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Reports of the Academy of Sciences of the USSR
1966. Volume 170, No. 6
UDC 517.941.9
MATHEMATICS
A. O. KRAVITSKII
ON EXPANSION IN A SERIES IN EIGENFUNCTIONS OF A NON-SELF-ADJOINT BOUNDARY-VALUE PROBLEM
(Presented by Academician A. A. Dorodnitsyn on 20 I 1966)
\(1^\circ\). On the interval \([0,a]\) the boundary-value problem
\[ Ly=-y''+q(x)y=s^2y, \tag{1′} \]
\[ y(0)=0,\qquad y'(a)+isy(a)=0, \tag{1″} \]
is considered, where \(q(x)\) is a complex-valued function, and \(s\) is the spectral parameter. Problem (1) arises in various questions of mathematical physics. Redheffer, who studied it in connection with scattering theory, showed that if \(q(x)\) in a left half-neighborhood of the point \(a\) satisfies the condition
\[ q(x)\sim c_\mu(a-x)^\mu,\quad x\to a-0;\qquad \mu\geqslant 0,\quad c_\mu\ne 0, \tag{2} \]
then the problem has a discrete spectrum \(s_n\), and the system of eigenfunctions
\[ \varphi_n(x),\qquad n=\pm1,\ \pm2,\ldots, \tag{3} \]
is complete in \(L_2(0,a)\)*.
Let us point out a remarkable feature of problem (1): for \(q(x)\equiv0\) the spectrum is absent altogether.
From the completeness, established by Redheffer, of the system (3), generally speaking, the convergence of an expansion in a series with respect to this system does not follow. The present article is devoted to the study of this question.
Note that the expansion
\[ f(x)=\sum_{n=-\infty}^{\infty}c_n\varphi_n(x),\quad x\in[0,a), \]
whose convergence is established below, is the expansion of the integral
\[ f(x)=\int_0^\infty F(\lambda)\varphi(x,\lambda)\,d\rho(\lambda),\qquad 0\leqslant x<\infty, \]
generated by the operator \(Ly=-y''+q(x)y\) in \(L_2(0,+\infty)\) with finite potential \(q(x)\), into a series by residues and, consequently, makes it possible to replace the Fourier integral by a series in the system of solutions (3), which in a number of cases turns out to be very convenient.
\(2^\circ\). Let us formulate the results obtained by us.
Theorem 1. Let in (2) \(\mu\) be equal to \(0\) or \(1\), and let \(f(x)\) be twice differentiable on \([0,a]\), \(f(0)=0\), and for \(\mu=1\) also \(f'(a)=0\). Then \(f(x)\) is expanded into the series
\[ f(x)=\sum_{n=-\infty}^{\infty}c_n\varphi_n(x)= \]
\[ =\frac12\sum_{n=-\infty}^{\infty}\varphi_n(x) \left[ \int_0^a f(t)\varphi_n(t)\,dt -i f(a)\frac{\varphi_n(a)}{s_n} \right] \bigg/ \left[ \int_0^a \varphi_n^2(t)\,dt -i\frac{\varphi_n^2(a)}{2s_n} \right], \tag{4} \]
\[ \text{* In } (^{1}) \text{ it is proved, moreover, that if the functions (3) are continued smoothly to the interval } [a,2a] \text{ by the formula } C_ne^{-is_nx}, \text{ then the system thus arising is complete in } L_2(0,2a). \]
converging everywhere on the half-open interval \([0,a)\) and uniformly inside this interval. Moreover,
\[ \left| f(x)-\sum_{n=-N_1}^{N_2} c_n\varphi_n(x)\right|\leq \mathrm{const}\cdot N^{-\delta/b} \]
uniformly in \(x\in[0,a-\delta]\), where \(b=2a/\mu+2\), and \(N=\min(N_1,N_2)\).*
For the proof we use the method of contour integration (see \((^3)\)). Denote by \(y_1(x,s)\) and \(y_2(x,s)\) the solutions of equation \((1')\) satisfying the conditions
\[ y_1(0,s)=0,\qquad y_1'(0,s)=1;\qquad y_2(a,s)=1,\qquad y_2'(a,s)=-is, \]
and by \(\omega(s)=W(y_1,y_2)\) the Wronskian determinant of these solutions:
\[ \omega(s)=-y_1'(a,s)-isy_1(as)=-y_2(0,s). \]
It can be shown that, as \(s\to\infty\), \(s=\sigma+i\tau\), the following formulas hold:
