \hfilneg]
{Maksim Sokolov}
\address{Maksim S. Sokolov \hfill\break
ICTP Affiliated Center, Mechanics and Mathematics Department,
National University of Uzbekistan,
Tashkent 700095, Uzbekistan}
\email{sokolovmaksim@hotbox.ru}
\date{}
\thanks{Published May 30, 2005.}
\thanks{Supported by the grants AC-84 and MISC-03/14 from
the Abdus Salam International \hfill\break\indent
Center for Theoretical Physics}
\subjclass[2000]{34L05, 47B25, 47B37, 47A16}
\keywords{Vector-operator; cyclic vector; spectral representation;
\hfill\break\indent
ordered representation; multiplicity; unitary transformation}
\begin{abstract}
We discuss the division method for subspectra which appears to be
one of the key approaches in the study of spectral properties of
self-adjoint differential vector-operators, that is operators
generated as a direct sum of self-adjoint extensions on an
Everitt-Markus-Zettl multi-interval system. In the current work we
show how the division method may be applied to obtain the ordered
spectral representation and Fourier-like decompositions for
self-adjoint differential vector-operators, after which we obtain
the analytical decompositions for the measurable (relative to a
spectral parameter) generalized eigenfunctions of a self-adjoint
differential vector-operator.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{defn}[thm]{Definition}
\newcommand{\norm}[1]{\|#1\|^2}
\section{Introduction}
\subsection*{Problem Overview}
We begin with a physical example of a Schr\"odinger
vector-operator. Gesztesy and Kirsch \cite{shrodinger} in
particular considered a Schr\"odinger operator generated by the
Hamiltonian
\begin{equation}\label{hamiltonian}
H=-\frac{d^2}{dx^2}+ \big(s^2-\frac14\big)\frac1{\cos^2x}\,,\quad s>0\,.
\end{equation}
Since the potential in Hamiltonian has a countable number of
singularities on a discreet set $X$ in $\mathbb{R}$, leading to
spoiling of the local integrability, it is impossible to apply the
standard methods of the theory of ordinary differential operators.
In order to proceed and build a self-adjoint extension of a
minimal operator generated by (\ref{hamiltonian}) on
$\mathbb{R}\setminus X$, one may take self-adjoint extensions
$T_i$, generated by the same Hamiltonian (\ref{hamiltonian}) in
the coordinate spaces $L^2(-\frac\pi 2+i\pi,\frac\pi 2+i\pi)$,
$i\in\mathbb{Z}$, and then consider the direct sum operator
$\oplus_{i\in\mathbb{Z}}T_i$ in the space
$$\label{space for sh op}
\oplus_{i\in\mathbb{Z}}L^2\big(-\frac\pi 2+i\pi,\frac\pi 2+i\pi\big)\,.
$$
This direct sum operator appears to be one of the possible
self-adjoint extensions of the minimal operator considered on
$\mathbb{R}\setminus X$. Moreover in the case $s\geqslant 1$ the
minimal on $\mathbb{R}\setminus X$ operator appears to be
essentially self-adjoint and its only self-adjoint extension is a
direct sum operator.
This physical example gave birth to the theory of general
differential direct sum operators, or in the text below
vector-operators. Beginning from 1992, the theory of differential
vector-operators has been investigated in connection with their
non-spectral properties in a Hilbert space (\cite{everittshubin},
\cite{everittzettl,everittmarkus} and in complete locally convex
spaces \cite{ashuroveveritt,ashuroveveritt1}. The interest in such
a theory is explained by its numerous applications in theoretical
physics and pure mathematics. Thus, physical applications may be
found in a single or a multi-particle quantum mechanics,
especially in problems where a quantum system is split into a
number of disconnected subsystems under the influence of a
potential. For applications in quantum mechanics see also the
respective references in \cite{everittmarkus}.
As it was shown in the fundamental works \cite{everittzettl} and
\cite{everittmarkus}, a differential vector-operator is an object
which resembles an ordinary differential operator by its general
properties, but in fact it has much more complicated structure.
Although the bigger part of studies concerned only non-spectral
properties of differential vector-operators, there has been some
development of their spectral theory recently. Some results
describing position of spectra of Schr\"odinger vector-operators
were presented in 1985 in \cite{shrodinger} and the most recent
results for general quasi-differential vector-operators belong to
Sobhy El-Sayed Ibrahim \cite{ibrahim,ibrahim1}.
The internal spectral structure of abstract self-adjoint
vector-operators was first investigated in \cite{sokolov}, for
which see also \cite{sokolov2, sokolov1}. The structure of
coordinate operators as differential operators played the key role
in \cite{sokolov3} where the unitary transformation making the
ordered representation was described in terms of generalized
eigenfunctions of a differential vector-operator. These
generalized eigenfunctions appear to be only measurable relative
to the spectral parameter, therefore it is an essential problem to
obtain their decomposition over some set of analytical kernels.
