Sturm-Liouville equation

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)

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)