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UDC 517.9
MATHEMATICS
I. S. SARGSYAN
EXPANSION IN EIGENFUNCTIONS OF A ONE-DIMENSIONAL DIRAC SYSTEM
(Presented by Academician I. N. Vekua on 27 V 1965)
1. Let \(\mathscr L\) denote the matrix operator
\[ \mathscr L \equiv \begin{pmatrix} p(x) & d/dx\\ -d/dx & r(x) \end{pmatrix}, \]
where \(p(x)\) and \(r(x)\) are real-valued functions defined on the half-line \((0,\infty)\) and summable on every finite interval. Further, let \(y(x)\) denote a two-component vector-function
\[ y(x)= \begin{pmatrix} y_1(x)\\ y_2(x) \end{pmatrix}. \]
Then the equation (\(\lambda\) is a parameter)
\[ (\mathscr L-\lambda)y=0 \]
is equivalent to the system of two compatible ordinary differential equations of the first order
\[ dy_2/dx+p(x)y_1=\lambda y_1, \tag{1} \]
\[ -dy_1/dx+r(x)y_2=\lambda y_2. \tag{2} \]
The system (1)—(2) is a one-dimensional analogue of the stationary system in the relativistic quantum theory of Dirac \((^1)\)
\[ (W-V+mc^2)X_1-ic\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)X_4-ic\frac{\partial X_3}{\partial z}=0, \]
\[ (W-V+mc^2)X_2-ic\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)X_3+ic\frac{\partial X_4}{\partial z}=0, \]
\[ (W-V-mc^2)X_3-ic\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)X_2-ic\frac{\partial X_1}{\partial z}=0, \tag{3} \]
\[ (W-V-mc^2)X_4-ic\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)X_1+ic\frac{\partial X_2}{\partial z}=0, \]
where \(V=V(x,y,z)\) is the potential function, \(c\) is the speed of light, \(W\) is the energy, and \(m\) is the mass of the particle.
Indeed, if \(X_1(x,y,z)=X_4(x,y,z)\equiv 0\), and the functions \(V(x,y,z)\), \(X_2(x,y,z)\), and \(X_3(x,y,z)\) do not depend on \(x\) and \(z\), then system (3) takes the form
\[ \{W-V(y)+mc^2\}X_2+c\frac{dX_3}{dx}=0, \tag{4} \]
\[ \{W-V(y)-mc^2\}X_3-c\frac{dX_2}{dx}=0. \]
Let the units of measurement be chosen so that the speed of light \(c=1\). If we put
\(p(y)=V(y)+m\), \(r(y)=V(y)-m\), \(\lambda=W\), then systems (1)—(2) and (4) will differ only in notation.
We adjoin to system (1)—(2) the boundary condition
\[ y_2(0)-hy_1(0)=0, \tag{5} \]
where \(h\) is an arbitrary real number.
Consider the eigenvalue problem (1) + (2) + (5).
In the present note, by a method analogous to the method of B. M. Levitan (3), we study the question of expanding an arbitrary vector-function \(f(x)=\{f_1(x), f_2(x)\}\) with square integrable norm on the half-line \((0,\infty)\),
\[ \int_0^\infty \{f_1^2(x)+f_2^2(x)\}\,dx<+\infty, \]
in the eigenfunctions of the problem (1) + (2) + (5).
- Denote by \(\varphi(x,\lambda)=\{\varphi_1(x,\lambda),\varphi_2(x,\lambda)\}\) the solution of the system (1)—(2) satisfying the initial conditions
\[ \varphi_1(0,\lambda)=1,\qquad \varphi_2(0,\lambda)=h. \tag{5'} \]
It is obvious that the vector-function \(\varphi(x,\lambda)\) satisfies condition (5).
It is known (2) that, for a given \(h\), to each boundary-value problem (1) + (2) + (5) there corresponds a unique nondecreasing, bounded on every finite interval, left-continuous function \(\rho(\lambda)\) \((-\infty<\lambda<\infty)\), generating an isometric mapping of the space of vector-functions \(f(x)\subset L^2(0,\infty)\) onto the space \(L^2_{\{\rho(\lambda)\}}(-\infty,\infty)\) by the formulas
\[ F(\lambda)=\int_0^\infty \{f_1(x)\varphi_1(x,\lambda)+f_2(x)\varphi_2(x,\lambda)\}\,dx, \tag{6} \]
\[ f_1(x)=\int_{-\infty}^{\infty} F(\lambda)\varphi_1(x,\lambda)\,d\rho(\lambda), \tag{7} \]
\[ f_2(x)=\int_{-\infty}^{\infty} F(\lambda)\varphi_2(x,\lambda)\,d\rho(\lambda), \tag{8} \]
where the integrals (6) and (7)—(8) converge respectively in the metrics of the spaces \(L^2_{\{\rho(\lambda)\}}(-\infty,\infty)\) and \(L^2(0,\infty)\), and Parseval’s equality holds:
\[ \int_0^\infty \{f_1^2(x)+f_2^2(x)\}\,dx = \int_{-\infty}^{\infty} F^2(\lambda)\,d\rho(\lambda). \]
Put \((i,k=1,2)\)
\[ \Theta_{ik}(x,s;\lambda)= \begin{cases} \displaystyle \int_0^\lambda \varphi_i(x,\lambda)\varphi_k(s,\lambda)\,d\rho(\lambda), & \lambda>0,\\[1.2em] \displaystyle -\int_\lambda^0 \varphi_i(x,\lambda)\varphi_k(s,\lambda)\,d\rho(\lambda), & \lambda<0,\\[1.2em] 0, & \lambda=0. \end{cases} \]
The square matrix of second order \(\Theta(x,s;\lambda)=\{\Theta_{ik}(x,s;\lambda)\}\) is called the spectral matrix of the boundary-value problem (1) + (2) + (5).
