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An ordinary differential equation of the second order
 
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>
+
$$-\frac{d}{dx} \left\{ p(x) \frac{dy}{dx} \right\} + l(x)y = \lambda r(x) y,$$
 +
where $x$ varies in a given finite or infinite interval $(a, b)$, $p(x)$, $l(x)$, $r(x)$ are given coefficients, $\lambda$ is a complex parameter, and $y$ is the sought solution. If $p(x), r(x)$ are positive, $p(x)$ has a first derivative and $p(x) r(x)$ has a second derivative, then by the Liouville substitution (see
 +
[[#References|[1]]]) this equation may be reduced to the standard form
  
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
+
$$-y'' = q(x)y = \lambda y, \qquad a < x < b. \tag{1}$$
 +
It is assumed that the complex function $q$ is measurable on $(a, b)$ 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/s09075010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$-y'' + q(x)y = \lambda y + f(x), \qquad a < x < b, \tag{2}$$
 +
where $f$ is a given function.
  
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
+
If $f$ is measurable on $(a, b)$ and summable on each of the subintervals in it, then for all complex numbers $c_0, c_1$ and any interior point $x_0$, equation (2) has on $(a, b)$ one and only one solution $y(x, \lambda)$ satisfying the conditions $y(x_0, \lambda) = c_0$, $y'(x_0, \lambda) = c_1$. For any $x \in (a, b)$ the function $y(x, \lambda)$ is an entire analytic function of $\lambda$. As $x_0$ one can take one of the end-points of $(a, b)$ (if this end-point is regular, cf.
 +
[[Sturm–Liouville operator|Sturm–Liouville operator]]).
  
<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>
+
Let $y_1(x, \lambda)$ and $y_2(x, \lambda)$ be two arbitrary solutions of (1). Their
 
+
[[Wronskian|Wronskian]]
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075014.png" /> is a given function.
+
$$W(y_1, y_2) = y_1(x, \lambda) y_2'(x, \lambda) - y_1'(x, \lambda) y_2(x, \lambda)$$
 
+
is independent of $x$ and vanishes if and only if these solutions are linearly dependent. The general solution of (2) is of the form
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>
 
  
 +
$$y(x, \lambda) = a_1 y_2(x, \lambda) + a_2 y_2(x, \lambda) + \int_{x_0}^x R(x, \xi, \lambda) f(\xi) \, d\xi,$$
 
where
 
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>
+
$$R(x, \xi, \lambda) = \frac{1}{W(y_1, y_2)} \{y_1(x, \lambda) y_2(\xi, \lambda) - y_1(\xi, \lambda)y_2(x, \lambda)\},$$
 +
$a_1, a_2$ are arbitrary constants and $y_1(x, \lambda), y_2(x, \lambda)$ are linearly independent solutions of (1).
  
<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
  
The following fundamental theorem of Sturm (see [[#References|[1]]]) is true: Let two equations
+
$$u''+q_1(x) u = 0, \tag{3}$$
  
<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>
+
$$v'' + q_2(x) v = 0 \tag{4}$$
 +
be given. If $q_1(x), q_2(x)$ are real and $q_1(x) < q_2(x)$ on the entire interval $(a, b)$, 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.
  
<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>
+
The following theorem is known as the comparison theorem (see
 
+
[[#References|[1]]]): Let the left-hand end-point of $(a,b)$ be finite, let $u(x)$ be a solution of (3) satisfying the conditions $u(a) = \sin \alpha$, $u'(a) = \cos \alpha$, and let $v(x)$ be a solution of (4) with the same conditions; let, moreover, $q_1(x) , q_2(x)$ on the whole interval $(a, b)$. Then, if $u(x)$ has $m$ zeros on $(a, b)$, $v(x)$ will have at least $m$ zeros and the $k$-th zero of $v(x)$ will be less than the $k$-th zero of $u(x)$.
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).
 
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
+
There are the following types of operator transforms for equation (1). Let $y(x, \lambda)$ 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>
 
  
 +
$$-y'' + q(x) y = \lambda^2 y ,\qquad -a < x < a, \quad a \le \infty , \tag{5}$$
 
satisfying the conditions
 
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>
+
$$y(0, \lambda) = 1, \quad y'(0, \lambda) = i \lambda. \tag{6}$$
 
 
 
It turns out that this solution has the following representation:
 
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>
+
$$y(x, \lambda) = e^{i\lambda x} = \int_{-x}^x K(x, t) e^{i\lambda t} \, dt,$$
 +
where $K(x, t)$ is a continuous function independent of $\lambda$; moreover,
  
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,
+
$$K(x, x) = \frac 12 \int_0^x q(t) \, dt, \qquad K(x, -x) = 0.$$
 +
The integral operator $I + K$ 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/s09075063.png" /></td> </tr></table>
+
$$(I + K) f = f(x) + \int_{-x}^x K(x, t) f(t) \, dt$$
 +
is called an operator transform (a transmutation operator), and preserves the conditions at the point $x= 0$. It transforms the function $e^{i\lambda x}$ (a solution of the simplest equation $y'' = \lambda^2 x$ with the conditions (6)) into the solution of (5) under the same conditions at the point $x=0$. Let $\phi_h(x, \lambda)$ and $\phi_\infty(x, \lambda)$ be the solutions of (5) satisfying
  
The integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090750/s09075064.png" /> defined by
+
$$\phi_h(0, \lambda) = 1, \qquad \phi_h'(0, \lambda) = h.$$
 
 
<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>
 
  
 +
$$\phi_\infty(0, \lambda) = 0, \qquad \phi_\infty'(0, \lambda) = 1.$$
 
These solutions have the representations
 
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>
+
$$\phi_h(x, \lambda) = \cos \lambda x + \int_0^x K_h(x, t) \cos \lambda t \, dt,$$
  
<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>
+
$$\phi_\infty(x, \lambda) = \frac{\sin \lambda x}{\lambda} + \int_0^x K_\infty(x, t) \frac{\sin \lambda t}{\lambda} \, dt,$$
 +
where $K_h(x, t)$ and $K_\infty(x, t)$ are continuous functions.
  
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 $\lambda$ in the upper half-plane, $\text{Im } \lambda \ge 0$, the equation (5), considered on the half-line $0 \le x < \infty$ under the conditions $\int_0^\infty x |q(x)| \, dx < \infty$, has a solution $y(x, \lambda)$ that can be represented in the form
  
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
+
$$y(x, y) = e^{i \lambda t} + \int_x^\infty K(x, t) e^{i \lambda t} \, dt,$$
 
+
where $K(x, t)$ 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/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>
 
  
 +
$$|K(x, t)| \le \frac 12 \sigma\left(\frac{x+t}{2}\right)\exp\left\{\sigma_1(x) - \sigma_1\left(\frac{x+t}{2}\right)\right\},$$
 
in which
 
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>
+
$$\sigma(x) = \int_x^\infty |q(t)| \, dt, \qquad \sigma_1(x) = \int_x^\infty \sigma(t) \, dt.$$
 
 
 
Moreover,
 
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>
+
$$K(x, x) = \frac 12 \int_x^\infty q(t) \, dt.$$
 
 
 
====References====
 
====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>
+
<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>
  
  
Line 100: Line 106:
  
 
====References====
 
====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>
+
<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>

Latest revision as of 05:22, 23 July 2018

An ordinary differential equation of the second order

$$-\frac{d}{dx} \left\{ p(x) \frac{dy}{dx} \right\} + l(x)y = \lambda r(x) y,$$ where $x$ varies in a given finite or infinite interval $(a, b)$, $p(x)$, $l(x)$, $r(x)$ are given coefficients, $\lambda$ is a complex parameter, and $y$ is the sought solution. If $p(x), r(x)$ are positive, $p(x)$ has a first derivative and $p(x) r(x)$ has a second derivative, then by the Liouville substitution (see [1]) this equation may be reduced to the standard form

$$-y'' = q(x)y = \lambda y, \qquad a < x < b. \tag{1}$$ It is assumed that the complex function $q$ is measurable on $(a, b)$ and summable on each of the subintervals in it. At the same time one also considers the non-homogeneous equation

$$-y'' + q(x)y = \lambda y + f(x), \qquad a < x < b, \tag{2}$$ where $f$ is a given function.

If $f$ is measurable on $(a, b)$ and summable on each of the subintervals in it, then for all complex numbers $c_0, c_1$ and any interior point $x_0$, equation (2) has on $(a, b)$ one and only one solution $y(x, \lambda)$ satisfying the conditions $y(x_0, \lambda) = c_0$, $y'(x_0, \lambda) = c_1$. For any $x \in (a, b)$ the function $y(x, \lambda)$ is an entire analytic function of $\lambda$. As $x_0$ one can take one of the end-points of $(a, b)$ (if this end-point is regular, cf. Sturm–Liouville operator).

Let $y_1(x, \lambda)$ and $y_2(x, \lambda)$ be two arbitrary solutions of (1). Their Wronskian $$W(y_1, y_2) = y_1(x, \lambda) y_2'(x, \lambda) - y_1'(x, \lambda) y_2(x, \lambda)$$ is independent of $x$ and vanishes if and only if these solutions are linearly dependent. The general solution of (2) is of the form

$$y(x, \lambda) = a_1 y_2(x, \lambda) + a_2 y_2(x, \lambda) + \int_{x_0}^x R(x, \xi, \lambda) f(\xi) \, d\xi,$$ where

$$R(x, \xi, \lambda) = \frac{1}{W(y_1, y_2)} \{y_1(x, \lambda) y_2(\xi, \lambda) - y_1(\xi, \lambda)y_2(x, \lambda)\},$$ $a_1, a_2$ are arbitrary constants and $y_1(x, \lambda), y_2(x, \lambda)$ are linearly independent solutions of (1).

