F. A. BEREZIN

ASYMPTOTICS OF THE EIGENFUNCTIONS OF THE MANY-PARTICLE SCHRÖDINGER EQUATION

(Presented by Academician I. G. Petrovskii, January 15, 1965)

MATHEMATICS

1. Consider a system of \(n\) identical one-dimensional particles with pair interaction. The Schrödinger equation of such a system has the form

\[ \left\{-\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2}\right)+\sum_{i<j} v(x_i-x_j)\right\}\psi=k^2\psi. \tag{1} \]

The expression standing in braces is the energy operator of the system, which we shall denote by \(H_n\). Throughout the article it is assumed that the potential \(v(x)\) satisfies the conditions *

\[ v(x)=\int_{\mu_0}^{\infty} e^{-\mu |x|}\xi(\mu)\,d\mu,\qquad \int_{\mu_0}^{\infty}|\xi(\mu)|\,d\mu<\infty,\qquad \mu_0>0. \tag{2} \]

Among the solutions of equation (1) there are functions symmetric and antisymmetric with respect to permutations of \(x_1,\ldots,x_n\). In this paper the asymptotics of the symmetric and antisymmetric solutions of equation (1) is found.

1. The variables separate for the operator \(H_2\), and therefore there exists a complete system of eigenfunctions of the form

\[ \psi(x_1,x_2\mid p_1,p_2)= e^{\frac{i}{2}(p_1+p_2)(x_1+x_2)} \widetilde{\psi}\left(\frac{x_1-x_2}{\sqrt{2}}\,\middle|\,\frac{p_1-p_2}{\sqrt{2}}\right), \tag{3} \]

where \(\widetilde{\psi}(x\mid p)\) is an eigenfunction of the Sturm–Liouville operator
\(-d^2/dx^2+v(x)\) with eigenvalue equal to \(p^2\).

The symmetric eigenfunctions of the operator \(H_2\) of the form (3) constitute a complete system in the space of symmetric functions of two variables with summable square, and the antisymmetric ones—in the space of antisymmetric functions.

Using known results on the Sturm–Liouville operator (see, for example, \((^1,^2)\)), it is not difficult to show that the symmetric and antisymmetric eigenfunctions of the form (3) can be normalized so that inside the angle \(x_1-x_2>\varepsilon |x_1+x_2|\) \((\varepsilon>0)\) they have the form

\[ \begin{aligned} \psi_s(x_1,x_2\mid p_1,p_2) &=c_s(p_1-p_2)e^{i(p_1x_1+p_2x_2)} +c_s(p_2-p_1)e^{i(p_2x_1+p_1x_2)}+o(1),\\ \psi_a(x_1,x_2\mid p_1,p_2) &=c_a(p_1-p_2)e^{i(p_1x_1+p_2x_2)} -c_a(p_2-p_1)e^{i(p_2x_1+p_1x_2)}+o(1), \end{aligned} \tag{4} \]

where \(c_s(p)=c_s(-p)\), \(c_a(p)=-c_a(-p)\). The eigenvalue in both cases is equal to \(p^2=p_1^2+p_2^2\).

* Probably these conditions are more restrictive than is necessary for the validity of the theorems stated below (see in this connection \((^4)\), where the case \(v(x)=\lambda\delta(x)\) is considered).

Theorem 1. Denote by \(\Gamma\) the polyhedral angle
\(x_1>x_2>\cdots>x_n\), and by \(\Gamma_1\) an arbitrary closed polyhedral angle with vertex at the origin, all points of which, except the vertex, lie strictly inside \(\Gamma\). There exist symmetric \(\psi_s\) and antisymmetric \(\psi_a\) solutions of equation (1), having inside \(\Gamma_1\) the form:

\[ \psi_s(x\mid p)=\prod_{\alpha<\beta} c_s(p_\alpha-p_\beta) e^{i(p_1x_1+\cdots+p_nx_n)}+\cdots+ O\!\left(\frac{1}{\sqrt{|x|}}\right), \tag{5_s} \]

\[ \psi_a(x\mid p)=\prod_{\alpha<\beta} c_a(p_\alpha-p_\beta) e^{i(p_1x_1+\cdots+p_nx_n)}+\cdots+ O\!\left(\frac{1}{\sqrt{|x|}}\right). \tag{5_a} \]

The functions \(c_s, c_a\) here are the same as in (4); the dots replace the terms obtained from the one written by all possible permutations of \(p_1,\ldots,p_n\), where in \((5_s)\) all terms enter with a plus sign, while in \((5_a)\) with a plus sign if the permutation is even and with a minus sign otherwise. The eigenvalue in both cases is equal to
\(p^2=p_1^2+\cdots+p_n^2\).

