UDC 517.9

MATHEMATICS

I. S. SARGSYAN

ASYMPTOTIC BEHAVIOR OF THE SPECTRAL MATRIX OF A ONE-DIMENSIONAL DIRAC SYSTEM

(Presented by Academician I. N. Vekua, May 27, 1965)

1. In relativistic Dirac quantum theory the wave function \(\Psi\) satisfies an equation of the form

\[ (p_0 + Vc^{-1} + \alpha_x p_x + \alpha_y p_y + \alpha_z p_z + \beta)\Psi = 0, \tag{1} \]

where \(V = V(x,y,z)\) is the potential function, \(c\) is the speed of light,

\[ p_0 = i\hbar c^{-1}\frac{\partial}{\partial t},\qquad p_x = -i\hbar\frac{\partial}{\partial x},\qquad p_y = -i\hbar\frac{\partial}{\partial y},\qquad p_z = -i\hbar\frac{\partial}{\partial z}, \]

\(\hbar\) is Planck’s constant, and the operators \(\alpha_x,\alpha_y,\alpha_z\) and \(\beta\) satisfy the conditions

\[ \alpha_x^2=\alpha_y^2=\alpha_z^2=1,\qquad \alpha_x\alpha_y+\alpha_y\alpha_x=\alpha_x\alpha_z+\alpha_z\alpha_x= \alpha_y\alpha_z+\alpha_z\alpha_y=0, \]

\[ \beta^2=m^2c^2,\qquad \alpha_x\beta+\beta\alpha_x= \alpha_y\beta+\beta\alpha_y= \alpha_z\beta+\beta\alpha_z=0, \tag{2} \]

and, finally, \(m\) is the mass of the particle. These formulas follow from formula (19), § 53.1 of [1], if in the latter formula we set \(A=0\) and \(e\Phi=V\). Let the operators \(\alpha_x,\alpha_y,\alpha_z\), and \(\beta\) be square \(4\times4\) matrices. Then the wave function \(\Psi\) will be a \(4\times1\) matrix, i.e., a four-component function

\[ \Psi=\{\Psi_1,\Psi_2,\Psi_3,\Psi_4\},\qquad \Psi_k=\Psi_k(x,y,z;t)\quad (k=1,2,3,4), \]

and therefore equation (1) is equivalent to a system of four partial differential equations for the functions \(\Psi_k=\Psi_k(x,y,z;t)\) \((k=1,2,3,4)\). If we put \(\Psi=e^{-iWt/\hbar}X\), where \(W\) is the energy, and the four-component function \(X\) depends only on \(x,y\), and \(z\), then equation (1) takes the form

\[ \{(W-V)c^{-1}+\alpha_xp_x+\alpha_yp_y+\alpha_zp_z+\beta\}X=0. \tag{3} \]

Here \(W\) plays the role of the eigenvalue parameter.

1. We shall consider the one-dimensional case of system (3), i.e., assume that the functions \(V\) and \(X\) depend only on \(x\). Then from system (3) we obtain

\[ \{(W-V)c^{-1}+\alpha p_x+\beta\}X=0, \tag{4} \]

where

\[ p_x=-i\hbar\frac{d}{dx},\qquad \alpha^2=1,\qquad \beta^2=m^2c^2,\qquad \alpha\beta+\beta\alpha=0. \tag{5} \]

In this case \(\alpha\) and \(\beta\) are \(2\times2\) matrices, while the function \(X\) is now two-component and may be regarded as a \(2\times1\) matrix. Therefore equation (4) is equivalent to two simultaneous ordinary differential equations of first order for the components \(X_1(x)\) and \(X_2(x)\) of the function \(X(x)\).

Let

\[ \alpha= \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}, \qquad \frac{\beta}{mc}= \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}, \qquad X(x)= \begin{pmatrix} X_1(x)\\ X_2(x) \end{pmatrix}. \]

Then conditions (5) are satisfied, and therefore from equation (4) it follows that

\[ \{W - V(x)\}c^{-1}X_1 - \hbar X_2' + mcX_1 = 0, \]
\[ \{W - V(x)\}c^{-1}X_2 + \hbar X_1' - mcX_2 = 0. \tag{6} \]

Suppose that the units of measurement are chosen so that the speed of light \(c\) and Planck’s constant \(\hbar\) are equal to 1. Then, also putting

\[ p(x)=V(x)-m,\quad r(x)=V(x)+m,\quad \lambda=W, \]

we can write system (6) in the form

\[ dX_2/dx + p(x)X_1 = \lambda X_1, \]
\[ -dX_1/dx + r(x)X_2 = \lambda X_2. \tag{7} \]

Let us adjoin to system (7) the boundary condition

\[ X_2(0)-hX_1(0)=0, \tag{8} \]

where \(h\) is an arbitrary real number.

