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MATHEMATICS
M. Kh. ZAKHAR-ITKIN
INVESTIGATION OF A DIFFERENTIAL EQUATION OF STURM—LIOUVILLE TYPE
(Presented by Academician P. S. Novikov on 26 VI 1964)
1. An equation of Sturm—Liouville type
\[ z^{\mathrm{IV}}(x)+[\lambda-q(x)]z(x)=0,\qquad a\leq x\leq b, \tag{1} \]
is considered under the assumptions: 1) continuity of the function \(q(x)\) on \((a,b)\) and 2) existence of the finite limits \(\lim\limits_{x\to a+0} q(x)\), \(\lim\limits_{x\to b-0} q(x)\) (similarly to \((*)\)).
A solution of equation (1) satisfying certain boundary conditions is an entire function of \(\lambda\) for all \(x\in[a,b]\). To each eigenvalue \(\lambda\) there correspond four linearly independent eigenfunctions of equation (1): \(\varphi(x,\lambda)\), \(\psi(x,\lambda)\), \(\vartheta(x,\lambda)\), \(\eta(x,\lambda)\). Their Wronskian \(W[\varphi,\psi,\vartheta,\eta]\) does not depend on \(x\) and is an entire function of \(\lambda\). We shall consider the third-order minors of the Wronskian, denoting by \(A[\varphi^{(i)}]\) the minor complementary to the element \(\varphi^{(i)}(x,\lambda)\) in the Wronskian, \(i=0,1,2,3\). These minors, which are functions of \(x\) and \(\lambda\), will be differentiated with respect to \(x\).
Lemma 1.
\[ \{A[\varphi^{(i)}]\}^{(j)}=A[\varphi^{(i-j)}],\quad i=1,2,3;\quad j\leq i,\qquad \{A[\varphi]\}'=[q(x)-\lambda]A[\varphi''']. \]
Lemma 2. \(A[\varphi'''](x,\lambda)\) is an eigenfunction of equation (1), linearly independent of \(\varphi(x,\lambda)\), corresponding to the eigenvalue \(\lambda\).
Lemma 3. If the boundary conditions
\[ \varphi^{(k+2)}(x,\lambda)=-p(\lambda)\varphi^{(k)}(x,\lambda),\qquad \psi^{(k+2)}(x,\lambda)=-p(\lambda)\psi^{(k)}(x,\lambda), \]
\[ \vartheta^{(k+2)}(x,\lambda)=p(\lambda)\vartheta^{(k)}(x,\lambda),\qquad \eta^{(k+2)}(x,\lambda)=p(\lambda)\eta^{(k)}(x,\lambda) \]
are satisfied at the point \(x=a\) (or at the point \(x=b\)), \(k=0,1\), then the determinants
\[ \left| \begin{matrix} \vartheta'(x,\lambda) & \eta'(x,\lambda)\\ \vartheta''(x,\lambda) & \eta''(x,\lambda) \end{matrix} \right|, \qquad \left| \begin{matrix} \varphi'(x,\lambda) & \psi'(x,\lambda)\\ \varphi''(x,\lambda) & \psi''(x,\lambda) \end{matrix} \right| \]
do not depend on \(x\),
\[ A[\varphi''']=2 \left| \begin{matrix} \vartheta'(x,\lambda) & \eta'(x,\lambda)\\ \vartheta''(x,\lambda) & \eta''(x,\lambda) \end{matrix} \right| \psi(x,\lambda), \]
\[ A[\psi''']=2 \left| \begin{matrix} \vartheta'(x,\lambda) & \eta'(x,\lambda)\\ \vartheta''(x,\lambda) & \eta''(x,\lambda) \end{matrix} \right| \varphi(x,\lambda), \]
\[ A[\vartheta''']=2 \left| \begin{matrix} \varphi'(x,\lambda) & \psi'(x,\lambda)\\ \varphi''(x,\lambda) & \psi''(x,\lambda) \end{matrix} \right| \eta(x,\lambda), \]
\[ A[\eta''']=2 \left| \begin{matrix} \varphi'(x,\lambda) & \psi'(x,\lambda)\\ \varphi''(x,\lambda) & \psi''(x,\lambda) \end{matrix} \right| \vartheta(x,\lambda), \]
\[ W[\varphi,\psi,\vartheta,\eta] = -4 \left| \begin{matrix} \varphi'(x,\lambda) & \psi'(x,\lambda)\\ \varphi''(x,\lambda) & \psi''(x,\lambda) \end{matrix} \right| \cdot \left| \begin{matrix} \vartheta'(x,\lambda) & \eta'(x,\lambda)\\ \vartheta''(x,\lambda) & \eta''(x,\lambda) \end{matrix} \right|. \]
If the boundary values of the functions \(\varphi(x,\lambda)\), \(\vartheta(x,\lambda)\) at \(x=a\) and the boundary values of the functions \(\psi(x,\lambda)\), \(\eta(x,\lambda)\) at \(x=b\) do not depend on \(\lambda\), then the determinants
\[ \left| \begin{matrix} \varphi' & \psi'\\ \varphi'' & \psi'' \end{matrix} \right| \quad\text{and}\quad \left| \begin{matrix} \vartheta' & \eta'\\ \vartheta'' & \eta'' \end{matrix} \right| \]
have real and simple zeros.
