MATHEMATICAL PHYSICS

F. A. BEREZIN

A TRACE FORMULA FOR THE MANY-PARTICLE SCHRÖDINGER EQUATION

(Presented by Academician I. G. Petrovskii, 21 III 1964)

In 1937, Beth and Uhlenbeck established a formula expressing
\(\operatorname{sp}(e^{-\beta H}-e^{-\beta H_0})\), where \(H_0=-d^2/dx^2\), \(H=-d^2/dx^2+v(x)\), in terms of scattering phases \(\bigl(v(x)\) is assumed to be a sufficiently rapidly decreasing function\bigr) (see (1), pp. 254–257). Subsequently this result was generalized (2), and the resulting expression of the trace in terms of the scattering operator received the name of trace formula. In the present paper trace formulas are established for the many-particle Schrödinger equation. These formulas are closely connected with quantum statistics and are different for different statistics.

1. The Boltzmann case. Consider, in three-dimensional space, a cube \(A\) of volume \(V\). Denote by \(L_2(n,A)\) the Hilbert space of functions of \(n\) variables, each of which ranges over the cube \(A\). In \(L_2(n,A)\) consider the operator \(H_n\):

\[ H_n=-\sum_1^n \Delta_k+g\sum_{1\le i<j\le n} v(x_i-x_j),\qquad x_k=(x_k^{(1)},x_k^{(2)},x_k^{(3)}), \tag{1} \]

where \(\Delta_k\) is the Laplace operator in the variables \(x_k\). Periodic boundary conditions are meant; \(g\) is a real parameter introduced for convenience.

The Boltzmann statsum is the series

\[ \Xi=1+\sum_1^\infty \frac{\xi^n}{n!}\operatorname{sp} e^{-\beta H_n}. \tag{2} \]

The parameter \(\xi\) is called the activity. Denote by \(b_n\) the coefficient in the expansion of \(\frac{1}{V}\ln \Xi\) in powers of \(\xi\). It is not hard to verify that

\[ b_n(V)=\frac{1}{V}\sum_{n_1+2n_2+\cdots=n} (-1)^{\,n_1+n_2+\cdots-1} \frac{(n_1+n_2+\cdots-1)!}{n_1!\,n_2!\cdots(1!)^{n_1}(2!)^{n_2}\cdots} Z_1^{n_1}Z_2^{n_2}\cdots, \tag{3} \]

where \(Z_k=\operatorname{sp}\exp(-\beta H_k)\).

Partition the set of \(n\) elements into nonintersecting subsets \(\Lambda_1,\Lambda_2,\ldots\). In what follows, sums over all such partitions are often encountered. We agree to denote by \(\Lambda\) the partition itself and by \(n_k=n_k(\Lambda)\) the number of subsets containing \(k\) elements. Thus, \(\sum kn_k=n\). Obviously, the number of different partitions with one and the same set of numbers \(n_k\) is
\(n!(n_1!\,n_2!\cdots(1!)^{n_1}(2!)^{n_2}\cdots)^{-1}\). With each partition associate the operator \(H_\Lambda\), equal to

\[ H_\Lambda=-\sum \Delta_k+g\sum_p \sum_{\substack{i<j\\ i,j\in\Lambda_p}} v(x_i-x_j); \tag{4} \]

\(\Lambda_p\) ranges over the collection of subsets entering the partition \(\Lambda\).

It is obvious that the operator (4) is an operator with separated variables and that

\[ \operatorname{sp}\exp(-\beta H_\Lambda) = \bigl(\operatorname{sp}\exp(-\beta H_1)\bigr)^{n_1} \bigl(\operatorname{sp}\exp(-\beta H_2)\bigr)^{n_2}\cdots. \tag{5} \]

Taking (5) into account, we obtain for \(b_n(V)\) the expression

\[ b_n(V)=\frac{1}{n!\,V}\operatorname{sp} \left\{ \sum_\Lambda (-1)^{\,n_1+n_2+\cdots-1} (n_1+n_2+\cdots-1)! \,e^{-\beta H_\Lambda} \right\}. \tag{6} \]

Along with \(b_n(V)\), consider the expression

\[ \tilde b_n(V)=\frac{1}{n!V}\operatorname{sp}\left\{\sum_\Lambda (-1)^{n_1+n_2+\cdots-1}(n_1+n_2+\cdots-1)!\,F(H_\Lambda)\right\}, \tag{6'} \]

where \(F\) is some function.

