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A [[Self-adjoint operator|self-adjoint operator]] generated by a differential expression

$$ 
l[f]  =  -(p(x)f ^ { \prime } )  ^  \prime  + q(x)f,\ \ 
x \in (a, b),
$$

and suitable boundary conditions in the Hilbert space  $  L _ {2} (a, b) $, 
where  $  (a, b) $
is a finite or infinite interval,  $  p  ^  \prime  , p, q $
are continuous real functions, and  $  p(x) > 0 $
for all  $  x \in (a, b) $(
sometimes any operator generated by a quasi-differential expression analogous to  $  l $
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|Sturm–Liouville problem]] on a finite interval.

A point  $  a $
is called a regular end-point if  $  a $
is finite,  $  p(a) \neq 0 $
and  $  p  ^  \prime  , p, q \in C(a, b) $. 
Otherwise this point is called a singular end-point. The expression  $  l $
is called regular or singular depending on whether both end-points of  $  (a, b) $
are regular or not.

Let  $  D _ {1} $
be the set of functions  $  f \in L _ {2} (a, b) $
for which  $  f ^ { \prime } $
is absolutely continuous and  $  l[f] \in L _ {2} (a, b) $, 
let  $  D _ {0} $
be the subset of  $  D _ {1} $
consisting of the functions with compact support. Further, let  $  L _ {1} :  f \rightarrow l[f] $, 
$  f \in D _ {1} $, 
and let  $  L _ {0} $
be the closure of the operator  $  L _ {0}  ^  \prime  :  f \rightarrow l[f] $, 
$  f \in D _ {0} $; 
$  L _ {0} $
is a symmetric operator, and  $  L _ {0}  ^ {*} = L _ {1} $. 
A Sturm–Liouville operator is an extension (restriction) of the operator  $  L _ {0} $
$  (L _ {1} ) $.

1) Let  $  l $
be regular, let the vectors  $  ( \alpha _ {i} , \alpha _ {i}  ^  \prime  , \beta _ {i} , \beta _ {i}  ^  \prime  ) $, 
$  i=1, 2 $, 
be linearly independent and let

$$ \tag{1 }
p(b)( \overline{ {\beta _ {i}  ^  \prime  }}\; \beta _ {j} - \overline{ {\beta _ {i} }}\; \beta _ {j}  ^  \prime  ) - p(a)(
\overline{ {\alpha _ {i}  ^  \prime  }}\; \alpha _ {j} - \overline{ {\alpha _ {i} }}\; \alpha _ {j}  ^  \prime  )  = \ 
0,\ \ 
i, j = 1, 2.
$$

Then the set of all functions  $  f \in D _ {1} $
that satisfy the conditions

$$ \tag{2 }
p(b)( \beta _ {i}  ^  \prime  f ^ { \prime } (b) - \beta _ {i} f(b)) - p(a)( \alpha _ {i}  ^  \prime  f ^ { \prime } (a) - \alpha _ {i} f(a))  =  0,
$$

$  i = 1, 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):

$$ \tag{3 }
f(a)  \cos  \phi - f ^ { \prime } (a)  \sin  \phi  =  0,\ \ 
\phi \in [0, \pi ],
$$

$$ \tag{4 }
f(b)  \cos  \theta - f ^ { \prime } (b)  \sin  \theta  =  0,\  \theta \in [0, \pi ],
$$

and the mixed boundary conditions

$$ \tag{5 }
f(a)  =  \nu f(b) ,\ \ 
f ^ { \prime } (a)  =  \delta f ^ { \prime } (b),
$$

where  $  \nu \overline \delta \; = p(b)/p(a) $. 
In particular, if  $  p(a) = p(b) $, 
then for  $  \nu = \delta = 1 $
the conditions (5) are called periodic, and for  $  \nu = \delta = -1 $
anti-periodic (or semi-periodic).

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

$  2 _ {1} $) 
Let  $  a $
be regular and  $  b $
be singular, and let the number of independent solutions of the equation  $  l[f] = i f $
belonging to  $  L _ {2} (a, b) $
be equal to 1. Then the expression  $  l $
is said to belong to the case of a Weyl limit point at  $  b $. 
The domain of definition of the Sturm–Liouville operator is determined by the boundary condition (3).

$  2 _ {2} $) 
If the number of linearly independent solutions of  $  l[f] = i f $
belonging to  $  L _ {2} (a, b) $
is 2, then the expression  $  l $
is said to belong to the case of a Weyl limit circle at  $  b $. 
The deficiency indices of the operator  $  L _ {0} $
are  $  (2, 2) $
in this case. The domain of definition of a Sturm–Liouville operator is described similarly to 1), replacing conditions (2) as follows:  $  p(b) $
is replaced by  $  p(a) $, 
$  f(b) $
and  $  f ^ { \prime } (b) $
are replaced by  $  (Sf  ) _ {1} (b) $
and  $  (Sf  ) _ {2} (b) $, 
respectively, where

$$ 
(Sf  ) _ {1} (b)  =  \lim\limits _ {x \rightarrow b }  p(x)[fu _ {2} ](x),\ \ 
(Sf  ) _ {2} (b)  =  \lim\limits _ {x \rightarrow b }  p(x)[u _ {1} f](x);
$$

here  $  [ \phi \psi ](x) $
is the [[Wronskian|Wronskian]] of the functions  $  \phi $
and  $  \psi $
at the point  $  x $, 
$  u _ {i} $, 
$  i = 1, 2 $, 
are the solutions of the equation  $  l[f] =0 $
with the initial conditions  $  u _ {i}  ^ {(j-1)} (0) = \delta _ {ij} $, 
$  i, j = 1, 2 $, 
and  $  \delta _ {ij} $
are the Kronecker symbols.

The resolvent kernel of a Sturm–Liouville operator is a [[Carleman kernel|Carleman kernel]]; moreover, the resolvent in cases 1) and  $  2 _ {2} $) 
is a [[Hilbert–Schmidt integral operator|Hilbert–Schmidt integral operator]], but in  $  2 _ {1} $) 
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  $  2 _ {2} $)) 
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  $  L _ {2} (a, b) $.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,   "Lineare Differentialoperatoren" , Akademie Verlag  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Akhiezer,   I.M. Glazman,   "Theory of linear operators in Hilbert space" , '''2''' , Pitman  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.M. Levitan,   I.S. Sargsyan,   "Introduction to spectral theory: selfadjoint ordinary differential operators" , Amer. Math. Soc.  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Marchenko,   "Sturm–Liouville operators and applications" , Birkhäuser  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.M. Glazman,   "Direct methods of qualitative spectral analysis of singular differential operators" , Israel Program Sci. Transl.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V. Hutson,   J. Pym,   "Applications of functional analysis and operator theory" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E.C. Titchmarsh,   "Eigenfunction expansions associated with second-order differential equations" , '''1''' , Clarendon Press  (1946)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  N. Levinson,   "Criteria for the limit-point case for second order linear differential operators"  ''Časopis Pěst. Mat. Fys.'' , '''74'''  (1949)  pp. 17–20</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  A.Ya. Povzner,   "On differential equations of Sturm–Liouville type on the half-line"  ''Mat. Sb.'' , '''23''' :  1  (1948)  pp. 3–52  (In Russian)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  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 &amp; Weksell  (1977)  pp. 62–81</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,   "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)  pp. Chapt. 10, §3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.M. Levitan,   I.S. Sargsyan,   "Sturm–Liouville and Dirac operators" , Kluwer  (1991)  (Translated from Russian)</TD></TR></table>