Theorem 2. In the case where the operator
\(-d^2/dx^2+v(x)\) has no discrete spectrum, the functions \((5_s)\), \((5_a)\) form complete systems in the spaces consisting respectively of symmetric or antisymmetric square-summable functions of \(n\) variables.

For brevity, introduce the subscript \(\nu\), taking the values \(s\) and \(a\). Consider the symmetric and antisymmetric eigenfunctions of scattering theory:

\[ \psi_\nu^-(x\mid p)=\lim_{\varepsilon\to+0} i\varepsilon (H_n-p^2-i\varepsilon)^{-1}\varphi_{\nu0}(x\mid p), \tag{6} \]

where
\[ \psi_{s0}(x\mid p)=\sum \exp\{i(p_{k_1}x_1+\cdots+p_{k_n}x_n)\}, \]
\[ \psi_{a0}(x\mid p)=\sum \pm \exp\{i(p_{k_1}x_1+\cdots+p_{k_n}x_n)\}. \]
In both cases the sums are over all permutations; in the second case the plus or minus sign is put according to the parity of the permutation.

Theorem 3. The functions \(\psi_{s-}, \psi_{a-}\), for \(x\in\Gamma_1\), are connected with the functions \((5_s)\), \((5_a)\) by the relations \((\nu=s,a)\):

\[ \psi_{\nu-}(x\mid p)= \left[\prod_{\alpha<\beta}\bar c_\nu(|p_\alpha-p_\beta|)\right]^{-1} \psi_\nu(x\mid p)+ O\!\left(\frac{1}{\sqrt{|x|}}\right). \tag{7} \]

Denote by \(G(x,y\mid E)\) the kernel of the operator
\((H_n-E-i0)^{-1}\). The following theorem establishes an important property of the function \(G\).

Theorem 4. Let \(f(x_1,\ldots,x_n)\) be a generalized function of the form
\[ f=\delta(x_i-x_j)\varphi(x_1,\ldots,x_n), \]
where \(\varphi\) is a classical function, constant on the hyperplanes
\(\sum a_{\alpha\beta}x_\beta=0,\ 1\leq \alpha\leq k\), such that

\[ \int |\varphi|\prod_\alpha \delta\!\left(\sum_\beta a_{\alpha\beta}x_\beta\right) dx_1,\ldots,dx_n<\infty. \]

Then, for \(x\in\Gamma_1\), the estimate holds

\[ \int (L_yG(x,y))f(y)d^ny = O\!\left(|x|^{-(n-k-1)/2}\right), \tag{8} \]

where \(L_y\) is an arbitrary differential operator of order at most one with constant coefficients, acting on \(G\) as on a function of \(y\).

In this theorem and in the preceding one, \(\Gamma_1\) is the same polyhedral angle as in Theorem 1.

3. We present the idea of the proof of Theorems 1 and 3. Consider, inside the polyhedral angle \(\Gamma_n\) defined by the inequalities
\(x_1>x_2>\cdots>x_n\), a solution of equation (1) of the special form

\[ \varphi_n(x\mid p)= e^{i(p,x)} \left( 1+ \int_{\Gamma_n} B_n(\mu\mid p)\, \delta(\mu_1+\cdots+\mu_n)e^{-(\mu,x)}d^n\mu \right). \tag{9} \]

The integral in (9) is extended to the polyhedral angle \(\Gamma_n'\), dual to \(\Gamma_n:\mu\in\Gamma_n'\), if \((\mu,x)\geq 0\) for \(x\in\Gamma_n\). The eigenvalue to which \(\varphi_n\) belongs is equal to \(p^2=p_1^2+\cdots+p_n^2\). From this point on we shall assume that \(p_1+\cdots+p_n=0\) (the general case is easily reduced to this). The function \(\varphi_n\) need not be extendable beyond \(\Gamma_n\), and therefore we shall call it a local solution. The function \(B_n(\mu\mid p)\), which determines the local solution, is naturally sought in the form

\[ B_n(\mu\mid p)=\sum_{k=2}^{n}\sum A_k(\mu_{i_1}\ldots \mu_{i_k}\mid p_{i_1},\ldots,p_{i_k}) \prod_{i_1\ldots i_k}\delta(\mu_1)\ldots\delta(\mu_n), \tag{10} \]

where \(A_k(\mu_1\ldots \mu_k\mid p_1\ldots p_k)\) are functions equal to zero if the vector \(\mu=(\mu_1,\ldots,\mu_k)\) does not belong to \(\Gamma_k'\); \(\prod_{i_1\ldots i_k}\delta(\mu_1)\ldots\delta(\mu_n)\) denotes the product of the functions \(\delta(\mu_i)\) over all \(i\) from \(i=1\) to \(i=n\), except \(i=i_1,\ldots,i_k\); the inner sum in (10) is taken over all permutations. For \(k<n\) the functions \(A_k\) are the same as those occurring in the expression for \(B_{n'}\), with \(n'<n\).