Let the vector-function \(f(x)=\{f_1(x),f_2(x)\}\subset L^2(0,\infty)\), i.e.
\[ \int_0^\infty \{f_1^2(x)+f_2^2(x)\}\,dx<+\infty. \]
Introduce the notation
\[ S_1(x,\lambda)=\int_0^\infty \{f_1(s)\Theta_{11}(x,s;\lambda)+f_2(s)\Theta_{12}(x,s;\lambda)\}\,ds, \tag{9} \]
\[ S_2(x,\lambda)=\int_0^\infty \{f_1(s)\Theta_{21}(x,s;\lambda)+f_2(s)\Theta_{22}(x,s;\lambda)\}\,ds. \tag{10} \]
By virtue of the definition of the functions \(\Theta_{ik}(x,s;\lambda)\) and equality (6), for the functions \(S_1(x,\lambda)\) and \(S_2(x,\lambda)\) we obtain the expressions
\[ S_1(x,\lambda)=\int_0^\lambda F(\lambda)\varphi_1(x,\lambda)\,d\rho(\lambda), \]
\[ S_2(x,\lambda)=\int_0^\lambda F(\lambda)\varphi_2(x,\lambda)\,d\rho(\lambda). \]
The functions \(S_1(x,\lambda)\) and \(S_2(x,\lambda)\) are segments of the expansions, respectively, of the functions \(f_1(x)\) and \(f_2(x)\) in the Fourier integral with respect to the eigenfunctions of problem (1) + (2) + (5).
Consider problem (1) + (2) + (5′) for \(p(x)=r(x)=0\) and \(h=0\). Then \(\varphi_1(x,\lambda)=\cos\lambda x\), \(\varphi_2(x,\lambda)=\sin\lambda x\). Therefore, in the case under consideration the spectral matrix, which we denote by \(\Theta^*(x,s;\lambda)\), is determined by the formulas
\[ \Theta^*_{11}(x,s;\lambda)=\frac{1}{\pi}\int_0^\lambda \cos\lambda x\cos\lambda s\,d\lambda, \]
\[ \Theta^*_{12}(x,s;\lambda)=\frac{1}{\pi}\int_0^\lambda \cos\lambda x\sin\lambda s\,d\lambda, \]
\[ \Theta^*_{21}(x,s;\lambda)=\frac{1}{\pi}\int_0^\lambda \sin\lambda x\cos\lambda s\,d\lambda, \]
\[ \Theta^*_{22}(x,s;\lambda)=\frac{1}{\pi}\int_0^\lambda \sin\lambda x\sin\lambda s\,d\lambda. \]
Consequently, the functions (analogously to the preceding)
\[ S_1^*(x,\lambda)=\int_0^\infty \{f_1(s)\Theta^*_{11}(x,s;\lambda)+f_2(s)\Theta^*_{12}(x,s;\lambda)\}\,ds, \]
\[ S_2^*(x,\lambda)=\int_0^\infty \{f_1(s)\Theta^*_{21}(x,s;\lambda)+f_2(s)\Theta^*_{22}(x,s;\lambda)\}\,ds \]
are segments of the expansions of the functions \(f_1(s)\) and \(f_2(s)\) in the ordinary Fourier integral.
Using the asymptotic estimates for the spectral matrix
\[
\Theta(x,s;\lambda)=\{\Theta_{ik}(x,s;\lambda)\}\quad (i,k=1,2),
\]
obtained in note \((^5)\), on the basis of the definition of the functions \(S_1(x,\lambda)\) and \(S_2(x,\lambda)\), i.e., equalities (9) and (10), the following is proved.
Lemma 1. If the coefficients \(p(x)\) and \(r(x)\) are summable on each finite interval, then for every fixed \(x\) and as \(|a|\to\infty\) the asymptotic estimates
\[ \bigvee_a^{a+1}\{S_i(x,\lambda)\}=o(1),\quad (i=1,2). \tag{11} \]
hold. The asymptotic estimates (11) hold uniformly in each finite interval of variation of \(x\).
