# Spectral analysis

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The investigation of the spectral properties of linear operators (cf. Linear operator), such as the geometry of the spectrum and its main parts, spectral multiplicity and the asymptotics of eigenvalues.

For operators on finite-dimensional spaces, the problem of determining the spectrum is equivalent to the problem of localizing the roots of the characteristic equation $\mathop{\rm det} ( A - \lambda I ) = 0$. For infinite-dimensional spaces the situation is much more complicated, although the apparatus of determinants can be set up and used successfully for the spectral analysis of some infinite-dimensional operators. In many cases the spectral analysis of an operator is based on an explicit construction of a functional calculus (respectively, by multiplication operators on function spaces, other model operators, as well as their restrictions or quotients). Various theorems on the form of the spectra of functions of one or several operators find wide application in spectral analysis. These may be very simple (the spectrum of a polynomial in an operator consists of the values of this polynomial on the spectrum of the operator, and the spectrum of the sum of two commuting operators is contained in the algebraic sum of their spectra) or very subtle (describing the spectra of functions of non-commuting operators, of functions of an operator having discontinuities at boundary points of its spectrum, the combined spectra of the image of many-valued mappings, images of approximate, point or deficiency spectra, etc.). Useful information about the spectrum of an operator can be extracted from its topological characteristics (for example, the spectrum of a continuous operator is compact, and the spectrum of a compact operator is at most countable and its non-zero points are isolated eigenvalues), from its behaviour with respect to a specific cone in the space (the principal eigenvalues of a positive operator), or from the scalar product (the spectrum of a self-adjoint operator is real, that of a positive Hermitian operator is non-negative, that of a dissipative operator lies in the upper half-plane, and that of a unitary operator lies on the unit circle). If the scalar product is not of fixed sign, but its index of indefiniteness $\kappa$ is finite, then the spectrum of an operator preserving the product (called a $J$- unitary operator) can have at most $2 \kappa$ points outside the unit circle. For $J$- self-adjoint and $J$- dissipative operators the situation is similar (see [5]).

Spectral characteristics can posses specific properties of stability (continuity) and these are studied in spectral perturbation theory (a branch of general perturbation theory). Thus, the spectrum is an upper semi-continuous function of the operator: Any neighbourhood of the spectrum of a bounded operator contains the spectra of all operators sufficiently close to it (in the case of unbounded operators a slight modification is required). This enables one to trace the variation of isolated points of the spectrum under small perturbations, and to express analytically (in the form of a power series in the parameter $\mu$) the eigenvalues of an operator $A + B \mu$ that lie in a neighbourhood of an isolated eigenvalue of finite multiplicity of $A$. In certain cases one can also estimate the variation of the number of eigenvalues of an operator in a given domain under the effect of a perturbation that is not necessarily small in norm but has fixed (finite) rank. The same circle of ideas contains Weyl's theorem (H. Weyl, 1909) on the invariance of the condensation spectrum (the complement in the spectrum of the set of isolated eigenvalues of finite multiplicity) of a self-adjoint operator under compact perturbations. It follows from this that the condensation spectrum of a self-adjoint operator $A$ coincides with its essential spectrum

$$\sigma _ {e} ( A) = \{ {\lambda \in \mathbf C } : { A - \lambda I \textrm{ is not a Fredholm operator } } \} ,$$

and the equation $\sigma _ {e} ( A + K ) = \sigma _ {e} ( A)$ holds whenever $A$ is closed and $K$ is compact. It also follows from Weyl's theorem that all self-adjoint extensions of a symmetric operator that have finite (and equal) deficiency numbers have identical essential spectrum. Weyl's theorem carries over to the case of relatively-compact perturbations ( $K$ is called a compact operator relative to $A$ if it maps every bounded set with bounded $A$- image into a compact set), which implies that the essential spectra of the self-adjoint extensions of a wide class of symmetric multi-dimensional differential operators coincide. Weyl's theorem has a converse (J. von Neumann, 1935): If two self-adjoint operators have identical essential spectra, then one of them is unitarily equivalent to a perturbation of the other by a compact operator (even one of Hilbert–Schmidt class) of arbitrarily small norm. There are generalizations of this result to the case of normal and essentially-normal operators, and also to representations of non-commutative $C ^ {*}$- algebras (cf. $C ^ {*}$- algebra).