Substituting \(\varphi_n\) into equation (1) and using the special form of the potential \(v(x)\), after simple transformations we obtain for the functions \(A_k\) integral equations close to Volterra equations. Like ordinary Volterra equations, these equations are uniquely solvable. From their analysis we obtain the following important properties of the functions \(A_k\): 1) \(A_n(\mu\mid p)=0\) inside the polyhedral angle \(\widetilde{\Gamma}_n'\subset\Gamma_n'\), whose sides are parallel to the sides of \(\Gamma_n'\), and whose vertex lies strictly inside \(\Gamma_n'\); 2) \(A_n(\mu\mid p)\), as a function of \(p\), admits analytic continuation into the domain \(\operatorname{Im}p\in\Gamma_n\) and, for fixed \(p\) in this domain, is bounded with respect to \(\mu\).

Let us now consider the symmetric function \(\Phi_n(x\mid p)\), defined inside \(\Gamma_n\) by the formula

\[ \Phi_n(x\mid p)=\prod_{\alpha<\beta} c_s(p_\alpha-p_\beta)\, \varphi_n(x_1,\ldots,x_n\mid p_1,\ldots,p_n)+\cdots, \tag{11} \]

where the ellipsis replaces the sum of terms obtained from the one written by all possible permutations of \(p_1,\ldots,p_n\). The function (11) is continuous in the whole space and everywhere, except on the planes \(x_\alpha=x_\beta\), satisfies equation (1). The jump of the normal derivative on the plane \(x_\alpha=x_\beta\) will be denoted by \(f_{\alpha\beta}(x)\); \(f_{\alpha\beta}(x)\) turns out to be a linear combination of the functions

\[ f_{\alpha\beta\mid k} = \int \bigl[i(p_\alpha+i\mu_\alpha)-i(p_\beta+i\mu_\beta)\bigr] A_k(\mu\mid p)e^{-(\mu,x)} \times \delta(\mu_1+\cdots+\mu_k)\delta(\mu_{k+1})\ldots\delta(\mu_n)\,d^n\mu \tag{12} \]

and of the functions obtained from (12) by permutations of \(p_1,\ldots,p_n\) and by those permutations of \(x_i\) for which \(x_{i_1}>x_{i_2}>\cdots>x_{i_k}\).

The function \(\Phi_n(x\mid p)\) is constructed with the following calculation, so that in the formation of \(f_{\alpha\beta}\) the terms \(f_{\alpha\beta\mid k}\) enter only for \(k\geq 3\). We note that the function \(f_{\alpha\beta\mid k}\) depends on \(k-1\) variables: in formula (12) there participate \(k\) variables; moreover, according to (12),
\(f_{\alpha\beta\mid k}(x_1+h,\ldots,x_k+h)=f_{\alpha\beta\mid k}(x_1,\ldots,x_k)\). This property reduces the number of variables on which \(f_{\alpha\beta\mid k}\) actually depends to \(k-1\). The value of \(f_{\alpha\beta\mid k}\) on the hyperplane \(x_\alpha=x_\beta\) is, therefore, a function of \(k-2\) variables. It is obvious that \(f_{\alpha\beta\mid k}\) decreases exponentially when \(x_1+\cdots+x_k=0\) and \(x_1\geq x_2\geq\cdots\geq x_k\).

We shall now seek the eigenfunction in the form \(\psi=\Phi_n+\varepsilon\). Substituting \(\psi\) into equation (1), we find for \(\varepsilon\) the equation

\[ (H_n-p^2)\varepsilon=\sum_{\alpha<\beta}\delta(x_\alpha-x_\beta)f_{\alpha\beta}(x). \tag{13} \]

One solution of this equation is
\(\varepsilon=(H_n-p^2-i0)^{-1}g\), where \(g\) denotes the right-hand side of (13). Thus,
\(\varepsilon(x)=\int G(x,y\mid p^2)g(y)\,d^ny\), and according to (8) \((L_y=1)\), \(\varepsilon=O(|x|^{-1/2})\) inside \(\Gamma_1\).

We note that the function \(f_{\alpha\beta\gamma k}\) gives a contribution to \(\varepsilon\) of order \(|x|^{-(k-2)/2}\). The antisymmetric case is considered analogously.