Denote by \(\varphi(x,\lambda)=\{\varphi_1(x,\lambda),\varphi_2(x,\lambda)\}\) the solution of system (7) satisfying the initial conditions

\[ \varphi_1(0,\lambda)=1,\quad \varphi_2(0,\lambda)=h. \tag{8′} \]

It is obvious that the vector-function \(\varphi(x,\lambda)=\{\varphi_1(x,\lambda),\varphi_2(x,\lambda)\}\) satisfies the boundary condition (8).

It is known \((^{2-3})\) that for a given \(h\), to each boundary-value problem (7)—(8) there corresponds a unique* nondecreasing, bounded on every finite interval, left-continuous function \(\rho(\lambda)\) \((-\infty<\lambda<\infty)\), which generates an isometric mapping of the space of vector-functions \(f(x)=\{f_1(x),f_2(x)\}\subset \mathcal L^2(0,\infty)\) square-integrable on the half-line \((0,\infty)\):

\[ \int_0^\infty \{f_1^2(x)+f_2^2(x)\}\,dx<+\infty, \]

onto the space \(\mathcal L_{\{\rho(\lambda)\}}(-\infty,\infty)\) by the formulas

\[ F(\lambda)=\int_0^\infty \{f_1(x)\varphi_1(x,\lambda)+f_2(x)\varphi_2(x,\lambda)\}\,dx, \tag{9} \]

\[ f_1(x)=\int_{-\infty}^{\infty} F(\lambda)\varphi_1(x,\lambda)\,d\rho(\lambda), \tag{10} \]

\[ f_2(x)=\int_{-\infty}^{\infty} F(\lambda)\varphi_2(x,\lambda)\,d\rho(\lambda), \tag{11} \]

where the integrals (9) and (10)—(11) converge in the metrics of the spaces \(\mathcal L_{\{\rho(\lambda)\}}^2(-\infty,\infty)\) and \(\mathcal L^2(0,\infty)\), respectively, and Parseval’s equality holds:

\[ \int_0^\infty \{f_1^2(x)+f_2^2(x)\}\,dx = \int_{-\infty}^{\infty} F^2(\lambda)\,d\rho(\lambda). \]

1. Let us introduce the following terminology: the function \(\rho(\lambda)\) will be called the spectral function of problem (7)—(8); the matrix of second order \(\Theta(x,s;\lambda)=\{\Theta_{ik}(x,s;\lambda)\}\) \((i,k=1,2)\), where the functions \(\Theta_{ik}(x,s;\lambda)\)

\[ \text{* The uniqueness of the function } \rho(\lambda) \text{ was proved by B. M. Levitan.} \]

are defined by the formulas

\[ \Theta_{ik}(x,s;\lambda)= \begin{cases} \displaystyle \int_{0}^{\lambda}\varphi_i(x,\lambda)\varphi_k(s,\lambda)\,d\rho(\lambda), & \lambda>0,\\[1.1em] \displaystyle -\int_{\lambda}^{0}\varphi_i(x,\lambda)\varphi_k(s,\lambda)\,d\rho(\lambda), & \lambda<0,\\[1.1em] 0, & \lambda=0, \end{cases} \tag{12} \]

we shall call the spectral matrix of problem (7)—(8).

In the present note, by means of a method analogous to B. M. Levitan’s method (⁴), the asymptotic behavior of the spectral matrix \(\Theta(x,s;\lambda)\) as \(|\lambda|\to\infty\) is studied. We shall compare the spectral matrix \(\Theta(x,s;\lambda)\) with the spectral matrix \(\Theta^*(x,s;\lambda)\) of a simpler problem, namely, problem (7)—(8) with \(p(x)=r(x)\equiv0\) and \(h=0\). In this case \(\varphi_1(x,\lambda)=\cos\lambda x\), \(\varphi_2(x,\lambda)=\sin\lambda x\), and therefore the matrix \(\Theta^*(x,s;\lambda)\) is defined by the formulas

\[ \Theta^*_{11}(x,s;\lambda) =\frac1\pi\int_0^\lambda \cos\lambda x\cos\lambda s\,d\lambda =\frac1{2\pi}\left\{\frac{\sin\lambda(x-s)}{x-s} +\frac{\sin\lambda(x+s)}{x+s}\right\}, \]

\[ \Theta^*_{12}(x,s;\lambda) =\frac1\pi\int_0^\lambda \cos\lambda x\sin\lambda s\,d\lambda = \]

\[ =-\frac1{2\pi}\left\{\frac{\cos\lambda(x-s)}{x-s} +\frac{\cos\lambda(x+s)}{x+s}\right\} +\frac1\pi\,\frac{x}{x^2-s^2}, \]

\[ \Theta^*_{21}(x,s;\lambda) =\frac1\pi\int_0^\lambda \sin\lambda x\cos\lambda s\,d\lambda = \]

\[ =\frac1{2\pi}\left\{\frac{\cos\lambda(x-s)}{x-s} -\frac{\cos\lambda(x+s)}{x+s}\right\} -\frac1\pi\,\frac{x}{x^2-s^2}, \]

\[ \Theta^*_{22}(x,s;\lambda) =\frac1\pi\int_0^\lambda \sin\lambda x\sin\lambda s\,d\lambda =\frac1{2\pi}\left\{\frac{\sin\lambda(x-s)}{x-s} -\frac{\sin\lambda(x+s)}{x+s}\right\}. \]