The Weyl–von Neumann theorem shows that the essential spectrum is the unique spectral characteristic of self-adjoint operators which is stable under compact perturbations, and that the continuous and point spectra are extremely unstable. At the same time, the absolutely-continuous spectrum $\sigma _ {ac} ( A)$( which is the spectrum of the restriction of $A$ to the subspace $H _ {ac} ( A)$ of all vectors $x \in H$ for which the function $\lambda \mapsto ( E _ {A} ( \lambda ) x , x )$ is absolutely continuous) also has a certain stability: it is invariant under nuclear perturbations. This is one of the fundamental results of the theory of wave operators, closely connected with the quantum-mechanical theory of scattering (see [2]). The wave operator $W ( A , B )$ for a pair of self-adjoint operators $A , B$ is the isometric linear mapping

$$x \rightarrow \lim\limits _ {t \rightarrow \infty } \ \mathop{\rm exp} ( i t B ) \mathop{\rm exp} ( - i t A ) x ,$$

defined on the closed subspace $\Sigma ( A , B )$ of all vectors $x \in H$ for which the limit exists. The relations $W ( A , B ) A = B W ( A , B )$ and $W ( A , B ) \Sigma ( A , B ) = \Sigma ( B , A )$ show that $W ( A , B )$ realizes a unitary equivalence of $A$ and $B$ if $\Sigma ( A , B ) = \Sigma ( B , A ) = H$. The condition of nuclearity of $B - A$ implies the inclusions $H _ {ac} ( A) \subset \Sigma ( A , B )$ and $H _ {ac} ( B) \subset \Sigma ( B , A )$, and thus the unitary equivalence of the absolutely-continuous parts of $A$ and $B$, which in turn ensures that their spectral characteristics are the same.

There is another approach to the problem of proving the unitary equivalence (or, in the case of non-self-adjoint operators, the similarity) of a perturbed operator and an unperturbed one. In this approach one writes the conditions of similarity of two operators $A$ and $A + K$ in the form of a linear operator equation $A V - V A = V K$. Then one looks for a linear operator $\Gamma$ that is left inverse to the multiplication operator $X \mapsto A X - X A$( that is, $A \Gamma ( X) - \Gamma ( X) A = X$) and for which the operator $\Gamma _ {K} : X \mapsto \Gamma ( X K )$ is a contraction in the space of operators. If one succeeds in finding such an operator $\Gamma$, then as $V$ one may take the operator $( I + \Gamma _ {K} ) ^ {-} 1 I$, having verified beforehand its invertibility. By this method one can successfully investigate a wide class of normal operators with discrete and continuous spectrum, quasi-nilpotent operators, weighted shift operators, and, what is especially important for applications, multi-dimensional integro-differential operators.

The spectral analysis of operators generated by analytic operations on function spaces (such as differential, integral and difference operators) assumes a description of their spectra in terms of the parameters (the coefficients) of the corresponding operation. The wide applicability of perturbation theory in such problems is explained by the fact that one can often successfully isolate the principal part and perturbation part in the same terms (by redistributing the coefficients). For example, let $G$ be a domain in $\mathbf R ^ {n}$, let $q$ be a real potential, i.e. a numerical function on $G$, and let $A _ {q} ( G)$ be the Schrödinger operator defined on $L _ {2} ( G)$ by the differential operation $l _ {q} ( u) = - \Delta \mu + q u$ and by the most stringent boundary conditions (a minimal operator). In this case $A _ {q} ( G)$ is symmetric. It is natural to assume that $- \Delta$( or, more precisely, $A _ {0} ( G)$) is an unperturbed operator, and that multiplication by $q$ is a perturbation. Such a representation has useful consequences in case the potential is small in some sense. Thus, if $q ( M) \rightarrow 0$ as $G \ni M \rightarrow \infty$, then Weyl's theorem ensures that the essential spectra of $A _ {q}$ and $A _ {0}$ coincide (and coincide with the essential spectra of their self-adjoint extensions). If $G$ is "sufficiently large" and the potential is square-integrable, then $\sigma _ {l} ( A _ {q} ) \supset [ 0 , \infty )$, and if, in addition,

$$\sum _ {| j | \leq n + 1 } \int\limits _ { G } ( 1 + r ^ {2} ) | \partial ^ {j} q | d V$$

