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MATHEMATICAL PHYSICS
G. M. ZHISLIN, A. G. SIGALOV
ON THE MATHEMATICAL THEORY OF ATOMIC SPECTRA
(Presented by Academician V. I. Smirnov, 15 I 1965)
The classification of atomic spectra is based on the symmetry properties of the solutions of the equation
\[ H\psi=\lambda\psi, \]
where
\[ H=H_n=T_n+V_n+W_n;\qquad T_n=-a\sum_{i=1}^{n}\Delta_i,\quad a>0; \]
\[ V_n=-b\sum_{i=1}^{n}|r_i|^{-1},\quad b\geqslant 0;\qquad W_n=c\sum_{i\ne j}|r_i-r_j|^{-1},\quad c\geqslant 0; \]
\[ r_i=(x_i,y_i,z_i),\qquad n\geqslant 2. \]
This equation has three symmetry groups: the permutation group \(S_n\), the rotation group \(\mathfrak R_0\), and the inversion group \(\mathfrak B\). If \(k,l,\omega\) are indices of irreducible representations of these groups, then their direct product
\[
\mathfrak P=S_n\times \mathfrak R_0\times \mathfrak B
\]
has irreducible representations with index \(\sigma=(k,l,\omega)\), determining the possible types of symmetry of the function \(\psi\). In the literature on the theory of atomic spectra it is stated that, regarding the solutions \(\psi^{kl\omega}\), only their symmetry properties are known (see, for example, \((^1)\)). Here the question is about the symmetry of solutions whose existence has not been proved. In the present note we give results showing the existence of an infinite sequence of eigenvalues for each type of symmetry \(\sigma=(k,l,\omega)\). At the same time, infinite series of eigenvalues of the operator \(H_n,\ n\geqslant 2\), corresponding to certain symmetry classes \(\sigma\), are found which lie on the limiting spectrum of the operator \(H_n\) of other symmetry classes (Theorem 3)\(*\).
- The index \(\sigma=(\Sigma,l,\omega)\) will denote irreducible representations of the group \(\mathfrak R_0\times\mathfrak B\), and the index \((\Sigma,l,\Sigma)\) irreducible representations of the group \(\mathfrak R_0\), etc. Together with the projection operators \(P^{(k)},\ \overline P^{(k)}\), defined by the group \(S_n\) (see \((^2)\)), we shall also consider the operators
\[ P^{\Sigma,l,\Sigma}=(2l+1)\int_{\mathfrak R_0}\chi^l(R)^*T_R\,dR, \]
where \(\chi^l(R),\ R\in\mathfrak R_0\), are the characters of the irreducible representation of the group \(\mathfrak R_0\) \((l=0,1,\ldots)\), and \(T_R\) is the rotation operator \((^3)\). The operators \(P^{\Sigma,\Sigma,\omega}\), \(\omega=\pm1\), are defined analogously. Put
\[ P^\sigma=P^{k,\Sigma,\Sigma}P^{\Sigma,l,\Sigma}P^{\Sigma,\Sigma,\omega} \]
for \(\sigma=(k,l,\omega)\). The operators \(P^\sigma\) are projection operators, and
\[ P^{\sigma_1}P^{\sigma_2}=P^{\sigma_2}P^{\sigma_1}=0 \]
if \(\sigma_1\ne\sigma_2\). The operators \(\overline P^\sigma\) arise when the representation \(D^k\) of the group \(S_n\) is replaced by the associated representation \(\overline D^{(k)}\). Let \(\mathfrak H_n=\mathfrak H\) be the Hilbert space of complex-valued functions defined in the entire \(3n\)-dimensional Euclidean space
\[ R_n=\{x,y,z,\ldots,x_n,y_n,z_n\}. \]
Put
\[ \mathfrak H^\sigma=P^\sigma\mathfrak H,\qquad \overline{\mathfrak H}^{\sigma}=\overline P^\sigma\mathfrak H. \]
By the Pauli principle only the spaces \(\overline{\mathfrak H}^{\sigma}\) have physical meaning. Since
\(*\) If only permutation symmetry is taken into account, this can be proved only for \(n\geqslant 3\) (see \((^2)\)).
the exposition for \(\overline{\mathfrak H}^{\sigma}\) does not differ essentially from the exposition for \(\mathfrak H^{\sigma}\); we shall consider only \(\mathfrak H^{\sigma}\).
- By \(H_n^{\sigma}\) we shall denote the extension of the operator \(H\), considered only on \(C_f^2 \cap \mathfrak H_n^{\sigma}\), to a self-adjoint one. Put
\[ W_2^1(D_n^{\sigma})=\{\psi\in\mathfrak H_n^{\sigma},\ \|\operatorname{grad}\psi\|<\infty\}; \]
\[ L[\psi]=a\|\operatorname{grad}\psi\|^2+(V_n\psi,\psi)+(W_n\psi,\psi); \]
\[ \lambda_0^{\sigma}=\lambda_0(D_n^{\sigma})=\inf L[\psi],\qquad \psi\in W_2^1(D_n^{\sigma}),\quad \|\psi\|=1. \]
For a given \(\sigma=(k,l,\omega)\) define \(\sigma_1\) by the equalities
\[ \mathfrak H^{\sigma_1}=\mathfrak H^{k,\Sigma,\Sigma \theta}\mathfrak H^{k,0,\omega_l}, \qquad \omega_l=(-1)^{l+1}\omega\quad \text{for } l>0; \]
\[ \mathfrak H^{\sigma_1}=\mathfrak H^{k,\Sigma,\Sigma \theta} \sum_{m=0}^{\infty}\mathfrak H^{k,m,\omega_m},\quad \text{for } l=0. \]
We define the symmetry type \(\sigma_2\) by decreasing by one the index \(k\) in the definition of \(\sigma_1\). Put
\[ \mu_{n-1}^{\sigma}=\min\{\lambda_0(D_{n-1}^{\sigma_1}),\lambda_0(D_{n-1}^{\sigma_2})\}. \]
Where the terms depending on \(k\) lose their meaning because of violation of the inequalities \(0\leq k\leq [n/2]\), we shall regard these terms as absent.