Lemma. If the coefficients \(p(x)\) and \(r(x)\) are summable on every finite interval and \((x_0,x_1)\) is an arbitrary finite interval, then there exists a constant \(C=C(x_0,x_1)\) such that, for all values \(x\) and \(s\) belonging to the interval \((x_0,x_1)\), and for all \(a\),

\[ \bigvee_a^{a+1}\{\Theta_{ik}(x,s;\lambda)\}<C \qquad (i,k=1,2). \tag{13} \]

Estimate (13) for the matrix \(\Theta^*(x,s;\lambda)\) is obtained directly, if one takes into account the explicit form of the functions \(\Theta^*_{ik}(x,s;\lambda)\) \((i,k=1,2)\).

Estimate (13) makes it possible to apply Tauber’s theorem for Fourier integrals of V. A. Marchenko (⁵). As a result, the following theorem can be proved:

Theorem 1. If the coefficients \(p(x)\) and \(r(x)\) satisfy Dini’s condition, then for every fixed \(x\) and \(s\) the following asymptotic formulas hold

\[ \lim_{\lambda\to\infty} \left\{ \Theta_{11}(x,s;\lambda)-\Theta_{11}(x,s;-\lambda) -\frac1\pi\left[ \frac{\sin\lambda(x-s)}{x-s} +\frac{\sin\lambda(x+s)}{x+s} \right] \right. \]

\[ \left. +\frac4\pi\,\frac{\sin\lambda(x-s)}{x-s}\, \sin^2\left[ \frac14\int_s^x \{p(\tau)+r(\tau)\}\,d\tau \right] \right\}=0, \tag{14} \]

\[ \lim_{\lambda\to\infty}\left\{\Theta_{12}(x,s;\lambda)-\Theta_{12}(x,s;-\lambda)- \frac{2}{\pi}\frac{\sin\lambda(x-s)}{x-s} \sin\left[\frac{1}{2}\int_s^x \{p(\tau)+r(\tau)\}\right]\right\}=0, \tag{15} \]

\[ \lim_{\lambda\to\infty}\left\{\Theta_{21}(x,s;\lambda)-\Theta_{21}(x,s;-\lambda)+ \frac{2}{\pi}\frac{\sin\lambda(x-s)}{x-s} \sin\left[\frac{1}{2}\int_s^x \{p(\tau)+r(\tau)\}\,d\tau\right]\right\}=0, \tag{16} \]

\[ \lim_{\lambda\to\infty}\left\{\Theta_{22}(x,s;\lambda)-\Theta_{22}(x,s;\lambda)- \frac{1}{\pi}\left[\frac{\sin\lambda(x-s)}{x-s}-\frac{\sin\lambda(x+s)}{x+s}\right]+ \frac{4}{\pi}\frac{\sin\lambda(x-s)}{x-s} \sin^2\left[\frac{1}{4}\int_s^x \{p(\tau)+r(\tau)\}\,d\tau\right]\right\}=0. \tag{17} \]

The asymptotic formulas (14)—(17) hold uniformly in each finite domain of variation of the variables \(x\) and \(s\).

From Theorem 1, more precisely from formula (14), for \(x=s=0\) it follows (by virtue of (8′) and (12))

Theorem 2. If the coefficients \(p(x)\) and \(r(x)\) satisfy Dini’s condition, then the asymptotic formula
\[ \lim_{\lambda\to\infty}\left\{\rho(\lambda)-\rho(-\lambda)-\frac{2}{\pi}\lambda\right\}=0 \]
holds.

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 discussing the results.

Moscow State University
named after M. V. Lomonosov

Received
27 V 1965

REFERENCES

1. N. F. Mott, I. N. Sneddon, Wave Mechanics and its Applications, Oxford, 1948.
2. E. C. Titchmarsh, Proc. London Math. Soc. (3), 11, 159 (1961).
3. E. C. Titchmarsh, Proc. London Math. Soc. (3), 11, 169 (1961).
4. B. M. Levitan, Izv. Acad. Sci. USSR, Ser. Math., 16, 325 (1952).
5. V. A. Marchenko, Izv. Acad. Sci. USSR, Ser. Math., 19, 381 (1955).