If \(F\) is a function analytic in a neighborhood of the real axis and \(H\) is a self-adjoint operator, then

\[ F(H)=-\frac{1}{2\pi i}\int F(z)(z-H)^{-1}\,dz. \]

(The integral is taken over a contour enclosing the spectrum of \(H\); the contour is traversed clockwise.) Taking this into account, after simple transformations we obtain for \(\tilde b_n(V)\), for \(n>1\), the expression

\[ b_n(V)=\frac{-1}{2\pi i\cdot n!\cdot V}\times \]

\[ \times \operatorname{sp}\int F(z)\frac{d}{dz}\sum_\Lambda (-1)^{n_1+\cdots-1}(n_1+\cdots-1)!\, \ln\left(1-(z-H_n^{(0)})^{-1}V_\Lambda\right)\,dz, \tag{7} \]

where \(H_n^{(0)}\) denotes the operator \(H_n^{(0)}=-\sum\Delta_k\), \(V_\Lambda=H_\Lambda-H_n^{(0)}\). In expression (7) it is possible to pass to the limit as \(V\to\infty\).* The operator under the trace sign in (7) will be denoted by \(T_V\). As \(V\to\infty\), \(T_V\) tends to a limit, which we denote by \(T\). After the Fourier transform the operator \(T\) is given by a kernel of the form

\[ K(p_1\ldots p_n\mid q_1\ldots q_n)\, \delta(p_1+\cdots+p_n-q_1\cdots q_n),\quad p_i=(p_i^{(1)},p_i^{(2)},p_i^{(3)}). \tag{8} \]

The regularized trace of the operator given by a kernel of the form (8) will mean the expression

\[ \operatorname{sp}_1 T=\int K(p_1\ldots p_n\mid p_1\ldots p_n)\,d^{3n}p. \tag{9} \]

Theorem 1. Let \(v(x)\) satisfy the estimate \(\int |v(x)|\,d^3x<\infty\), and let \(F(E)\) be a function analytic in a neighborhood of the real axis and satisfying the estimate

\[ \int_0^\infty |F(E+i\alpha)|E^{3n-1}\,dE<\infty. \]

Then there exists the limit \(\tilde b_n=\lim_{V\to\infty}\tilde b_n(V)\), which is equal to

\[ \tilde b_n=\frac{-1}{2\pi i\cdot n!}\times \]

\[ \times \operatorname{sp}_1\int F(z)\frac{d}{dz}\sum_\Lambda (-1)^{n_1+n_2+\cdots-1}(n_1+n_2+\cdots-1)! \ln\left(1-(z-H_n^{(0)})^{-1}V_\Lambda\right)\,dz. \tag{10} \]

The sum is extended over all partitions \(\Lambda\) of a set of \(n\) elements into disjoint subsets; \(n_k\) is the number of subsets containing \(k\) elements.

The operator under the trace sign has the form (8), and therefore has no trace in the ordinary sense. It is not hard to verify, however, that it leaves invariant the set of functions of the form \(f(p_1,\ldots,p_n)\delta(p_1+\cdots+p_n)\). If on this set one introduces the scalar product by the formula

\[ (f,f)=\int |f|^2\delta(p_1+\cdots+p_n)\,d^{3n}p, \]

then one obtains the Hilbert space \(L_2^0(n)\). If the function \(v(x)\) decreases sufficiently rapidly, then the operator under the trace sign in (10) has in \(L_2^0(n)\) a trace in the ordinary sense. The coefficient \(\tilde b_n\) is easily expressed through the trace of this operator in \(L_2^0(n)\).

Consider \(n\) points, some of which are joined by lines. The resulting complex will be called filled if, together with every closed polygon whose boundary has no self-intersections, it contains all diagonals of this polygon. The number \(n\) will be called the order of the complex. A complex of order \(n\) will be called

\[ \text{* The existence of the limit } b_n=\lim_{V\to\infty} b_n(V) \text{ was first established in the work of Lee and Yang } (^{3}). \]

monolithic if it consists of an \(n\)-gon and all its diagonals. It is obvious that every connected filled complex of order \(n\) consists of several monolithic complexes of smaller order, and any two of these monolithic complexes either have no common vertices or have exactly one common vertex. We shall call these monolithic subcomplexes the components of the original complex. Every filled complex is completely determined by its components. Therefore a complex of \(k\) components is naturally denoted by \(M=\{M_1,\ldots,M_n\}\), where \(M_k\) is the set of vertices of the \(k\)-th component.

