Sturm-Liouville operator

A self-adjoint operator generated by a differential expression

and suitable boundary conditions in the Hilbert space , where is a finite or infinite interval, are continuous real functions, and for all (sometimes any operator generated by a quasi-differential expression analogous to is called so). Starting in 1830, J.Ch. Sturm and J. Liouville published a number of fundamental studies on the theory of the Sturm–Liouville problem on a finite interval.

A point is called a regular end-point if is finite, and . Otherwise this point is called a singular end-point. The expression is called regular or singular depending on whether both end-points of are regular or not.

Let be the set of functions for which is absolutely continuous and , let be the subset of consisting of the functions with compact support. Further, let , , and let be the closure of the operator , ; is a symmetric operator, and . A Sturm–Liouville operator is an extension (restriction) of the operator .

1) Let be regular, let the vectors , , be linearly independent and let

(1)

Then the set of all functions that satisfy the conditions

(2)

, is the domain of definition of some Sturm–Liouville operator. Conversely, the domain of definition of any Sturm–Liouville operator can be determined in this way.

Among the boundary conditions, an important place is occupied by the separated boundary conditions (or boundary conditions of Sturm type):

(3)

(4)

and the mixed boundary conditions

(5)

where . In particular, if , then for the conditions (5) are called periodic, and for anti-periodic (or semi-periodic).

2) Let be singular. The case when both end-points are singular can be reduced to the case of one singular end-point by splitting.

) Let be regular and be singular, and let the number of independent solutions of the equation belonging to be equal to 1. Then the expression is said to belong to the case of a Weyl limit point at . The domain of definition of the Sturm–Liouville operator is determined by the boundary condition (3).

) If the number of linearly independent solutions of belonging to is 2, then the expression is said to belong to the case of a Weyl limit circle at . The deficiency indices of the operator are in this case. The domain of definition of a Sturm–Liouville operator is described similarly to 1), replacing conditions (2) as follows: is replaced by , and are replaced by and , respectively, where

here is the Wronskian of the functions and at the point , , , are the solutions of the equation with the initial conditions , , and are the Kronecker symbols.

The resolvent kernel of a Sturm–Liouville operator is a Carleman kernel; moreover, the resolvent in cases 1) and ) is a Hilbert–Schmidt integral operator, but in ) this is not necessarily the case.

The spectral expansion of a Sturm–Liouville operator in the case of a discrete spectrum (for example, in 1) and )) is similar to the Fourier expansion in eigenfunctions of the Sturm–Liouville problem, and in the other cases it contains eigenfunctions that are not in .

Problems of finding conditions on the coefficients and under which the Sturm–Liouville operator would have a discrete spectrum, or fills the whole line, and under which would be of limit-point or limit-circle type, are of great interest. Completely general necessary and sufficient conditions for and , which ensure that belongs to the limit-circle or limit-point type, are unknown (1984).

References

[1]	M.A. Naimark, "Lineare Differentialoperatoren" , Akademie Verlag (1960) (Translated from Russian)
[2]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 2 , Pitman (1980) (Translated from Russian)
[3]	B.M. Levitan, I.S. Sargsyan, "Introduction to spectral theory: selfadjoint ordinary differential operators" , Amer. Math. Soc. (1975) (Translated from Russian)
[4]	V.A. Marchenko, "Sturm–Liouville operators and applications" , Birkhäuser (1986) (Translated from Russian)
[5]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[6]	I.M. Glazman, "Direct methods of qualitative spectral analysis of singular differential operators" , Israel Program Sci. Transl. (1965) (Translated from Russian)
[7]	V. Hutson, J. Pym, "Applications of functional analysis and operator theory" , Acad. Press (1980)
[8]	E.C. Titchmarsh, "Eigenfunction expansions associated with second-order differential equations" , 1 , Clarendon Press (1946)
[9]	G.A. Mirzoev, "Description of the self-adjoint extensions of quasi-regular operators generated by differential expressions with two terms" Math. Notes , 29 : 2 (1981) pp. 116–121 Mat. Zametki , 29 : 2 (1981) pp. 225–233
[10]	A.M. Molchanov, "On conditions for discreteness of the spectrum of a second-order differential equation" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 169–199 (In Russian)
[11]	N. Levinson, "Criteria for the limit-point case for second order linear differential operators" Časopis Pěst. Mat. Fys. , 74 (1949) pp. 17–20
[12]	R.S. Ismagilov, "Conditions for self-adjointness of differential operators of higher order" Soviet Math. Dokl. , 3 : 1 (1962) pp. 279–283 Dokl. Akad. Nauk SSSR , 142 : 6 (1962) pp. 1239–1242
[13]	A.Ya. Povzner, "On differential equations of Sturm–Liouville type on the half-line" Mat. Sb. , 23 : 1 (1948) pp. 3–52 (In Russian)
[14]	W. Everitt, "On the deficiency index problem for ordinary differential equations 1910–1976" G. Berg (ed.) et al. (ed.) , Differential Equations (Proc. Internat. Conf. Uppsala) , Almqvist & Weksell (1977) pp. 62–81

References

[a1]	Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)
[a2]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963) pp. Chapt. 10, §3
[a3]	B.M. Levitan, I.S. Sargsyan, "Sturm–Liouville and Dirac operators" , Kluwer (1991) (Translated from Russian)