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MATHEMATICS
R. G. BARANTSEV
TWO EXPANSION THEOREMS CONNECTED WITH BOUNDARY-VALUE PROBLEMS FOR THE EQUATION \(\psi_{\sigma\sigma}-K(\sigma)\psi_{\theta\theta}=0\)
(Presented by Academician V. I. Smirnov, 3 VI 1957)
- Let, in the strip \(S\{\sigma_0\le \sigma\le \sigma_1,\ -\infty<\theta<+\infty\}\), \(K(\sigma)\ge \varepsilon>0\), and let \(\theta=s(\sigma)\) be a certain curve \(s\) with endpoints at the points \((\sigma_0,\theta_0)\), \((\sigma_1,\theta_1)\). Consider the problem
\[ \left. \begin{gathered} \psi_{\sigma\sigma}-K(\sigma)\psi_{\theta\theta}=0;\qquad \psi|_s=\bar\psi(\sigma);\\ \psi_\theta|_s \begin{cases} =\bar\psi_1(\sigma), & \text{if } s \text{ is oriented along the } \sigma\text{-axis},\\ \text{is not prescribed}, & \text{if } s \text{ is a characteristic}; \end{cases}\\ \psi(\sigma_0,\theta)a_0+\psi_\sigma(\sigma_0,\theta)b_0=0,\qquad \psi(\sigma_1,\theta)a_1+\psi_\sigma(\sigma_1,\theta)b_1=0, \end{gathered} \right\} \tag{\(C_0\)} \]
where \(\bar\psi(\sigma)\), \(\bar\psi_1(\sigma)\) are given functions; \(a_0,b_0,a_1,b_1\) are constants such that \(a_0^2+b_0^2\ne 0\), \(a_1^2+b_1^2\ne 0\).
By replacing \((\sigma,\theta)\) by \((x,t)\) and \(\psi\) by \(v\) according to the formulas
\[ cx=\int_{\sigma_0}^{\sigma}\sqrt{K}\,d\sigma,\qquad ct=\theta-\theta_0,\qquad c=\int_{\sigma_0}^{\sigma_1}\sqrt{K}\,d\sigma,\qquad v(x,t)=\psi(\sigma,\theta)K^{1/4}(\sigma), \]
problem \((C_0)\) is reduced to the form \((^1)\)
\[ \left. \begin{gathered} v_{xx}-v_{tt}+N(x)v=0;\qquad v|_{t=l(x)}=p(x);\\ v_t|_{t=l(x)} \begin{cases} =q(x), & \text{if } |l'(x)|<1,\\ \text{is not prescribed}, & \text{if } l(x)\equiv \pm x; \end{cases}\\ v(0,t)\cos\alpha+v_x(0,t)\sin\alpha=0,\qquad 0<\alpha\le \pi;\\ v(1,t)\cos\beta+v_x(1,t)\sin\beta=0,\qquad 0<\beta\le \pi, \end{gathered} \right\} \tag{C} \]
where
\[ N(x)=-K^{-1/4}(\sigma)\frac{d^2K^{1/4}(\sigma)}{dx^2}. \]
- Let \(s_n\) and \(B_n(x)\) be the eigenvalues and normalized eigenfunctions of the following Sturm–Liouville problem:
\[ B_n''+\bigl[s_n+N(x)\bigr]B_n=0, \tag{1} \]
\[ B_n(0)\cos\alpha+B_n'(0)\sin\alpha=0,\qquad B_n(1)\cos\beta+B_n'(1)\sin\beta=0. \tag{2} \]
Put \(\lambda_{\pm n}=\pm\sqrt{s_n}\). We assume that zero is not an eigenvalue.
If the solution of problem \((C)\) is sought in the form
\[ v=\sum_{n=-\infty}^{\infty}{}' c_nB_n(x)\exp(-i\lambda_n t), \tag{3} \]
then the satisfaction of the initial conditions on \(l\) leads to the expansion of \(p(x)\) and \(q(x)\) in the series
\[ p(x)\approx \sum_{n=-\infty}^{\infty}{}' c_n \bar z_n(x), \tag{4} \]
\[ q(x)\approx \sum_{n=-\infty}^{\infty}{}' c_n(-i\lambda_n)\bar z_n(x), \tag{5} \]
where
\[ \bar z_n(x)=B_n(x)\exp[-i\lambda_n l(x)]. \tag{6} \]
Substituting \(B_n(x)\) from (6) into (1) and (2), we obtain for \(\bar z_n(x)\) the equation
\[ \bar z_n''+2i\lambda_n l'\bar z_n' +\bar z_n\{N+i\lambda_n l''+\lambda_n^2(1-l'^2)\}=0 \tag{7} \]
and the boundary conditions
\[ \begin{aligned} &\bar z_n(0)[\cos\alpha+i\lambda_n l'(0)\sin\alpha]+\bar z_n'(0)\sin\alpha=0,\\ &\bar z_n(1)[\cos\beta+i\lambda_n l'(1)\sin\beta]+\bar z_n'(1)\sin\beta=0. \end{aligned} \tag{8} \]
Thus, in order to determine the coefficients \(c_n\) of the series (3), it is necessary to expand \(p(x)\) and \(q(x)\) in the form (4), (5) in terms of the eigenfunctions of the non-self-adjoint system (7), (8).
