Directing functionals, method of

A special method for proving theorems about eigen function expansions of self-adjoint differential operators. In the special case of a second-order singular differential operator on the positive semi-axis, the corresponding theorem was first obtained by H. Weyl [1]. The general theorem for a differential operator of order $ 2n $ was first proved by M.G. Krein [2], who used the method now known as the method of directing functionals. The result may be formulated as follows (see [3]). Let $ l ( y) $ be a self-adjoint differential expression of order $ 2n $ on an interval $ ( a, b) $, let $ u _ {1} ( x, \lambda ) \dots u _ {2n} ( x, \lambda ) $ be the system of solutions of the equation

$$ l ( y) = \lambda y, $$

satisfying the initial conditions

$$ u _ {j} ^ {[ k - 1] } ( x _ {0} ) = \ \left \{ \begin{array}{ll} 1 & \textrm{ if } j = k, \\ 0 & \textrm{ if } j \neq k, \\ \end{array} \right .$$

where $ x _ {0} $ is a fixed point in $ ( a, b) $ and $ u _ {j} ^ {[ k - 1] } $ is the $ ( k - 1) $- th quasi-derivative of $ u _ {j} $. Then, for any self-adjoint extension $ L $ of the operator generated by $ l ( y) $, there exists a matrix-valued distribution function

$$ \sigma ( \lambda ) = \ ( \sigma _ {jk} ( \lambda )),\ \ j, k = 1 \dots 2n, $$

such that, for any function $ f \in L _ {2} ( a, b) $,

$$ \tag{1 } \phi _ {j} ( \lambda ) = \ \int\limits _ { a } ^ { b } f ( x) u _ {j} ( x, \lambda ) dx, $$

$$ \tag{2 } f ( x) = \int\limits _ {- \infty } ^ {+ \infty } \sum _ {j, k = 1 } ^ { 2n } \phi _ {j} ( \lambda ) u _ {k} ( x, \lambda ) d \sigma _ {jk} ( \lambda ), $$

where the integrals in (1) and (2) are assumed to be convergent in the sense of the metrics in $ L _ {2} ( \sigma ) $ and $ L _ {2} ( a, b) $, respectively. Under these assumptions, one has the following analogue of Parseval's equality:

$$ \int\limits _ { a } ^ { b } | f ( x) | ^ {2} dx = \ \int\limits _ {- \infty } ^ { + \infty } \sum _ {j, k = 1 } ^ { 2n } \phi _ {j} ( \lambda ) \overline{ {\phi _ {k} ( \lambda ) }}\; \ d \sigma _ {jk} ( \lambda ). $$

The functionals $ \phi ( \lambda ) $, defined on functions in $ L _ {2} ( a, b) $ with compact support, are called the directing functionals of $ l ( y) $.

The generalization and further development of the method of directing functionals gave rise to the concept of rigged Hilbert spaces and generalized eigen element expansions (see [4], [5], [6]).

References

[1]	H. Weyl, "Ueber gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen" Math. Ann. , 68 (1910) pp. 220–269
[2]	M.G. Krein, "On a general method for decomposing Hermitian positive kernels into elementary factors" Dokl. Akad. Nauk SSSR , 53 : 1 (1946) pp. 3–6 (In Russian)
[3]	M.A. Naimark, "Lineare Differentialoperatoren" , Akademie Verlag (1960) (Translated from Russian)
[4]	Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)
[5]	I.M. Gel'fand, G.E. Shilov, "Some problems in differential equations" , Moscow (1958) (In Russian)
[6]	I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1964) (Translated from Russian)
[7]	B.M. Levitan, "Eigenfunction expansions of second-order differential equations" , Moscow-Leningrad (1950) (In Russian)