The function
\[ u(x,\lambda)= \frac{A[\varphi''']}{W}\int_a^x \varphi(y,\lambda)f(y)\,dy + \frac{A[\psi''']}{W}\int_x^b \psi(y,\lambda)f(y)\,dy + \]
\[ + \frac{A[\vartheta''']}{W}\int_a^x \vartheta(y,\lambda)f(y)\,dy + \frac{A[\eta''']}{W}\int_x^b \eta(y,\lambda)f(y)\,dy \]
is a solution of the equation
\[ x^{\mathrm{IV}}(x)+[\lambda-q(x)]x(x)=f(x), \qquad \text{where } f(x)\in L[a,b]. \]
Under the assumptions made in Lemma 3,
\[ \begin{aligned} x(x,\lambda)=& -\frac{\psi(x,\lambda)}{2 \begin{vmatrix} \varphi'&\psi'\\ \varphi''&\psi'' \end{vmatrix}} \int_a^x \varphi(y,\lambda)f(y)\,dy -\frac{\varphi(x,\lambda)}{2 \begin{vmatrix} \varphi'&\psi'\\ \varphi''&\psi'' \end{vmatrix}} \int_x^b \psi(y,\lambda)f(y)\,dy\\ &-\frac{\eta(x,\lambda)}{2 \begin{vmatrix} \vartheta'&\eta'\\ \vartheta''&\eta'' \end{vmatrix}} \int_a^x \vartheta(y,\lambda)f(y)\,dy -\frac{\vartheta(x,\lambda)}{2 \begin{vmatrix} \vartheta'&\eta'\\ \vartheta''&\eta'' \end{vmatrix}} \int_x^b \eta(y,\lambda)f(y)\,dy . \end{aligned} \]
\[ \sum_{\lambda_k}\operatorname{Res} x(x,\lambda) \]
(\(\lambda_k\) are simple poles at which at least one of the relations
\(\varphi(x,\lambda_k)=c_k\psi(x,\lambda_k)\), \(\vartheta(x,\lambda_k)=d_k\eta(x,\lambda_k)\)
is fulfilled) represents the expansion of the function \(f(x)\) in eigenfunctions of equation (1).