- Writing the equation and boundary conditions for the function
\(z_n(x)=B_n(x)\exp[i\lambda_n l(x)]\), analogous to (7), (8), it is easy to obtain from (3) the following orthogonality relation:
\[ I_{m,n}\equiv \int_0^1 \{l'(z_n'\bar z_m-\bar z_n z_m') +i(1-l'^2)(\lambda_n+\bar\lambda_m)z_n\bar z_m\}\,dx=0, \qquad m\ne n. \tag{9} \]
With the help of (9), also using the fact that \(I_{n,n}=2i\lambda_n\), the coefficients \(c_n\) of the series (3) can formally be determined in the form
\[ c_n=\frac{i}{2\lambda_n}\int_0^1 \{-p\,l''z_n+z_n[p'l'+(1-l'^2)(q-i\lambda_n p)]\}\,dx, \qquad |l'(x)|<1; \tag{10} \]
\[ c_n=\frac{i}{2\lambda_n}[pz_n']_0^1-\frac{i}{\lambda_n}\int_0^1 pz_n'\,dx, \qquad l(x)\equiv x. \tag{11} \]
- Let us denote the partial sums of the series (4), (5) with coefficients (10), respectively, by \(S_n^{(p)}(x)\) and \(S_n^{(q)}(x)\), and the partial sum of the series (4) with coefficients (11) by \(\Sigma_n^{(p)}(x)\).
Theorem 1. If on \([0,1]\) \(|l'(x)|<1\) and the functions \(N(x)\), \(l''(x)\), \(p'(x)\), \(q(x)\) have bounded variation, then as \(n\to\infty\):
in case A \((0<\alpha<\pi,\ 0<\beta<\pi)\)
\[ S_n^{(p)}(x)\to p(x), \qquad 0\le x\le 1; \]
\[ S_n^{(q)}(x)\to \frac{q(x-0)+q(x+0)}{2}, \qquad 0<x<1; \]
\[ S_n^{(q)}(0)\to q(0+), \qquad S_n^{(q)}(1)\to q(1-); \]
in case B \((\alpha=\beta=\pi)\)
\[ S_n^{(p)}(x)\to p(x), \qquad 0<x<1; \]
\[ S_n^{(q)}(x)\to \frac{q(x-0)+q(x+0)}{2}, \qquad \text{if } p(0)l(x)=p(1)[l(1)-l(x)]=0, \]
\[ 0<x<1; \]
\[ S_n^{(p)}(0)=S_n^{(p)}(1)=S_n^{(q)}(0)=S_n^{(q)}(1)=0; \]
in case C \((\alpha=\pi,\ 0<\beta<\pi)\)
\[ S_n^{(p)}(x)\to p(x), \qquad 0<x\leqslant 1; \]
\[ S_n^{(q)}(x)\to \frac{q(x-0)+q(x+0)}{2}, \qquad \text{if } p(0)l(x)=0,\quad 0<x<1; \]
\[ S_n^{(q)}(1)\to q(1-), \qquad \text{if } p(0)l(1)=0; \]
\[ S_n^{(p)}(0)=S_n^{(q)}(0)=0; \]
in case D \((0<\alpha<\pi,\ \beta=\pi)\)
\[ S_n^{(p)}(x)\to p(x), \qquad 0\leqslant x<1; \]
\[ S_n^{(q)}(x)\to \frac{q(x-0)+q(x+0)}{2}, \qquad \text{if } p(1)[l(1)-l(x)]=0,\quad 0<x<1; \]
\[ S_n^{(q)}(0)\to q(0+), \qquad \text{if } p(1)l(1)=0; \]
\[ S_n^{(p)}(1)=S_n^{(q)}(1)=0. \]
This theorem generalizes Langer’s results \((^2)\). In case B, for \(p(0)=p(1)=0\), it was proved in \((^3)\).
