MATHEMATICAL PHYSICS

G. M. ZHISLIN, A. G. SIGALOV

ON THE MIXED SPECTRUM OF CERTAIN MULTIDIMENSIONAL DIFFERENTIAL OPERATORS OF QUANTUM MECHANICS

(Presented by Academician V. I. Smirnov, 24 IV 1964)

Consider the operator

\[ H_n = T_n + V_n + W_n, \]

\[ T_n = - a \sum_{i=1}^{n} \Delta_i, \qquad V_n = - b \sum_{i=1}^{n} |r_i|^{-1}, \qquad W_n = c \sum_{i \ne j} |r_{ij}|^{-1}, \]

where \(r_i=(x_i,y_i,z_i)\), \(r_{ij}=r_i-r_j\), acting in the Hilbert space \(\mathfrak H_n\) of complex-valued functions of \(3n\) independent variables \(r_1,\ldots,r_n\), \(n \ge 1\). For \(a=\hbar^2/2m\), \(b=ze^2\), \(c=e^2\), \(H_n\) is the energy operator for an atom or ion with a fixed nucleus. If the interaction term \(W_n\) is omitted in \(H_n\), then as \(n\) increases an ever greater number of eigenvalues lie on the continuous spectrum. However, the methods used up to now have yielded only isolated eigenvalues of the full operator \(H_n\) \((^{1,2})\). In this note it is shown that the symmetry properties of the operator \(H_n\) make it possible to distinguish eigenvalues belonging to one symmetry class on the limiting spectrum of another symmetry class. In this way it is established that, for \(n \ge 4\), all eigenvalues of the operator \(H_n\) corresponding to physically realizable symmetry lie on its limiting spectrum, except, possibly, for some exceptional values.

1. Let \(D_n^{(k)}\) be the irreducible representations of the permutation group of \(n\) symbols \(S_n\); \(\chi_n^{(k)}(R)\), \(R \in S_n\), their characters; \(\overline{D}_n^{(k)}\), \(\overline{\chi}_n^{(k)}\) the associated irreducible representations and their characters; \(k=0,1,\ldots,[n/2]\)*. For \(n=2\), in general expressions containing the indices \(n,k\), one should put \(D_2^{(1)}=\overline{D}_2^{(1)}=D_2^{(0)}\); \(D_1^{(0)}=\overline{D}_1^{(0)}\) is the identity representation. In accordance with the Pauli principle, the representations \(\overline{D}_n^{(k)}\), as well as the representation \(D_2^{(0)}\), have physical meaning for the operator \(H_n\) (see \((^3)\), Chap. 22). If

\[ R = \begin{pmatrix} i_1,\ldots,i_n\\ 1,\ldots,n \end{pmatrix} \in S_n, \]

then \(T_R \psi(r_1,\ldots,r_n)=\psi(r_{i_1},\ldots,r_{i_n})\) is a unitary operator in \(\mathfrak H_n\). Put

\[ P_n^{(k)} = \frac{l_n^{(k)}}{h_n} \sum_{R \in S_n} \chi_n^{(k)}(R) T_R, \qquad \mathfrak H_n^{(k)} = P_n^{(k)} \mathfrak H; \]

\[ \overline{P}_n^{(k)} = \frac{l_n^{(k)}}{h_n} \sum_{R \in S_n} \overline{\chi}_n^{(k)} T_R, \qquad \overline{\mathfrak H}_n^{(k)} = \overline{P}_n^{(k)} \mathfrak H, \]

where \(l_n^{(k)}\) is the common dimension of the representations \(D_n^{(k)}\), \(\overline{D}_n^{(k)}\); \(h_n\) is the order of the group \(S_n\); \(P_n^{(k)}\), \(\overline{P}_n^{(l)}\) \((k,l=1,\ldots,[n/2])\) are pairwise orthogonal projection operators, with the exception of certain pairs for which both operators are equal.

\[ \text{* We use the notation of Wigner’s book } (^{3}), \text{ adding to it the lower index } n. \]

1. Let \(C_f^2\) be the set of all finite twice continuously differentiable functions from \(\mathfrak H\). The operator \(H_n\) is bounded below on \(C_f^2\) (2). We extend it to a self-adjoint operator, retaining the previous notation. For \(\psi \in C_f^2\),
   \[ H_n \bar P_n^{(k)} \psi = \bar P_n^{(k)} H_n \psi . \]
   Put
   \[ C_f^2(\bar D_n^{(k)}) = C_f^2 \cap \overline{\mathfrak H}_n^{(k)} = \bar P_n^{(k)} C_f^2 . \]
   By \(\bar H_n^{(k)}\) we denote the extension of the operator \(H_n\), considered only on \(C_f^2(\bar D_n^{(k)})\), to a self-adjoint operator acting in \(\overline{\mathfrak H}_n^{(k)}\).