We give the idea of the proof of Theorem 3. For simplicity we restrict ourselves to the symmetric case and \(n=3\). Expand the function \(\psi_0=e^{i(p_1x_1+\cdots+p_3x_3)}+\cdots\) in the functions \(\left(\prod_{i>j}\bar c(|p_i-p_j|)\right)^{-1}\psi_s(x\mid p)\):

\[ \psi_0(x\mid p)=\int a(p,q)\left(\prod_{i<j}\bar c(|q_i-q_j|)\right)^{-1}\psi_s(x\mid q)\delta(q_1+q_2+q_3)\,d^3q. \tag{14} \]

From (14) and (7) we find that

\[ \psi_{s-}(x\mid p)=\int \lim_{\varepsilon\to+0} i\varepsilon\, \frac{a(p,q)}{q^2-p^2-i\varepsilon}\, \frac{\psi_s(x\mid q)}{\prod_{i<j}\bar c(|q_i-q_j|)} \,\delta(q_1+q_2+q_3)\,d^3q. \tag{15} \]

Starting from (14) and \((5_s)\), we obtain for \(a(p,q)\) the expression

\[ a(p,q)=\frac{1}{(2\pi i)^2}\, \frac{1}{\prod_{i<j}c(|q_i-q_j|)} \left\{ \frac{\theta(p_1-p_2)\theta(p_2-p_3)} {(p_1-q_1-i0)(p_3-q_3+i0)} \prod_{i<j}c(q_i-q_j)+\cdots \right\}+ \]

\[ +\alpha(p,q), \tag{16} \]

where \(\theta(p)=1\) for \(p>0\) and \(\theta(p)=0\) for \(p<0\); the ellipsis replaces the terms obtained from the written one by symmetrization with respect to \(p_\alpha\) and \(q_\beta\); \(\alpha(p,q)\) is the Fourier transform of the remainder term:

\[ \alpha(p,q)=\frac{\alpha_1(p,q)}{p^2-q^2-i0} +\delta(p^2-q^2)\alpha_2(p,q)+\alpha_3(p,q), \tag{17} \]

\(\alpha_i(p,q)\) are classical bounded functions.

The identity holds (see \((3)\))

\[ \lim_{\varepsilon\to+0} i\varepsilon\, \frac{\delta(p_1+p_2+p_3-q_1-q_2-q_3)} {(p_1^2+p_2^2+p_3^2-q_1^2-q_2^2-q_3^2+i\varepsilon)(p_1-q_1-i0)(p_3-q_3+i0)} = \]

\[ =(2\pi i)^2\theta(p_1-p_2)\theta(p_2-p_3) \delta(p_1-q_1)\delta(p_2-q_2)\delta(p_3-q_3). \]

Using this formula and formula (15), we find that

\[ \psi_{s-}(x\mid p)= \frac{\psi_s(x\mid p)} {\prod_{i<j}\bar c(|p_i-p_j|)} + \]

\[ +\int \delta(p^2-q^2)\alpha_2(p,q)\delta(q_1+q_2+q_3)\,d^3q = \frac{\psi_s(x\mid p)} {\prod_{i<j}\bar c(|p_i-p_j|)} +O(|x|^{-1/2}). \]

Finally, we derive an equation whose investigation makes it possible to obtain an estimate of the kernel of the operator \(G_n=(H_n-p^2-i0)^{-1}\). For simplicity we restrict ourselves to the case \(n=3\) and omit the index \(n\). Put \(G=G_0+G_0TG_0\), where \(G_0\) is the operator obtained from \(G\) for \(v(x)=0\). The operator \(T\) satisfies the equation
\[ (E-K_{12}-K_{13}-K_{23})T=V_{12}+V_{13}+V_{23}=T_0, \]
where \(V_{ij}\) is the operator of multiplication by \(V(x_i-x_j)\), and \(K_{ij}=V_{ij}G_0\). Put \((E-K_{ij})^{-1}=E+L_{ij}\). Multiplying both sides of the equation for \(T\) by \((E+L_{12})(E+L_{13})(E+L_{23})\) and taking into account that
\[ L_{ij}-K_{ij}-L_{ij}K_{ij}=(E+L_{ij})(E-K_{ij})-E=0, \]
we find the final equation
\[ T=(E+L_{12})(E+L_{13})(E+L_{23})T_0+RT. \]

Received
6 I 1965

CITED LITERATURE

1. Z. S. Agranovich, V. A. Marchenko, The Inverse Problem of Scattering Theory, Kharkov, 1960.
2. L. D. Faddeev, UMN, 14, 4, 57 (1959).
3. F. A. Berezin, V. P. Sushko, ZhETF, 48, no. 5 (1965).
4. F. A. Berezin, G. P. Pokhil, V. M. Finkelberg, Vestn. Mosk. Univ., no. 1, 21 (1964).