- Let \(\Phi(x,\lambda)\), \(\Psi(x,\lambda)\), \(\Theta(x,\lambda)\), \(H(x,\lambda)\) be linearly independent eigenfunctions of equation (1) for \(q(x)\equiv 0\). They may be chosen so that
\(A[\Phi''']=A[\Phi'']=A'[\Phi']=0\), \(A[\Phi]=-s^6\) at some point of the interval \([a,b]\). For example:
\[ \Phi(y,\lambda)=\frac{1}{\sqrt{2}}\operatorname{ch}[s(y-x)], \]
\[ \Theta(y,\lambda)=\frac{1}{\sqrt{2}}\{\operatorname{ch}[s(y-x)]-\cos[s(y-x)]\}, \]
\[ \Psi(y,\lambda)=\frac{1}{\sqrt{2}}\operatorname{sh}[s(y-x)], \]
\[ H(y,\lambda)=\frac{1}{\sqrt{2}}\{\operatorname{sh}[s(y-x)]-\sin[s(y-x)]\},\qquad s=\sqrt[4]{\lambda}. \]
By fourfold integration by parts in
\[ \int_a^x A[\Phi''']\,\varphi^{\mathrm{IV}}(y)\,dy \]
one establishes the relation
\[ \int_a^x A[\Phi''']\,[\varphi^{\mathrm{IV}}(y)+\lambda\varphi(y)]\,dy = W[\varphi,\Psi,\Theta,H]\big|_x^a, \]
which may be rewritten in the form
\[ \int_a^x A[\Phi'''](y,\lambda)q(y)\varphi(y,\lambda)\,dy = \]
\[ = W[\varphi(y,\lambda),\Psi(y,\lambda),\Theta(y,\lambda),H(y,\lambda)]\big|_{y=a} +s^6\varphi(x,\lambda); \]
expanding the Wronskian along the first column, we obtain:
\[ \begin{aligned} \varphi(x,\lambda)=& \varphi(a,\lambda)\frac{\operatorname{ch}[s(a-x)]+\cos[s(a-x)]}{2} -\varphi'(a,\lambda)\frac{\operatorname{sh}[s(a-x)]+\sin[s(a-x)]}{2s}\\ &+\varphi''(a,\lambda)\frac{\operatorname{ch}[s(a-x)]-\cos[s(a-x)]}{2s^2} -\varphi'''(a,\lambda)\frac{\operatorname{sh}[s(a-x)]-\sin[s(a-x)]}{2s^3}\\ &+\frac{1}{2s^3}\int_a^x\{\sin[s(y-x)]-\operatorname{sh}[s(y-x)]\}q(y)\varphi(y,\lambda)\,dy . \end{aligned} \tag{2} \]
This integral equation, asymptotic in \(s\), is the main one in the subsequent considerations. Analogous formulas are easily obtained for \(\psi(x,\lambda)\), \(\vartheta(x,\lambda)\), \(\eta(x,\lambda)\).
Theorem 1. The solution of the boundary-value problem for equation (1) is an eigenfunction of the integral equation (2).
Analogous asymptotic formulas can be obtained for solutions of the boundary-value problem of the equation \(z^{(2n)}(x)+[\lambda-q(x)]z(x)=0\), \(n\) arbitrary. The larger \(n\) is, the smaller asymptotically is the magnitude of the integral term in comparison with the nonintegral one, which is the solution of the boundary-value problem for the equation \(z^{(2n)}(x)+\lambda z(x)=0\), i.e., the smaller is the influence of the function \(q(x)\).
If, in formula (2), we restrict ourselves to the first two terms (and proceed in the same way in the asymptotic expressions for \(\psi(x,\lambda)\), \(\vartheta(x,\lambda)\), \(\eta(x,\lambda)\), considering the boundary values \(\vartheta(x,\lambda)\) given at \(x=a\) and the boundary values \(\psi(x,\lambda)\), \(\eta(x,\lambda)\) given at \(x=b\)), and also require
\[ 1)\quad \left|\begin{matrix} \varphi(a) & \vartheta(a)\\ \varphi'(a) & \vartheta'(a) \end{matrix}\right| \cdot \left|\begin{matrix} \psi(b) & \eta(b)\\ \psi'(b) & \eta'(b) \end{matrix}\right| \ne 0, \]
\[ 2)\quad \left|\begin{matrix} \psi(b) & \eta(b)\\ \psi'(b) & \eta'(b) \end{matrix}\right| \cdot \left|\begin{matrix} \varphi'(a) & \vartheta'(a)\\ \varphi''(a) & \vartheta''(a) \end{matrix}\right| - \left|\begin{matrix} \varphi(a) & \vartheta(a)\\ \varphi'(a) & \vartheta'(a) \end{matrix}\right| \cdot \left|\begin{matrix} \psi'(b) & \eta'(b)\\ \psi''(b) & \eta''(b) \end{matrix}\right| =0, \]
then we obtain the asymptotic formula for the Wronskian
\[ W[\varphi(x,\lambda),\psi(x,\lambda),\vartheta(x,\lambda),\eta(x,\lambda)] = \]
\[ = \lambda \left\{ \left|\begin{matrix} \varphi(a) & \vartheta(a)\\ \varphi'(a) & \vartheta'(a) \end{matrix}\right| \cdot \left|\begin{matrix} \psi(b) & \eta(b)\\ \psi'(b) & \eta'(b) \end{matrix}\right| \operatorname{ch}[s(a-b)]\cos[s(a-b)] +O\left(\frac{e^{|s|}}{s^2}\right) \right\}. \]
Theorem 2. Under the assumptions made, the zeros of the Wronskian form two sequences
\[ s_k=\frac{k\pi}{a-b}+O(k^{-2}),\qquad t_k=\frac{k\pi i}{a-b}+O(k^{-2}),\qquad k=0,\ \pm1,\ \pm2,\ldots \]
For both sequences, \(\lambda_k=\left(\dfrac{k\pi}{a-b}\right)^4+O(k)\) are real.