2. For functions \(\psi \in \mathfrak H\) having, in each bounded domain, generalized derivatives in the sense of S. L. Sobolev, put, for \(n \ge 1\),
   \[ W_2^1(\bar D_n^{(k)})=\{\psi\in\overline{\mathfrak H}_n^{(k)},\ \|\operatorname{grad}\psi\|<\infty\}, \]
   where
   \[ \|\operatorname{grad}\psi\|^2 =\sum_i \|\operatorname{grad}_i\psi\|^2 =\int\left\{\sum_i\left(\left|\frac{\partial\psi}{\partial x_i}\right|^2+ \left|\frac{\partial\psi}{\partial y_i}\right|^2+ \left|\frac{\partial\psi}{\partial z_i}\right|^2\right)\right\}d\Omega; \]
   \[ L_n[\psi]=a\|\operatorname{grad}\psi\|^2+(V_n\psi,\psi)+(W_n\psi,\psi); \]
   \[ \lambda_0(\bar D_n^{(k)})=\inf L_n[\psi],\qquad \psi\in W_2^1(\bar D_n^{(k)}),\qquad \|\psi\|=1; \]
   \[ \bar\mu_{n-1}^{(k)}=\min\{\lambda_0(\bar D_{n-1}^{(k)}),\lambda_0(\bar D_{n-1}^{(k-1)})\},\qquad 0<k\le \left[\frac n2\right], \]
   excluding \(k=n/2\) for even \(n\),
   \[ \bar\mu_{n-1}^{(0)}=\lambda_0(\bar D_{n-1}^{(0)}),\qquad \bar\mu_{n-2}^{(n/2)}=\lambda_0(\bar D_{n-1}^{(n/2-1)}). \]

The corresponding quantities relating to the representation \(D_n^{(k)}\) will be denoted by \(\lambda_0(D_n^{(k)})\), \(\mu_{n-1}^{(k)}\).

Theorem 1. For all \(n \ge 2\), the inequality
\[ \lambda_0(\bar D_n^{(k)}) \le \bar\mu_{n-1}^{(k)} \]
holds. In order that \(\lambda_0(\bar D_n^{(k)})\) be an isolated eigenvalue of finite multiplicity of the operator \(\bar H_n^{(k)}\), it is necessary and sufficient that the condition
\[ (\mathrm E)\quad \lambda_0(\bar D_n^{(k)})<\bar\mu_{n-1}^{(k)}. \]
be satisfied.

The points \(\lambda>\bar\mu_{n-1}^{(k)}\) form the entire limiting spectrum of the operator \(\bar H_n^{(k)}\). The condition (E) expresses the energetic disadvantage of such a departure of one particle, under which the system passes from the state of symmetry \(\bar D_n^{(k)}\) to a state of symmetry \(\bar D_{n-1}^{(k)}\) or \(\bar D_{n-1}^{(k-1)}\).

1. Let
   \[ \lambda_0^{(k)}\le \lambda_1^{(k)}\le \cdots \le \lambda_{p-1}^{(k)}\quad (p\ge 1) \]
   be eigenvalues, and let \(u_0,u_1,\ldots,u_{p-1}\) be the corresponding orthonormal eigenfunctions of the operator \(\bar H_n^{(k)}\);
   \[ Q_p=\{\psi\in W_2^1(\bar D_n^{(k)});\,(\psi,u_l)=0;\ l=0,1,\ldots,p-1\}, \]
   \[ \lambda_p(\bar D_n^{(k)})=\inf L[\psi],\qquad \psi\in Q_p,\qquad \|\psi\|=1. \]

Theorem 2. For \(n \ge 2\), the inequality
\[ \lambda_p(\bar D_n^{(k)})\le \bar\mu_{n-1}^{(k)}\quad (p\ge 1) \]
holds. In order that \(\lambda_p(\bar D_n^{(k)})\) be an isolated eigenvalue of finite multiplicity of the operator \(\bar H_n^{(k)}\), it is necessary and sufficient that the condition
\[ (\mathrm E_1)\quad \lambda_p(\bar D_n^{(k)})<\bar\mu_{n-1}^{(k)}. \]
be satisfied.