Let us pass to the expression of the coefficient \(\tilde b_n\) in terms of scattering operators. We denote by \(S_\Lambda\) the scattering operator corresponding to \(H_\Lambda\). We shall assume that the scattering is purely elastic. In this case the scattering operator is a unitary operator commuting with \(H_n^{(0)}\).* The logarithm \(S_\Lambda\) is a skew-Hermitian operator which, after the Fourier transform, is specified by a kernel of the form

\[ 2\pi i\delta\left(p_1^2+\cdots+p_n^2-q_1^2-\cdots-q_n^2\right) \delta(p_1+\cdots+p_n-q_1,\ldots,-q_n)\, \sigma_\Lambda(p_1\ldots p_n\mid q_1\ldots q_n), \tag{11} \]

where

\[ p_i^2=\left(p_i^{(1)}\right)^2+\left(p_i^{(2)}\right)^2+\left(p_i^{(3)}\right)^2. \]

To every monolithic complex of order \(n\), whose vertices form the set \(M\), we assign the function
\[ E_M(p_{i_1}\ldots p_{i_n})= \sigma(p_{i_1}\ldots p_{i_n}\mid p_{i_1}\ldots p_{i_n}), \]
where \(M=\{i_1,\ldots,i_n\}\) is the set of vertices of the complex,

\[ \sigma(p_1\ldots p_n\mid q_1\ldots q_n)= \]

\[ =\sum_\Lambda(-1)^{n_1+n_2+\cdots-1}(n_1+n_2+\cdots-1)!\, \sigma_\Lambda(p_1\ldots p_n\mid q_1\ldots q_n). \tag{12} \]

(The sum extends over all partitions \(\Lambda\) of the set of \(n\) elements into nonintersecting subsets; \(n_k\) is the number of subsets consisting of \(k\) elements.)

Theorem 2. Let \(v(x)\) be a finite function and let \(F(E)\) be a function defined for \(E\ge 0\) and satisfying, together with its first \(n\) derivatives, the estimate
\[ \int_0^\infty \left|F^{(k)}(E)\right|E^{3n-1}\,dE<\infty. \]

Then there exists the limit \(\tilde b_n=\lim_{V\to\infty}\tilde b_n(V)\), which is equal to

\[ \tilde b_n=\sum_M\int F^{(k)}(p_1^2+\cdots+p_n^2)\,E_{M_1}\cdots E_{M_k}\,d^{3n}p. \tag{13} \]

The sum extends over all connected filled complexes of order \(n\), \(M=\{M_1,\ldots,M_k\}\). The number of the derivative of the function \(F\) in the summand corresponding to the complex \(M\) is equal to the number of components of the complex.

2. Bose case. The statistical sum in this case is the series

\[ \Xi=1+\sum_1^\infty \xi^n\,\operatorname{sp} P_n e^{-\beta H_n}, \tag{14} \]

where \(P_n\) is the projection operator in \(L_2(n,A)\) onto the space of functions invariant under permutations of the arguments. Expanding
\[ \frac{1}{V}\ln\Xi \]
in powers of \(\xi\), we determine, as before, the coefficients \(b_n(V)\).

* For this, for example, nonnegativity of \(v(x)\) is sufficient.

** The scattering operator \(S_\Lambda\) is closely related to the operator \(1-(E+i0-H_n^{(0)})^{-1}V_\Lambda\). Let us perform the Fourier transform and denote by \(R(E\parallel p\mid q)\) the kernel of the operator
\[ \left(1-(E+i0-H_n^{(0)})^{-1}V_\Lambda\right) \left(1-(E-i0-H_n^{(0)})^{-1}V_\Lambda\right)^{-1}. \]
Then the kernel of the operator \(S_\Lambda\) is determined by the formula
\[ S_\Lambda(p\mid q)=R(p^2\parallel p\mid q). \]

Denote by \(\tilde b_n^0(V)\) the coefficients obtained from \(\tilde b_n(V)\) by replacing \(H_k\) by \(H_k^{(0)}\) \((k=1,2,\ldots)\).