\[ y_1(x,s)=\frac{\sin sx}{s}+O\left(\frac{e^{|\tau|x}}{s^2}\right), \tag{5} \]
\[ y_2(x,s)=e^{is(a-x)} \left[1+O\left(\frac{1}{s}\right)\right] +\frac{c_\mu\Gamma(\mu+1)}{(-2is)^{\mu+2}}e^{-is(a-x)} \left[1+O\left(\frac{1}{s}\right)\right], \tag{6} \]
\[ \omega(s)=-y_2(0,s)=-e^{isa} \left[1+O\left(\frac{1}{s}\right)\right] \left[1+O\left(\frac{1}{s}\right)-\frac{e^{-2isa}}{As^{\mu+2}}\right], \tag{7} \]
where \(A=-(-2i)^{\mu+2}/c_\mu\Gamma(\mu+1)\), so that the eigenvalues \(s_n\) asymptotically satisfy the equation
\[ e^{-2isa}=As^{\mu+2}\left[1+O(1/s)\right]. \]
Solving it, we obtain for \(s_n=\sigma_n+i\tau_n\)
\[ \sigma_n=\frac{n\pi}{a}-\frac{1}{2a}\arg A+O\left(\frac{\ln n}{n}\right), \]
\[ \tau_n=\frac{\mu+2}{2a}\ln |n|+\frac{1}{2a}\ln\left[\left(\frac{\pi}{a}\right)^{\mu+2}|A|\right]+O\left(\frac{1}{n}\right). \tag{8} \]
From these formulas it is evident, in particular, that, beginning with some \(n\), all eigenvalues of problem (1) are simple.
The principal asymptotic terms of formulas (8) were obtained by a different method by Reddhe \((^2)\). The method we use makes it possible to obtain for \(s_n\) any number of terms of the asymptotic expansion.
After this, consider, as usual, the nonhomogeneous equation \(Ly-s^2y=f(x)\). Its solution \(\Phi(x,s)\), satisfying the conditions \((1'')\), has the form
\[ \Phi(x,s)=\int_0^a G(x,t;s)f(t)\,dt= \]
\[ =\frac{y_2(x,s)}{-\omega(s)}\int_0^x y_1(t,s)f(t)\,dt +\frac{y_1(x,s)}{-\omega(s)}\int_x^a y_2(t,s)f(t)\,dt, \]
where \(G(x,t;s)\) is the Green’s function of the problem under consideration. Using smooth-
* For simplicity of notation we assume that the eigenvalues are simple. Below we shall show that problem (1) can have only a finite number of multiple eigenvalues.
ness of \(f(x)\) and the boundary conditions, one can prove the identity
\[ -\,s\Phi(x,s)-i\,\frac{y_1(x,s)}{\omega(s)}\,f(a) = \frac{f(x)}{s} + \frac{y_1(x,s)}{s\omega(s)}\,f'(a) - \frac{1}{s}G_sLf, \tag{9} \]
where \(G_s\) is the integral operator with kernel \(G(x,t;s)\).
As the contour of integration we choose the rectangle \(\gamma_N\) with vertices \((\pm R_N,T_N)\), \((\pm R_N,-T_N)\), where
\[ R_N=\frac{\pi}{a}\left(N+\frac12\right)-\frac{1}{2a}\arg A, \]
and \(T_N\) is such that \(T_N/\ln N\to\infty\), \(T_N/R_N\to0\) as \(N\to\infty\).
Using the asymptotic formulas (5)—(7), one can show that, under the conditions of the theorem, the integral over \(\gamma_N\) of the second and third terms on the right-hand side of (9) is \(O(N^{-\delta/b})\) on \([0,a-\delta]\). Therefore, by virtue of (9),
\[ \frac{1}{2\pi i}\oint_{\gamma_N} \left[ -s\Phi(x,s)-i\,\frac{y_1(x,s)}{\omega(s)}\,f(a) \right]ds \to f(x). \]
On the other hand, taking this integral by residues and using the relation
\[ \omega'(s_n)= \frac{2s_n}{\varphi_n(a)} \left( \int_0^a \varphi_n^2(t)\,dt - i\,\frac{\varphi_n^2(a)}{2s_n} \right), \]
we obtain the partial sum of the series (4). The theorem is proved.
3°. By the same method, but somewhat more complicatedly, one investigates the case of an arbitrary nonnegative integer \(\mu\) in formula (2). In this case stronger restrictions have to be imposed on the function being expanded. Denote by \(D_\mu\) (\(\mu\ge0\) an integer) the class of functions \(f(x)\) satisfying the following conditions:
a) the derivative \(f^{(\mu+2-k)}(x)\) exists and has bounded variation on the interval \([x_k,a]\), where \(x_k=\max[0,a-kb]\), \(b=2a/\mu+2\), \(1\le k\le 2+[\mu/2]\)*;
b) \(L^k f(0)=0\), \(0\le k\le[\mu/4]\), \(f^{(k)}(a)=0\), \(1\le k\le[\mu]\).