1. Theorem 3. If \(b,c\) in the expression \(H_n\) satisfy the condition
   \[ b>(n-1)c, \]
   then for \(k=0,1,\ldots,\left[\frac n2\right]\):

1) for \(\bar H_n^{(k)}\) there exists an infinite sequence of points of the discrete spectrum
\[ \lambda_p(\bar D_n^{(k)})\to \bar\mu_{n-1}^{(k)}\qquad (p\to\infty); \]

2) for \(n \ge 3\) one has
\[ \bar\mu_{n-1}^{(0)}=\bar\mu_{n-1}^{(1)}<\bar\mu_{n-1}^{(k)}, \]
except for \((n,k)=(3,1)\).

1. Theorems 1, 2, and 3, 1) are valid without changes for the representations \(D_n^{(k)}\). Assertion 3, 2), as applied to \(D_n^{(k)}\), is replaced by the following: for \(n \geqslant 3\), \(k \geqslant 2\), the inequality \(\mu_{n-1}^{(1)} < \mu_{n-1}^{(k)}\) holds.

2. Since for neutral atoms and positive ions \(Z \geqslant n\), the condition \(b > (n-1)c\) is always satisfied for them. We give some consequences of Theorems 1–2. Since \(\lambda_p(\bar D_n^{(k)}) \to \bar\mu_{n-1}^{(k)}\) \((p \to \infty)\), it follows, by 3, 2), that \(\lambda_p(\bar D_n^{(k)}) > \mu_{n-1}^{(0)}=\mu_{n-1}^{(1)}\) for all \(p\), except, possibly, for a finite number. By Theorem 1, all these \(\lambda_p(\bar D_n^{(k)})\) belong to the limiting spectrum of the operator \(H_n^{(l)}\), \(l=0,1\). For \(n=3\) (the lithium atom) \(D_3^{(1)}=\bar D_3^{(1)}\), \(\mu_2^{(0)}=\mu_2^{(1)}=\bar\mu_2^{(1)}<\bar\mu_2^{(0)}\). Therefore the realized eigenvalues \(\lambda_p(\bar D_3^{(0)})\) all, except perhaps for a finite number, lie in the limiting spectrum of the realized symmetry \(\bar D_3^{(1)}\). Apparently, an analogous situation holds for \(n>3\). It follows from Theorems 1–3 that, for all \(n\), there exists an infinite sequence of eigenvalues of the realized symmetry that do not lie in the realized limiting spectrum. Among them is the least of all, \(\lambda_0(\bar D_n^{(k)})\), corresponding to the ground state.

3. The proofs of Theorems 1 and 2 are based on the following proposition.

Theorem 4. Let \(\psi_m \in C_2^f(\bar D_n^{(k)})\),

\[ \|\psi_m\|_{L^2}+\|\operatorname{grad}\psi_m\|_{L^2}\leqslant M\quad (m=1,2,\ldots), \]

\[ \int_{\Omega}|\psi_m|^2\,d\Omega \to 0 \quad (m\to\infty) \]

for any bounded domain \(\Omega\). Then
\[ \lim_{m\to\infty} L_n[\psi_m]\geqslant \bar\mu_{n-1}^{(k)} . \]

This proposition means that an arbitrary decay of a system from a state of symmetry \(\bar D_n^{(k)}\) cannot be energetically more advantageous than the departure of one particle, under which the system passes into a state of symmetry \(\bar D_{n-1}^{(k)}\) or \(\bar D_{n-1}^{(k-1)}\).

For the proof of Theorem 3 it is established that the inequality \(b>(n-1)c\) implies the energy inequalities (E), \((E_1)\). The proof of Theorem 4 is based on the decomposition of configuration space used earlier in \((^2)\).

These results carry over, without essential difficulties, to other irreducible representations of the group \(S_n\), and also to potentials of a more general form than Coulomb potentials. The relations between the quantities \(\lambda_0(D_n^{(k)})\), \(\lambda_0(\bar D_n^{(k)})\) remain unclear, except for the cases indicated in Theorem 3, 2).

Received
20 IV 1964

References

1. T. Kato, Trans. Am. Math. Soc., 70, 2, 196 (1951).
2. G. M. Zhislin, Tr. Moskovsk. matem. obshch., 9, 81 (1960).
3. E. Wigner, Group Theory and Its Application to the Quantum-Mechanical Theory of Atomic Spectra, IL, 1961.