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

$$u''+q_1(x) u = 0, \tag{3}$$

$$v'' + q_2(x) v = 0 \tag{4}$$ be given. If $q_1(x), q_2(x)$ are real and $q_1(x) < q_2(x)$ on the entire interval $(a, b)$, 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 [1]): Let the left-hand end-point of $(a,b)$ be finite, let $u(x)$ be a solution of (3) satisfying the conditions $u(a) = \sin \alpha$, $u'(a) = \cos \alpha$, and let $v(x)$ be a solution of (4) with the same conditions; let, moreover, $q_1(x) , q_2(x)$ on the whole interval $(a, b)$. Then, if $u(x)$ has $m$ zeros on $(a, b)$, $v(x)$ will have at least $m$ zeros and the $k$-th zero of $v(x)$ will be less than the $k$-th zero of $u(x)$.

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 $y(x, \lambda)$ be the solution of

$$-y'' + q(x) y = \lambda^2 y ,\qquad -a < x < a, \quad a \le \infty , \tag{5}$$ satisfying the conditions

$$y(0, \lambda) = 1, \quad y'(0, \lambda) = i \lambda. \tag{6}$$ It turns out that this solution has the following representation:

$$y(x, \lambda) = e^{i\lambda x} = \int_{-x}^x K(x, t) e^{i\lambda t} \, dt,$$ where $K(x, t)$ is a continuous function independent of $\lambda$; moreover,

$$K(x, x) = \frac 12 \int_0^x q(t) \, dt, \qquad K(x, -x) = 0.$$ The integral operator $I + K$ defined by

$$(I + K) f = f(x) + \int_{-x}^x K(x, t) f(t) \, dt$$ is called an operator transform (a transmutation operator), and preserves the conditions at the point $x= 0$. It transforms the function $e^{i\lambda x}$ (a solution of the simplest equation $y'' = \lambda^2 x$ with the conditions (6)) into the solution of (5) under the same conditions at the point $x=0$. Let $\phi_h(x, \lambda)$ and $\phi_\infty(x, \lambda)$ be the solutions of (5) satisfying

$$\phi_h(0, \lambda) = 1, \qquad \phi_h'(0, \lambda) = h.$$

$$\phi_\infty(0, \lambda) = 0, \qquad \phi_\infty'(0, \lambda) = 1.$$ These solutions have the representations

$$\phi_h(x, \lambda) = \cos \lambda x + \int_0^x K_h(x, t) \cos \lambda t \, dt,$$

$$\phi_\infty(x, \lambda) = \frac{\sin \lambda x}{\lambda} + \int_0^x K_\infty(x, t) \frac{\sin \lambda t}{\lambda} \, dt,$$ where $K_h(x, t)$ and $K_\infty(x, t)$ are continuous functions.

A new type of operator transforms has been introduced (see [8]) that preserves the asymptotic behaviour of solutions at infinity; namely, it turned out that for all $\lambda$ in the upper half-plane, $\text{Im } \lambda \ge 0$, the equation (5), considered on the half-line $0 \le x < \infty$ under the conditions $\int_0^\infty x |q(x)| \, dx < \infty$, has a solution $y(x, \lambda)$ that can be represented in the form

$$y(x, y) = e^{i \lambda t} + \int_x^\infty K(x, t) e^{i \lambda t} \, dt,$$ where $K(x, t)$ is a continuous function satisfying the inequality

$$|K(x, t)| \le \frac 12 \sigma\left(\frac{x+t}{2}\right)\exp\left\{\sigma_1(x) - \sigma_1\left(\frac{x+t}{2}\right)\right\},$$ in which

$$\sigma(x) = \int_x^\infty |q(t)| \, dt, \qquad \sigma_1(x) = \int_x^\infty \sigma(t) \, dt.$$ Moreover,

$$K(x, x) = \frac 12 \int_x^\infty q(t) \, dt.$$

References

[1] B.M. Levitan, I.S. Sargsyan, "Introduction to spectral theory: selfadjoint ordinary differential operators" , Amer. Math. Soc. (1975) (Translated from Russian)
[2] M.A. Naimark, "Lineare Differentialoperatoren" , Akademie Verlag (1960) (Translated from Russian)
[3] B.M. Levitan, "Generalized translation operators and some of their applications" , Israel Program Sci. Transl. (1964) (Translated from Russian)
[4] V.A. Marchenko, "Sturm–Liouville operators and applications" , Birkhäuser (1986) (Translated from Russian)
[5] 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
[6] A.Ya. Povzner, "On Sturm–Liouville type differential equations on the half-line" Mat. Sb. , 23 : 1 (1948) pp. 3–52 (In Russian)
[7] 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)
[8] 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)
[9] B.M. Levitan, "Inverse Sturm–Liouville problems" , VNU (1987) (Translated from Russian)


Comments

References

[a1] R. Carroll, "Transformation theory and application" , North-Holland (1985)
[a2] B.M. Levitan, I.S. Sargsyan, "Sturm–Liouville and Dirac operators" , Kluwer (1991) (Translated from Russian)
How to Cite This Entry:
Sturm-Liouville equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sturm-Liouville_equation&oldid=16467
This article was adapted from an original article by G.Sh. GuseinovB.M. Levitan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article