From the definitions of the spectral matrix
\[
\Theta^*(x,s;\lambda)=\{\Theta^*_{ik}(x,s;\lambda)\}
\]
and the functions \(S_1^*(x,\lambda)\) and \(S_2^*(x,\lambda)\) it follows directly that
Lemma 2. As \(|a|\to\infty\), uniformly on the half-line \((0,\infty)\), the asymptotic estimate
\[ \bigvee_a^{a+1}\{S_i^*(x,\lambda)\}=o(1)\quad (i=1,2) \]
holds.
Lemmas 1 and 2 make it possible to apply Tauberian theorem for Fourier integrals of B. M. Levitan \((^4)\).
As a result, one can obtain the following theorem:
Theorem 1 (on equiconvergence). Let \(f(x)=\{f_1(x),f_2(x)\}\subset L^2(0,\infty)\). If the coefficients \(p(x)\) and \(r(x)\) are summable on every finite interval, then for each fixed \(x\) the equalities
\[ \lim_{\lambda\to\infty}\{[S_1(x,\lambda)-S_1(x,-\lambda)]-[S_1^*(x,\lambda)-S_1^*(x,-\lambda)]\}=0, \tag{12} \]
\[ \lim_{\lambda\to\infty}\{[S_2(x,\lambda)-S_2(x,-\lambda)]-[S_2^*(x,\lambda)-S_2^*(x,-\lambda)]\}=0. \tag{13} \]
hold.
The equalities (12) and (13) hold uniformly on every finite interval of variation of \(x\).
Since, by virtue of the definitions of the functions \(S_1(x,\lambda)\) and \(S_2^*(x,\lambda)\),
\[ S_1^*(x,\lambda)-S_1^*(x,-\lambda) = \frac1\pi\int_{-\lambda}^{\lambda} \left\{\int_0^\infty f_1(s)\cos\lambda s\,ds\right\} \cos\lambda x\,d\lambda, \]
\[ S_2^*(x,\lambda)-S_2^*(x,-\lambda) = \frac1\pi\int_{-\lambda}^{\lambda} \left\{\int_0^\infty f_2(s)\sin\lambda s\,ds\right\} \sin\lambda x\,d\lambda, \]
the equalities (12) and (13) have the form
\[ \lim_{\lambda\to\infty} \left\{ S_1(x,\lambda)-S_1(x,-\lambda) - \frac1\pi\int_{-\lambda}^{\lambda} \left[\int_0^\infty f_1(s)\cos\lambda s\,ds\right] \cos\lambda x\,d\lambda \right\}=0, \tag{14} \]
\[ \lim_{\lambda\to\infty} \left\{ S_2(x,\lambda)-S_2(x,-\lambda) - \frac1\pi\int_{-\lambda}^{\lambda} \left[\int_0^\infty f_2(s)\sin\lambda s\,ds\right] \sin\lambda x\,d\lambda \right\}=0. \tag{15} \]
The equalities (14) and (15) mean that, uniformly on every interval of the half-line \((0,\infty)\), the difference between the expansion of an arbitrary vector-function \(f(x)=\{f_1(x),f_2(x)\}\subset L^2(0,\infty)\) in the generalized Fourier integral with respect to the eigenfunctions of the one-dimensional Dirac operator and the expansion in the ordinary Fourier integral tends to zero.
Theorem 1 gives the final solution of the question of convergence of the expansion in eigenfunctions of the one-dimensional Dirac operator for a two-component vector-function with integrable square.
In particular, from Theorem 1 it follows that
Theorem 2 (on convergence). If the coefficients \(p(x)\) and \(r(x)\) are summable functions on every finite interval and the vector-function \(f(x)=\{f_1(x),f_2(x)\}\) belongs to the class \(L^2(0,\infty)\), then at every point where the local conditions of expandability of the vector-function \(f(x)\) into the ordinary Fourier integral are satisfied, the equalities
\[ \lim_{\lambda\to\infty}\{S_1(x,\lambda)-S_1(x,-\lambda)\}=f_1(x), \]
\[ \lim_{\lambda\to\infty}\{S_2(x,\lambda)-S_2(x,-\lambda)\}=f_2(x), \]
hold, i.e. the expansion of the vector-function \(f(x)=\{f_1(x),f_2(x)\}\) in the generalized Fourier integral with respect to the eigenfunctions of the one-dimensional Dirac operator tends to the value of the vector-function.
I take this opportunity to express my deep gratitude to B. M. Levitan for posing the problem, for his interest in the work, and for discussion of the results.
Moscow State University
named after M. V. Lomonosov
Received
27 V 1965
REFERENCES
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- B. M. Levitan, I. S. Sargsyan, UMN, 15, no. 1 (91), 3 (1960).
- I. S. Sargsyan, DAN, 166, no. 5 (1966).