Theorem 2 is analogous to the theorem on the growth of the eigenvalues of integral equations with a kernel of Green’s-function type (these eigenvalues are zeros of a certain entire function) (2).
3. Under the additional assumptions
\[ \varphi^{(k)}(a,\lambda)=-\frac{1}{s^2}\varphi^{(k+2)}(a,\lambda),\qquad \varphi^{(k)}(a,\lambda)=-\frac{1}{s^2}\varphi^{(k+2)}(a,\lambda), \]
\[ \vartheta^{(k)}(a,\lambda)=\frac{1}{s^2}\vartheta^{(k+2)}(a,\lambda),\qquad \eta^{(k)}(a,\lambda)=\frac{1}{s^2}\eta^{(k+2)}(a,\lambda),\qquad k=0,1, \]
\[ \frac{\psi'(b,\lambda)}{\psi(b,\lambda)} = \frac{\varphi'(a,\lambda)}{\varphi(a,\lambda)},\qquad \frac{\eta'(b,\lambda)}{\eta(b,\lambda)} = \frac{\vartheta'(a,\lambda)}{\vartheta(a,\lambda)} \]
we may assert the following:
\(\beta)\) For \(s=s_k,\ k=0,\ \pm1,\ \pm2,\ldots\), there is the asymptotic proportionality
\[ \eta(x,\lambda_k)=(-1)^k \frac{\eta(b,\lambda_k)}{\vartheta(a,\lambda_k)} \vartheta(x,\lambda_k) + O\left(\frac{e^{|s|}}{s^3}\right), \]
and, moreover, no asymptotic proportionality
\[ \varphi(x,\lambda_k)=c\psi(x,\lambda_k)+O(e^{|s|}/s^3) \]
is possible for any \(c\).
2) For \(s=t_k,\ k=0,\pm1,\pm2\ldots\), there is the asymptotic proportionality
\[ \psi(x,\lambda_k)=(-1)^k \frac{\psi(b,\lambda_k)}{\varphi(a,\lambda_k)} \varphi(x,\lambda_k) + O\left(\frac{e^{|s|}}{s^3}\right), \]
and, moreover, no asymptotic proportionality
\[ \vartheta(x,\lambda_k)=c\eta(x,\lambda_k)+O(e^{|s|}/s^3) \]
is possible for any \(c\).
