View source for Sturm-Liouville equation

An ordinary differential equation of the second order

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907501.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907502.png" /> varies in a given finite or infinite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907504.png" /> are given coefficients, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907505.png" /> is a complex parameter, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907506.png" /> is the sought solution. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907507.png" /> are positive, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907508.png" /> has a first derivative and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s0907509.png" /> has a second derivative, then by the Liouville substitution (see [[#References|[1]]]) this equation may be reduced to the standard form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

It is assumed that the complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075011.png" /> is measurable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075012.png" /> and summable on each of the subintervals in it. At the same time one also considers the non-homogeneous equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075014.png" /> is a given function.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075015.png" /> is measurable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075016.png" /> and summable on each of the subintervals in it, then for all complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075017.png" /> and any interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075018.png" />, equation (2) has on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075019.png" /> one and only one solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075020.png" /> satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075022.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075023.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075024.png" /> is an entire analytic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075025.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075026.png" /> one can take one of the end-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075027.png" /> (if this end-point is regular, cf. [[Sturm–Liouville operator|Sturm–Liouville operator]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075029.png" /> be two arbitrary solutions of (1). Their [[Wronskian|Wronskian]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075030.png" /></td> </tr></table>

is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075031.png" /> and vanishes if and only if these solutions are linearly dependent. The general solution of (2) is of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075032.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075033.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075034.png" /> are arbitrary constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075035.png" /> are linearly independent solutions of (1).

The following fundamental theorem of Sturm (see [[#References|[1]]]) is true: Let two equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

be given. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075038.png" /> are real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075039.png" /> on the entire interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075040.png" />, then between any two zeros of any non-trivial solution of the first equation there is at least one zero of each solution of the second equation.

The following theorem is known as the comparison theorem (see [[#References|[1]]]): Let the left-hand end-point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075041.png" /> be finite, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075042.png" /> be a solution of (3) satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075044.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075045.png" /> be a solution of (4) with the same conditions; let, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075046.png" /> on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075047.png" />. Then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075048.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075049.png" /> zeros on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075051.png" /> will have at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075052.png" /> zeros and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075053.png" />-th zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075054.png" /> will be less than the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075055.png" />-th zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075056.png" />.

One of the important properties of (1) is the existence of so-called operator transforms with a simple structure. Operator transforms arose from general algebraic considerations related to the theory of generalized shift operators (change of the basis).

There are the following types of operator transforms for equation (1). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075057.png" /> be the solution of

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

satisfying the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

It turns out that this solution has the following representation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075060.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075061.png" /> is a continuous function independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075062.png" />; moreover,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075063.png" /></td> </tr></table>

The integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075064.png" /> defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075065.png" /></td> </tr></table>

is called an operator transform (a transmutation operator), and preserves the conditions at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075066.png" />. It transforms the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075067.png" /> (a solution of the simplest equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075068.png" /> with the conditions (6)) into the solution of (5) under the same conditions at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075069.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075071.png" /> be the solutions of (5) satisfying

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075072.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075073.png" /></td> </tr></table>

These solutions have the representations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075074.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075075.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075077.png" /> are continuous functions.

A new type of operator transforms has been introduced (see [[#References|[8]]]) that preserves the asymptotic behaviour of solutions at infinity; namely, it turned out that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075078.png" /> in the upper half-plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075079.png" />, the equation (5), considered on the half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075080.png" /> under the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075081.png" />, has a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075082.png" /> that can be represented in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075083.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075084.png" /> is a continuous function satisfying the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075085.png" /></td> </tr></table>

in which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075086.png" /></td> </tr></table>

Moreover,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075087.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  M.A. Naimark,   "Lineare Differentialoperatoren" , Akademie Verlag  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.M. Levitan,   "Generalized translation operators and some of their applications" , Israel Program Sci. Transl.  (1964)  (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">  J. Delsarte,   "Sur certaines transformations fonctionnelles rélatives aux équations linéaires aux dérivées partielles du second ordre"  ''C.R. Acad. Sci. Paris'' , '''206'''  (1938)  pp. 1780–1782</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.Ya. Povzner,   "On Sturm–Liouville type differential equations on the half-line"  ''Mat. Sb.'' , '''23''' :  1  (1948)  pp. 3–52  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B.M. Levitan,   "The application of generalized shift operators to linear second-order differential equations"  ''Uspekhi Mat. Nauk'' , '''4''' :  1  (1949)  pp. 3–112  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B.Ya. Levin,   "Transformations of Fourier and Laplace types by means of solutions of second order differential equations"  ''Dokl. Akad. Nauk SSSR'' , '''106''' :  2  (1956)  pp. 187–190  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.M. Levitan,   "Inverse Sturm–Liouville problems" , VNU  (1987)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Carroll,   "Transformation theory and application" , North-Holland  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.M. Levitan,   I.S. Sargsyan,   "Sturm–Liouville and Dirac operators" , Kluwer  (1991)  (Translated from Russian)</TD></TR></table>