Then the following is valid.
Theorem 2. For \(f\in D_\mu\), the series (4) converges to \(f(x)\) uniformly on \([0,a-\delta]\), and
\[ \left| f(x)-\sum_{n=-N_1}^{N_2} c_n\varphi_n(x) \right| \le \mathrm{const}\cdot N^{-\delta/b}, \qquad N=\min(N_1,N_2). \tag{10} \]
In connection with this theorem we make several remarks.
1) By somewhat strengthening the smoothness requirements on \(f(x)\), one can obtain uniform convergence of the series on the whole interval \([0,a]\).
2) The smoothness conditions imposed in the theorem on the function \(f(x)\) are essential. One can construct a function \(f(x)\) satisfying all smoothness requirements except one (for example, \(f^{(\mu)}(x)\) has a discontinuity at a point \(a-b<x_0<a\)) and for which the series (4) diverges on the whole interval \([0,a]\) or on some part of it.
3) It follows from Theorem 2 that when \(\mu=0\), in the condition of Theorem 1 it is sufficient to require one-time differentiability of \(f(x)\).
* Let us explain that the smoothness requirements on \(f(x)\) imposed by this condition weaken as one moves away from the right endpoint of the interval \([0,a]\); thus, on the interval \([a-b,a]\), \(f(x)\) has \(\mu+1\) derivatives, on the interval \([a-2b,a-b]\) it has \(\mu\) derivatives, and so on.
4°. Let us note that the system of functions (3) is linearly dependent on \([0,a)\). Namely, as can be shown, for any function \(f \in D_{\mu+2m-1}\), \(m=0,1,\ldots\), the relations
\[ \sum_{n=-\infty}^{\infty} s_n^{2j-1} c_n \varphi_n(x) \equiv 0,\qquad x \in [0,a),\qquad 0 \leqslant j \leqslant m, \tag{11} \]
hold, where \(c_n\) are determined from formula (4). (In addition, for \(j=0\) it is necessary to require that \(f(a)=0\).)
Equality (11) makes it possible to establish the following fact:
Theorem 3. The system of functions (3) is twice complete in \(L_2(0,a)\), and for any pair of functions \(\{f_0(x), f_1(x)\}\), \(f_0 \in D_{\mu+1}\), \(f_1 \in D_\mu\), \(f_1(a)=0\), the expansion
\[ f_i(x)=\sum_{n=-\infty}^{\infty} a_n s_n^i \varphi_n(x),\qquad i=0,1, \tag{12} \]
(see (13)), converging uniformly inside the interval \([0,a)\), is valid.
Indeed, by Theorem 1,
\[ f_0(x)=\sum_{n=-\infty}^{\infty} c_n^{(0)}\varphi_n(x),\qquad f_1(x)=\sum_{n=-\infty}^{\infty} c_n^{(1)}\varphi_n(x), \]
and according to (11)
\[ \sum_{n=-\infty}^{\infty} c_n^{(0)} s_n \varphi_n(x)\equiv 0,\qquad \sum_{n=-\infty}^{\infty} c_n^{(1)}\frac{\varphi_n(x)}{s_n}\equiv 0. \]
Therefore
\[ f_0(x)=\sum_{n=-\infty}^{\infty}\left(c_n^{(0)}+\frac{c_n^{(1)}}{s_n}\right)\varphi_n(x),\qquad f_1(x)=\sum_{n=-\infty}^{\infty}\left(c_n^{(0)}+\frac{c_n^{(1)}}{s_n}\right)s_n\varphi_n(x), \]
and the theorem is proved for
\[ a_n=c_n^{(0)}+c_n^{(1)}/s_n. \tag{13} \]
Unfortunately, in the general case we have not been able to prove the uniqueness of the expansion (12); however, if the expansion (12) converges uniformly on the closed interval \([0,a]\), then it is unique.
The author expresses deep gratitude to Prof. V. B. Lidskii for proposing the problem and for his constant attention to the work.
Moscow Institute of Physics and Technology
Received
17 I 1966
REFERENCES
- T. Regge, Nuovo Cimento, 9, No. 3 (1958); Sborn. per. Matematika, 7, 4 (1963).
- T. Regge, Nuovo Cimento, 8, No. 5 (1958); Sborn. per. Matematika, 7, 4 (1963).
- É. Ch. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I, Moscow, 1960.