In this case we obtain the asymptotic formulas
\[ \frac{A[\varphi''']}{W} = \frac{\cos[s(x-b)]}{2s^3\varphi(a)\sin[s(a-b)]} + O\left(\frac{e^{|s|}}{s^5}\right), \]
\[ \frac{A[\psi''']}{W} = \frac{\cos[s(a-x)]}{2s^3\psi(b)\sin[s(a-b)]} + O\left(\frac{e^{|s|}}{s^5}\right), \]
\[ \frac{A[\vartheta''']}{W} = \frac{\operatorname{ch}[s(x-b)]}{2s^3\vartheta(a)\operatorname{sh}[s(a-b)]} + O\left(\frac{e^{|s|}}{s^5}\right), \]
\[ \frac{A[\eta''']}{W} = \frac{\operatorname{ch}[s(a-x)]}{2s^3\eta(b)\operatorname{sh}[s(a-b)]} + O\left(\frac{e^{|s|}}{s^5}\right). \]
\[ \begin{aligned} \varkappa(x,\lambda)={}& \frac{1}{2s^3}\frac{\cos[s(x-b)]}{\sin[s(a-b)]} \int_a^x \cos[s(a-y)]f(y)\,dy+\\ &+\frac{1}{2s^3}\frac{\cos[s(a-x)]}{\sin[s(a-b)]} \int_x^b \cos[s(y-b)]f(y)\,dy+\\ &+\frac{1}{2s^3}\frac{\operatorname{ch}[s(x-b)]}{\operatorname{sh}[s(a-b)]} \int_a^x \operatorname{ch}[s(a-y)]f(y)\,dy+\\ &+\frac{1}{2s^3}\frac{\operatorname{ch}[s(a-x)]}{\operatorname{sh}[s(a-b)]} \int_a^b \operatorname{ch}[s(y-b)]f(y)\,dy +O\!\left(\frac{e^{|s|}}{s^5}\right), \end{aligned} \]
\[ \sum_{s_k,t_k}\operatorname{Res}\varkappa(x,\lambda) =\lim_{k\to\infty}\frac{1}{2\pi i}\int_{\Gamma_k}\varkappa(x,\lambda)\,d\lambda . \]
The contour of integration \(\Gamma_k\) is obtained under the mapping \(\lambda=s^4\) of the contour \(S_k\) of the \(s\)-plane; \(S_k\) is the boundary of a square with sides parallel to the coordinate axes, with side lengths \((2k+1)\pi\), the square being symmetric with respect to the coordinate axes.
\[ \begin{aligned} \sum_{s_k,t_k}\operatorname{Res}\varkappa(x,\lambda) ={}&\sum_{k=0}^{\infty}\frac{4}{(a-b)} \Biggl\{ \int_a^x \frac{\cos[s_k(x-b)]\cos[s_k(a-y)]} {\cos[s_k(a-b)]}\,f(y)\,dy+\\ &+\int_x^b \frac{\cos[s_k(a-x)]\cos[s_k(y-b)]} {\cos[s_k(a-b)]}\,f(y)\,dy+\\ &+\int_a^x \frac{\operatorname{ch}[t_k(x-b)]\operatorname{ch}[t_k(a-y)]} {\operatorname{ch}[t_k(a-b)]}\,f(y)\,dy+\\ &+\int_a^b \frac{\operatorname{ch}[t_k(a-x)]\operatorname{ch}[t_k(b-y)]} {\operatorname{ch}[t_k(a-b)]}\,f(y)\,dy +O\!\left(\frac{e^{|s_k|}}{s_k}\right) \Biggr\}. \end{aligned} \tag{3} \]
Theorem 3. The asymptotic formula (3) for the expansion of an arbitrary function \(f(x)\in L[a,b]\) in the eigenfunctions of equation (1), which is an analogue of the trigonometric Fourier series, is valid. If \(f(x)\) has bounded variation, then the expansion converges to
\[ \frac{1}{2}\,[f(x+0)+f(x-0)]. \]
- One may consider, analogously to what was set forth above, the equation
\[ z^{(2n)}(x)+[\lambda-q(x)]z(x)=0,\qquad a\leq x\leq b. \]
The preservation of properties of the Wronskian minors is verified; an asymptotic formula analogous to (2) is constructed for solutions of the boundary-value problem; \(\varkappa(x,\lambda)\) is constructed as the solution of the equation
\[ z^{(2n)}(x)+[\lambda-q(x)]z(x)=f(x) \]
for any function \(f(x)\in L[a,b]\). The zeros of the Wronskian \(W[\lambda]\) grow, \(\lambda_k=O(k^{2n})\), and are real.
Under additional assumptions on the boundary values of the eigenfunctions of equation (4), the expansion of an arbitrary \(f(x)\in L[a,b]\) in the eigenfunctions into a series analogous to the trigonometric Fourier series is constructed.
The author expresses gratitude to A. O. Gelfond for his guidance of the work.
Moscow State University
named after M. V. Lomonosov
Received
26 VI 1964
CITED LITERATURE
- E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, 1, Moscow, 1960.
- M. Kh. Zakhar-Itkin, DAN, 158, No. 5 (1964).