This problem is positively solved by Theorem \ref{kernelsdecomp}
of the current work.
\subsection*{Mathematical background}
Basic concepts of quasi-differential operators are well described
in \cite{everittzettl, everittmarkus}. A good reference for
operators with real coefficients is the book of Naimark
\cite{naimark}.
Let $\Omega$ be a finite or a countable set of indices. On
$\Omega$, we have a multi-interval differential
Everitt-Markus-Zettl system $\{I_i,\tau_i\}_{i\in\Omega}$, where
$I_i$ are arbitrary intervals of the real line and $\tau_i$ are
formally self-adjoint differential expressions of a finite order.
This EMZ system generates a family of Hilbert spaces $\{L^2(I_i)=
L^2_i\}_{i\in\Omega}$ and families of minimal
$\{T_{min,i}\}_{i\in\Omega}$ and maximal
$\{T_{max,i}\}_{i\in\Omega}$ differential operators. Consider a
respective family $\{T_i\}_{i\in\Omega}$ of self-adjoint
extensions. Further, we introduce a system Hilbert space
${\mathbf{L}^2}=\oplus_{i\in\Omega}{ L^2_i}$, consisting of the vectors
$\mathbf{f}=\oplus_{i\in\Omega}{f_i}$ such that $f_i\in
L^2_i$ and
$$
\norm{\mathbf{f}}=\sum_{i\in\Omega} \|f_i\|_i^2
=\sum_{i\in \Omega}\int_{I_i}|f_i|^2\,dx<\infty\,.
$$
In the space ${\mathbf{L}^2}$ consider the operator
$T:D(T)\subseteq{\mathbf{L}^2}\rightarrow{\mathbf{L}^2}$, defined
on the domain
$$
D(T)=\big\{\mathbf{f}\subseteq{\mathbf{L}^2}:\sum_{i\in\Omega}
\|T_if_i\|_i^2 <\infty\big\}
$$
by $ T\mathbf{f}=\oplus_{i\in\Omega}{T_if_i}$.
The operator $T$ is called a \emph{differential vector-operator}
generated by the self-adjoint extensions $T_i$, or a
\emph{self-adjoint differential vector-operator}. If $\Omega$ is
infinite, the vector-operator $T$ is called \emph{infinite}. The
operators $T_i$ are called \emph{coordinate} operators.
The abstract preliminaries for this work may be found, for
instance, in books \cite{reedsimon, danford}.
Fix $i\in\Omega$. For each $T_i$ there exists a unique resolution
of the identity $E_\lambda^i$ and a unitary operator $U_i$, making the
isometrically isomorphic mapping of the Hilbert space $ L^2_i$
onto the space $L^2(M_i,\mu_i)$, where the operator $T_i$ is represented
as a multiplication operator. Below, we remind the structure of
the mapping $U_i$.
We call $\phi\in L^2_i$ a \emph{cyclic vector} if for each $z\in
L^2_i$ there exists a Borel function $f$, such that
$z=f(T_i)\phi$. Generally, there is no a cyclic vector in $ L^2_i$
but there is a collection $\{\phi^k\}$ of them in $ L^2_i$, such
that $ L^2_i=\oplus_k{ L^2_i(\phi^k)}$, where $ L^2_i(\phi^k)$ are
$T_i$-invariant subspaces in $ L^2_i$ generated by the cyclic
vectors $\phi^k$. That is $
L^2_i(\phi^k)=\overline{\{f(T_i)\phi^k\}},
$ for a varying Borel function $f$, such that $\phi^k\in
D(f(T_i))$.
A vector $\phi\in L^2_i$ is called \emph{maximal} relative to the
operator $T_i$, if each measure $(E^i(\cdot)x,x)_i$, $x\in L^2_i$,
is absolutely continuous with respect to the measure
$(E^i(\cdot)\phi,\phi)_i$.
For each Hilbert space $ L^2_i$, there exist a unique (up to
unitary equivalence) decomposition $ L^2_i=\oplus_k
L^2_i(\varphi^k_i)$, where $\varphi^1_i$ is maximal in $ L^2_i$
relative to $T_i$, and a decreasing set of multiplicity sets
$e_k^i$, where $e_1^i$ is the whole line, such that $\oplus_k
L^2_i(\varphi^k_i)$ is equivalent with $\oplus_k
L^2(e^i_k,\mu_i)$, where the measure of the ordered representation
is defined as
$\mu_i(\cdot)=(E^i(\cdot)\varphi_i^1,\varphi_i^1)_i$. A spectral
representation of $T_i$ in $\oplus_k L^2(e^i_k,\mu_i)$ is called
the \emph{ordered representation} and it is unique, up to a
unitary equivalence. Two operators are called \emph{equivalent},
if they create the same ordered representation of their spaces.