is sufficiently small, then $A _ {q}$ and $A _ {0}$ are unitarily equivalent. In other cases one takes $A {\widetilde{q} }$ as a non-perturbed operator, where the potential $\widetilde{q}$ is "close" to $q$, but has a simpler structure. This enables one to prove, for example, that in the interval $( - \infty , a )$ the spectra of the self-adjoint extensions of $A _ {q}$ are finite, provided that $\lim\limits \inf q ( M) \rightarrow a$ as $M \rightarrow \infty$( in particular, $A _ {q}$ is semi-bounded and has a discrete spectrum when $q ( M) \rightarrow \infty$ as $G \ni M \rightarrow \infty$).

In the spectral analysis of symmetric differential operators (especially of one-dimensional such operators), another approach has been used, based not on spectral perturbation theory, but on a special form of the spectral decomposition theorem. A unitary transformation effecting a spectral representation of a differential operator can be realized (in the simplest case of an operator with a cyclic vector) by the integral operator

$$U f ( \lambda ) = \int\limits _ { G } u ( x , \lambda ) f ( x) d x ,$$

the kernel $u ( x , \lambda )$ of which at any $\lambda$ is a solution of the differential equation $l ( y) = \lambda y$, where $l$ is the original differential operation. This enables one to use the qualitative theory of differential equations for the spectral analysis of differential operators, and leads not only to a description of the geometry of the spectrum (here, the results of this approach correspond to, and in the multi-dimensional case even yield, those of perturbation theory), but also leads to convenient analytic expressions for the spectral characteristics, to refined results about the convergence of spectral decompositions, etc.

The functions $u ( x , \lambda )$ involved in the spectral decomposition of a differential operator are not, in the case of a continuous spectrum, its eigenfunctions, since they do not belong to $L _ {2} ( G)$. An abstract variant of the decomposition with respect to "generalized eigenfunctions" can be constructed within the limits of the theory of rigged Hilbert spaces (see [4]). A rigged Hilbert space is a triple $\Phi \subset H \subset \Phi ^ \prime$, where $H$ is a Hilbert space, $\Phi$ is a topological vector space continuously imbedded in $H$ and $\Phi ^ \prime$ is the dual of $\Phi$. An element $f \in \Phi ^ \prime$ is called a generalized eigenvector of an operator $A$ on $H$ if $A \Phi \subset \Phi$ and if $f ( A x - \lambda x ) = 0$ for all $x \in \Phi$( where $\lambda$ is the corresponding eigenvalue). For every self-adjoint operator $A$ one can select a rigging such that the set of generalized eigenvectors $\{ {f _ \lambda } : {\lambda \in \sigma ( A) } \}$ of $A$ is complete in the following sense: For any $x \in \Phi$,

$$\| x \| ^ {2} = \int\limits _ {\sigma ( A) } | f _ \lambda ( x) | ^ {2} d \rho ( \lambda ) ,$$

where $\rho$ is some measure on $\sigma ( A)$. If $A$ has a cyclic vector $x _ {0}$, then one can take $\rho$ to be $( E _ {A} ( \cdot ) x _ {0} , x _ {0} )$, where $E _ {A}$ is the spectral measure of $A$. Here, $f _ \lambda = ( d P _ \lambda x ) / ( d \rho ( - \infty , \lambda ) )$, and the limit is taken in the topology of $\Phi ^ \prime$.

For operators with a point spectrum, the question of the asymptotics of the eigenvalues is of paramount importance. In the case of a self-adjoint operator it is somewhat simpler to describe the asymptotic behaviour of the function $N ( \lambda )$ equal to the number of eigenvalues less than $\lambda$, or, equivalently, equal to the dimension of the spectral subspace corresponding to the interval $( - \infty , \lambda )$. The classical Weyl theorem states: For the Laplace operator with Dirichlet boundary conditions in a domain $\Omega \subset \mathbf R ^ {n}$, $N ( \lambda )$ is asymptotically equal to $r _ {n} ( 2 \pi ) ^ {-} n | \Omega | \lambda ^ {n/2}$, where $| \Omega |$ is the volume of $\Omega$ and $r _ {n}$ is the volume of the unit ball in $\mathbf R ^ {n}$.

#### References

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