Theorem 2. If \(l(x)\equiv x\) and the functions \(N(x)\), \(p(x)\) have bounded variation on \([0,1]\), then as \(n\to\infty\):
in case A \((0<\alpha<\pi,\ 0<\beta<\pi)\)
\[ \Sigma_n^{(p)}(x)\to \frac{p(x-0)+p(x+0)}{2}, \qquad 0<x<1; \]
\[ \Sigma_n^{(p)}(0)\to p(0+), \qquad \Sigma_n^{(p)}(1)\to p(1-); \]
in case B \((\alpha=\beta=\pi)\)
\[ \Sigma_n^{(p)}(x)\to \frac{p(x-0)+p(x+0)-p(0+)-p(1-)}{2}, \qquad 0<x<1; \]
\[ \Sigma_n^{(p)}(0)=\Sigma_n^{(p)}(1)=0; \]
in case C \((\alpha=\pi,\ 0<\beta<\pi)\)
\[ \Sigma_n^{(p)}(x)\to \frac{p(x-0)+p(x+0)-p(0+)}{2}, \qquad 0<x<1; \]
\[ \Sigma_n^{(p)}(0)=0, \qquad \Sigma_n^{(p)}(1)\to p(1-)-p(0+); \]
in case D \((0<\alpha<\pi,\ \beta=\pi)\)
\[ \Sigma_n^{(p)}(x)\to \frac{p(x-0)+p(x+0)-p(1-)}{2}, \qquad 0<x<1; \]
\[ \Sigma_n^{(p)}(0)\to p(0+)-p(1-), \qquad \Sigma_n^{(p)}(1)=0. \]
For \(\alpha=\beta=\pi\) (case B) we have an intersection with the results of Mishoe \((^4)\). The unusual fact noted in \((^4)\), that the sum of the expansion of \(p(x)\) at any point \(x\in[0,1]\) depends, generally speaking, on \(p(0+)\) and \(p(1-)\), occurs, as is seen from Theorem 2, only in those cases when \(\alpha\) or \(\beta\) is equal to \(\pi\).
- The proof of the theorems is carried out, as in \((^3)\), by the method of contour integration in the complex \(\lambda\)-plane over an infinitely large circle. With the aid of a modification of the generalized Fourier transform \((^3)\), one can show that
\[ S_n^{(p)}(x)=\frac{1}{2\pi}\oint_{C_n}\Phi(x,\lambda)\,d\lambda, \qquad S_n^{(q)}(x)=-\frac{i}{2\pi}\oint_{C_n}\lambda\Phi(x,\lambda)\,d\lambda, \]
where
\[ \Phi(x,\lambda)= \]
\[ = e^{-i\lambda l(x)} \left\{ \frac{\chi(x,\lambda)}{\omega(\lambda)} \int_{0}^{x} e^{i\lambda l(y)} \left[ \varphi(y,\lambda)\bigl(p'l' + q(1-l'^2)-i\lambda p\bigr) -\varphi'(y,\lambda)pl' \right]\,dy +\right. \]
\[ \left. +\frac{\varphi(x,\lambda)}{\omega(\lambda)} \int_{x}^{1} e^{i\lambda l(y)} \left[ \chi(y,\lambda)\bigl(p'l' + q(1-l'^2)-i\lambda p\bigr) -\chi'(y,\lambda)pl' \right]\,dy \right\}; \]
\[ \Sigma_n^{(p)}(x)=-\frac{1}{\pi}\oint_{C_n}\Psi(x,\lambda)\,d\lambda; \]
\[ \Psi(x,\lambda)=e^{-i\lambda x} \left\{ \frac{p(0+)}{2}\frac{\chi(x,\lambda)}{\omega(\lambda)}\sin\alpha -\frac{p(1-)}{2}\frac{\varphi(x,\lambda)}{\omega(\lambda)}e^{i\lambda}\sin\beta +\right. \]
\[ \left. +\frac{\chi(x,\lambda)}{\omega(\lambda)} \int_{0}^{x} e^{i\lambda y}p(\varphi'+i\lambda\varphi)\,dy +\frac{\varphi(x,\lambda)}{\omega(\lambda)} \int_{x}^{1} e^{i\lambda y}p(\chi'+i\lambda\chi)\,dy \right\}. \]
Here \(\varphi(x,\lambda)\), \(\chi(x,\lambda)\) are solutions of equation (1), where \(s=\lambda^2\), such that
\[ \varphi(0,\lambda)=\sin\alpha,\qquad \varphi'(0,\lambda)=-\cos\alpha; \]
\[ \chi(1,\lambda)=\sin\beta,\qquad \chi'(1,\lambda)=-\cos\beta; \]
\[ \omega(\lambda)=\varphi\chi'_x-\chi\varphi'_x. \]
\(C_n\) is a circle with center at the origin and radius \(R_n\), satisfying the inequality \(\lambda_n+\varepsilon\le R_n\le \lambda_{n+1}-\varepsilon\), \(\varepsilon>0\).
In passing to the limit \((n\to\infty)\), essential use is made of the asymptotic formulas for the functions \(\varphi,\chi,\varphi',\chi',\omega\), obtained by the method indicated in (5) (§1.7).
Leningrad State University
named after A. A. Zhdanov
Received
31 V 1957
CITED LITERATURE
- R. G. Barantsev, DAN, 114, No. 5 (1957).
- R. E. Langer, Trans. Am. Math. Soc., 31, 868 (1929).
- R. G. Barantsev, Vestn. LGU, 13, No. 1 (1958).
- L. J. Mishoe, On the Expansion of an Arbitrary Function in Terms of the Eigenfunctions of a Nonselfadjoint Differential System, Thesis, New York University, 1953.
- E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, 1946.