For $i\in\Omega$, we introduce a \emph{sliced union} of sets $M_i$
(see also preliminaries) as a set $M$, containing all $M_i$ on
different copies of $\cup_{i\in\Omega} M_i$. The sets $M_i$ do not
intersect in $M$, but they can \emph{superpose}, i.e. two sets
$M_i$ and $M_j$ superpose, if their projections in the set
$\cup_{i\in\Omega} M_i$ intersect.
For $z_i\in L^2_i$, $i\in\Omega$, define
$\widehat{\mathbf{z_i}}=\{0,\dots,0,z_i,0,\dots,0\}\in{\mathbf{L}^2}$,
where $z_i$ is on the $i$-th place.
For each $i\in\Omega$, let $\delta(T_i)$ denote the
\emph{subspectrum} of the operator $T_i$, i.e.
$\delta(T_i)=\sigma_{pp}(T_i)\cup\sigma_{cont}^*(T_i)$, where
$\sigma_{pp}(T_i)$ is the set of eigenvalues which may be open and
$\sigma_{cont}^*(T_i)$ is the continuous spectrum with a removed
set of spectral measure zero. $\sigma_{cont}^*(T_i)$ may be also
open. Note that $\overline{\delta(T_i)}=\sigma(T_i)$. For
instance, the subspectrum of an operator having the complete
system of eigenfunctions with eigenvalues being the rational
numbers of $[0,1]$ equals to $\mathbb{Q}\cap [0,1]$; the
subspectrum of an operator having the continuous spectrum [0,1] is
assumed to equal to (0,1) without loss of generality. The notion
of the subspetrum arises quite naturally. Indeed, let we are given
a self-adjoint operator $A$ with a simple spectrum
$\sigma(A)=[a,b]$. Choosing any point $\xi\in\sigma(A)$ we can
obtain $\sigma(A)=[a,\xi)\cup[\xi,b]$. If we are interested in
obtaining a formula $A_1\oplus A_2 = A$, where
$\sigma(A_1)=[a,\xi]$ and $\sigma(A_2)=[\xi, b]$, we have to
suppose that $\xi\not\in \sigma_{pp}(A)$. But if we pass to
subspectra, we will not need to care about inessential points
appearing as limit points.
Consider a projecting mapping $P:M\to \cup_{i\in\Omega} M_i$ such
that $P(\delta(T_i))=\delta(T_i)$. Let $\Omega=\cup_{k=1}^K A_k$,
$A_k\cap A_s=\emptyset$ for $k\neq s$ and
\[
A_k=\{s\in\Omega: \forall s,l\in A_k, s\neq l\,,
P(\delta(T_s))\cap P(\delta(T_l))=B_{sl}\,,
\]
where $\norm{E^t(B_{sl})\varphi_t}_t=0$ for any cyclic $
\varphi_t\in L^2_t,\,t=s,l\}$.
From all such divisions of $\Omega$ we choose and fix the
one, which contains the minimal number of $A_k$. In case when all
the coordinate spectra $\sigma(T_i)$ are simple, we define the
number $\Lambda=\min\{K\}$ as the \emph{spectral index} of the
vector-operator $T$.
The following two lemmas were proved in \cite{sokolov}.
\begin{lem}\label{irequality}
The identity resolution $\{E_\lambda\}$ of the vector-operator $T$
equals to the direct sum of the coordinate identity resolutions
$\{E_\lambda^i\}$, that is $\{E_\lambda\} = \oplus_{i\in\Omega}{\{E_\lambda^i\}} $
\end{lem}
\begin{lem}\label{spectral index}
Let each $T_i$ have a cyclic vector $a_i$ in $ L^2_i$. Then the
vector-operator $T$ has minimum $\Lambda$ cyclic vectors
$\{\mathbf{a}_k\}_{k=1}^\Lambda$, having the form $
\mathbf{a}_k=\sum_{i\in A_k}\widehat{\mathbf{a_i}}. $
\end{lem}
In the next section we will see what a spectral multiplicity of a
vector-operator is. Nevertheless, this notation is intuitively
clear. Running ahead, let us present here two examples, which will
show the difference between the spectral index and the spectral
multiplicity of the vector-operator $T$.
\noindent\textbf{Example 1.} We have a three-interval EMZ system
$\{I_i,\tau_i,1\}_{i=1}^3$ (a kinetic energy, a mirror kinetic
energy, an impulse):
\begin{gather*}
I_1=[0,+\infty),\quad \tau_1=-\big(\frac d{dt}\big)^2,\\
D(T_1)=\{f\in D(T_{max,1}): f(0)+kf'(0)=0,-\infty