\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \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}} \setcounter{page}{101} \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 $$\label{hamiltonian} H=-\frac{d^2}{dx^2}+ \big(s^2-\frac14\big)\frac1{\cos^2x}\,,\quad s>0\,.$$ 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