Consider a decomposition of \(\Lambda\) into \(n\) numbers into nonintersecting subsets \(\Lambda_1,\Lambda_2,\ldots\). Let \(n_k\), as before, denote the number of subsets containing \(k\) numbers. By \(P_{\Lambda_k}\) we shall denote the projection operator in \(L_2(n,\Lambda)\) onto the subspace of functions symmetric in the variables whose indices form the set \(\Lambda_k\). By \(P_\Lambda\) we denote the product of all \(P_{\Lambda_k}\), and by \(T_\Lambda\) the operator equal to \((1!)^{n_1}(2!)^{n_2}\cdots P_\Lambda\).

Theorem 3. Under the same assumptions on \(v(x)\) and \(F(z)\) as in Theorem 1, in the Bose case there exists the limit \(\tilde b_n=\lim\limits_{V\to\infty}\tilde b_n(V)\), which is equal to

\[ \tilde b_n=\tilde b_n^0-\frac{1}{2\pi i\cdot n!}\times \tag{15} \]

\[ \times \operatorname{sp}_1 \int F(z)\frac{d}{dz}\sum_{\Lambda}(-1)^{n_1+n_2+\cdots-1}(n_1+n_2+\cdots-1)!\,T_\Lambda \ln\left(1-(z-H_n^{(0)})^{-1}V_\Lambda\right)\,dz . \]

Here the trace in this formula is understood in the same sense as in Theorem 1.*

To each monolithic complex whose vertices form the set \(M=(i_1,\ldots,i_n)\), associate the function

\[ E_M^{(s)}(p_{i_1},\ldots,p_{i_n}) = \sigma^{(s)}(p_{i_1},\ldots,p_{i_n}\mid p_{i_1},\ldots,p_{i_n}), \]

where

\[ \sigma^{(s)}(p_1,\ldots,p_n\mid q_1,\ldots,q_n)= \]

\[ =\sum_{\Lambda}(-1)^{n_1+n_2+\cdots-1}(n_1+n_2+\cdots-1)!\,T_\Lambda \sigma_\Lambda(p_1\ldots p_n\mid q_1\ldots q_n) \]

and \(\sigma_\Lambda\) is the function defined above.

Theorem 4. Under the same assumptions as in Theorem 2,

\[ \tilde b_n=\tilde b_n^0+\frac{1}{n!}\sum\int F^{(k)}(p_1^2+\ldots+p_n^2)E_{M_1}^{(s)}\ldots E_{M_k}^{(s)}\,d^{3n}p . \tag{16} \]

The sum is taken over all connected complexes \(M=\{M_1,\ldots,M_k\}\); the order of the derivative of the function \(F\) in the term corresponding to the complex \(M\) is equal to the number of components of the complex.

In the Fermi case there hold formulas which, in writing, coincide with (15) and (16). The difference consists in the fact that the operator \(P_\Lambda\), which in the Bose case is the projection operator onto the space of symmetric functions, in the Fermi case is the projection operator onto the space of antisymmetric functions. Formulas (15), (16) are in an obvious way generalized to the case of particles of arbitrary spin.

In conclusion we emphasize once more that formulas (13) and (16) are valid in the case where there is only elastic scattering. Their extension to the general case is not automatic. As for formulas (10) and (15), they are valid for scattering of any character. A typical condition under which there is only elastic scattering is \(v(x)\ge 0\), \(g>0\). When the sign of \(g\) is changed, inelastic scattering arises. From formulas (10) and (15) it follows easily that \(b_n\) in both the Boltzmann and the Bose and Fermi cases is an entire function of \(g\).** Therefore, in the case \(v(x)\ge 0\), \(g<0\), the coefficients \(b_n\) can be obtained from (13) and (16) by analytic continuation in \(g\). The coefficients \(b_3\) were studied earlier by Pais and Uhlenbeck with the aid of perturbation theory \((^4)\).

Moscow State University
named after M. V. Lomonosov

Received
4 III 1964

CITED LITERATURE

\(^{1}\) L. D. Landau, E. M. Lifshitz, Statistical Physics, Moscow, 1950.
\(^{2}\) M. G. Krein, M. Sh. Birman, DAN, 144, No. 3 (1962).
\(^{3}\) T. O. Lie, C. N. Yang, Phys. Rev., 113, No. 5 (1959).
\(^{4}\) Pais, Uhlenbeck, Phys. Rev., 116, No. 2 (1959).

* In contrast to the Boltzmann case, the operator standing under the integral sign is not an operator with convergent trace even in the space \(L_2^{(0)}(n)\). Therefore in formula (15) one cannot use the invariant definition of the trace.

** The analytic properties of \(\tilde b_n\) as a function of \(g\) depend on the analytic properties of \(F\).