Theorem 1.
1) The inequality \(\lambda_0(D_n^{\sigma})\leq \mu_{n-1}^{\sigma}\) always holds.
2) In order that \(\lambda_0(D_n^{\sigma})\) be a point of the discrete spectrum of the operator \(H_n^{\sigma}\), it is necessary and sufficient that the condition \(\lambda_0(D_n^{\sigma})<\mu_{n-1}^{\sigma}\) be satisfied.
3) The points \(\lambda\geq \mu_{n-1}^{\sigma}\) form the limit spectrum of the operator \(H_n^{\sigma}\).
- Theorem 2. Let
\[ \lambda_0^{\sigma}\leq \lambda_1^{\sigma}\leq \cdots \leq \lambda_{p-1}^{\sigma} \qquad (p\geq 1) \]
be eigenvalues; \(u_0,u_1,\ldots,u_{p-1}\) the corresponding orthonormal eigenfunctions of the operator \(H_n\),
\[ Q_p=\{\psi\in W_2^1(D_n^{\sigma}),\ \|\psi\|=1,\ (\psi,u_s)=0;\ s=0,1,\ldots,p-1\}, \]
\[ \lambda_p^{\sigma}=\lambda_p(D_n^{\sigma})=\inf L[\psi], \qquad \psi\in Q_p\quad (p\geq 1). \]
1) The inequality \(\lambda_p(D_n^{\sigma})\leq \mu_{n-1}^{\sigma}\) always holds.
2) In order that \(\lambda_p(D_n^{\sigma})\) be a point of the discrete spectrum of the operator \(H_n^{\sigma}\), it is necessary and sufficient that the condition \(\lambda_p(D_n^{\sigma})<\mu_{n-1}^{\sigma}\) be satisfied.
- Theorem 3. Suppose the quantities \(b,c\) in the expression \(H_n\) satisfy the condition \(b>(n-1)c\). Then:
1) For \(n\geq 2\) and any symmetry type \(\sigma=(k,l,\omega)\), except for the type \(\sigma=(\Sigma,0,-1)\) which does not exist for \(n=2\), there exists an infinite sequence of points of the discrete spectrum \(\lambda_p^{\sigma}\) \((p=0,1,\ldots)\) of the operator \(H_n^{\sigma}\), accumulating at the value \(\mu_{n-1}^{\sigma}\).
2) \(\lambda_0^{0,0,+1}<\lambda_0^{\sigma}\) for \(\sigma\ne(0,0,+1)\).
3) For \(n=2\) and \(\sigma=(k,l,(-1)^{l+1})\), where \(k\) and \(l>0\) are arbitrary fixed indices, all eigenvalues \(\{\lambda_p^{\sigma}\}\) \((p=0,1,\ldots)\), with the exception, perhaps, of a finite number of them, lie on the limit spectrum of the operator \(H_2^{\sigma'}\) for any \(\sigma'=\{k_1,m,(-1)^m\}\), where \(k_1,m\) are arbitrary indices.
Corollary. By virtue of 1)—2), for \(n\geq 3\), for each \(\sigma=(k,l,\omega)\), \(k\geq 2\), all eigenvalues \(\lambda_p^{\sigma}\), except possibly a finite number of them, lie on the limit spectrum of the operator \(H_n^{0,0,1}\).
Remark. If \(H_2(\varepsilon)=T_2+V_2+\varepsilon W_2\), then for \(\varepsilon=0\) the symmetry type \(\sigma'=(k,m,(-1)^m)\) is possessed by all eigenfunctions obtained by symmetrization \((k=0)\) or antisymmetrization \((k=1)\) from functions of the form \(\psi_0(r_1)\cdot\psi_p(r_2)\), where \(\psi_p(r)\) are eigenfunctions of the operato-
for \(H_1\), \(\psi_0\) corresponds to the smallest eigenvalue. However, the eigenfunctions of the operator \(H_2(0)\) possessing symmetry \(\sigma'\) are not exhausted by this series. For \(H_2(1)\) an analogous fact cannot be established*, since the methods employed yield, for each type of symmetry, only the very lowest series of corresponding eigenvalues.
- The proof of Theorems 1 and 2 is based on the following proposition.
Theorem 4. Let \(\sigma=(k,l,\omega)\) be arbitrary. If \(\{\psi_m\}\in C_f^2\cap \bigcap \mathfrak L_n^\sigma u\)
\[ 1)\quad \int_{R_n}|\psi_m|^2\,d\Omega+\int_{R_n}|\operatorname{grad}\psi_m|^2\,d\Omega\leq c \quad (m=1,2,\ldots); \]
\[ 2)\quad \int_{\Omega}|\psi_m|^2\,d\Omega\to 0\quad (m\to\infty) \quad \text{for every bounded domain } \Omega\subset R_n, \]
then
\[ \lim_{n\to\infty} L_n[\psi_m]\geq \mu_{n-1}^{\sigma}. \]
The proofs of these theorems are carried out analogously to \((^1)\).
Scientific-Research Radiophysical Institute
at N. I. Lobachevsky Gorky State University
Received
13 I 1965
REFERENCES
- E. Wigner, Group Theory, IL, 1961.
- G. M. Zhislin, A. G. Sigalov, DAN, 157, No. 6, (1964).
- M. A. Naimark, Linear Representations of the Lorentz Group, Moscow, 1958.
* Although it apparently does hold.