MATHEMATICS

G. M. ZHISLIN

ON THE SPECTRUM OF THE SCHRÖDINGER OPERATOR

(Presented by Academician V. I. Smirnov on 19 V 1958)

§ 1. In the present note the form of the spectrum of the operator*

[
H=-\sum_{i=1}^{n} a_i \Delta_i
-2a_0 \sum_{\substack{i,j=1\ i<j}}^{n}
\left(
\frac{\partial^2}{\partial x_i \partial x_j}
+
\frac{\partial^2}{\partial y_i \partial y_j}
+
\frac{\partial^2}{\partial z_i \partial z_j}
\right)
-\sum_{i=1}^{n} b_i \frac{1}{r_i}
+\sum_{\substack{i,j=1\ i<j}}^{n} c_{ij}\frac{1}{r_{ij}},
\tag{1}
]

where (a_i=a_0+a_i'), (a_i'), (b_i), (c_{ij}) ((i,j=1,2,\ldots,n)) are arbitrary positive numbers; (a_0) is any nonnegative number.

In the case (a_0=0), i.e. without taking into account the motion of the nucleus, the author proved({}^{(1)}) the existence of a sequence of eigenvalues of the operator (\widetilde H)** when the conditions

[
b_i>\sum_{\substack{j=1\ j\ne i}}^{n} c_{ij},\qquad i=1,2,\ldots,n
\tag{2}
]

are satisfied (where (c_{ij}=c_{ji}) for (j<i)), to which atoms with any number of electrons and positive ions correspond.

In the present note this result is generalized to the case (a_0>0). In addition, the existence of the limiting spectrum*** of the operator (\widetilde H) is established. We use the notation and definitions introduced in ({}^{(1)}).

§ 2. Theorem. There exists a number (\mu_1<0) such that the entire limiting spectrum of the operator (\widetilde H) consists of all numbers (\lambda), (\lambda\ge \mu_1). Moreover, if conditions (2) are satisfied, then all points of the spectrum lying to the left of (\mu_1) form an increasing sequence of eigenvalues (\lambda^{(p)}), accumulating at (\mu_1), whose eigenfunctions (\psi_p) are differentiable any number of times and satisfy the equation (H\psi_p=\lambda^{(p)}\psi_p) at every point of the space (R_{3n}) that lies on none of the manifolds (r_i=0) ((i=1,2,\ldots,n)), (r_{ij}=0) ((i,j=1,2,\ldots,n;\ i\ne j)).

In what follows it is sufficient to consider only real functions. Let (\psi) and (\varphi) be arbitrary real functions respectively from

* The Schrödinger operator for atoms and ions is reduced to the form (1) with (a_0>0) if the motion of their nuclei is taken into account.

** (\widetilde H) is the self-adjoint extension of (H), obtained in the same way as in ({}^{(1)}).

*** For the definition of the limiting spectrum see ({}^{(2)}), p. 391.

(W_2^1(R_{3n})) and (W_2^1(R_{3n-3,i})). Introduce the following notation:

[
\begin{aligned}
L[\psi] &=
\sum_{j=1}^n a_j \int_{R_{3n}} |\operatorname{grad}j \psi|^2\,d\Omega
+2a_0 \int^n}} \sum_{\substack{l,j=1\ l<j}
(\operatorname{grad}l\psi,\operatorname{grad}_j\psi)\,d\Omega \
&\quad
-\sum\,d\Omega}^n b_j \int_{R_{3n}} \frac{|\psi|^2}{r_j
+\sum_{\substack{l,j=1\ l<j}}^n c_{lj}\int_{R_{3n}} \frac{|\psi|^2}{r_{lj}}\,d\Omega ,
\end{aligned}
\tag{3}
]

where

[
(\operatorname{grad}_l\psi,\operatorname{grad}_j\psi)
=
\frac{\partial\psi}{\partial x_l}\frac{\partial\psi}{\partial x_j}
+
\frac{\partial\psi}{\partial y_l}\frac{\partial\psi}{\partial y_j}
+
\frac{\partial\psi}{\partial z_l}\frac{\partial\psi}{\partial z_j};
]

[
\begin{aligned}
L^i[\varphi] &=
\sum_{\substack{j=1\ j\ne i}}^n
a_j \int_{R_{3n-3,i}} |\operatorname{grad}j\varphi|^2\,d\Omega
+2a_0 \int}
\sum_{\substack{l,j=1\ \ne i}}^n
(\operatorname{grad}l\varphi,\operatorname{grad}_j\varphi)\,d\Omega \
&\quad
-\sum^n}
b_j \int_{R_{3n-3,i}} \frac{|\varphi|^2}{r_j}\,d\Omega
+\sum_{\substack{l,j=1\ l\ne i,\ l<j,\ j\ne i}}^n
c_{lj}\int_{R_{3n-3,i}} \frac{|\varphi|^2}{r_{lj}}\,d\Omega ;
\end{aligned}
\tag{4}
]

[
\lambda_{3n-3,i}
=
\inf_{\substack{\varphi\in W_2^1(R_{2n-2,i}),\ |\varphi|=1}}
L^i[\varphi],
\qquad
n>1,
\qquad
\lambda_{0,1}=0.
]

Lemma. If, for a completely spreading sequence ({u_m}) from (W_2^1(R_{3n})),

[
|u_m|=1,\qquad
|u_m|{W_2^1(R<M,\qquad})
m=1,2,\ldots,
]

then

[
\lim L[u_m]\ge \min_{1\le i\le n}{\lambda_{3n-3,i}}.
]

§ 3. Let (\alpha_k) ((k=0,1,\ldots,l-1)) be (l) arbitrary distinct natural numbers, (1\le \alpha_k,\ l\le n);

[
H^{\alpha_0\alpha_1\ldots\alpha_{l-1}}
=
-\sum_{i=1}^{n}{}' a_i\Delta_i
-
]

[
-2a_0\sum_{\substack{i,j=1\ i<j}}^{n}{}'
\left(
\frac{\partial^2}{\partial x_i\partial x_j}
+
\frac{\partial^2}{\partial y_i\partial y_j}
+
\frac{\partial^2}{\partial z_i\partial z_j}
\right)
-\sum_{i=1}^{n}{}'\frac{b_i}{r_i}
+\sum_{\substack{i,j=1\ i<j}}^{n}{}' c_{ij}\frac{1}{r_{ij}}\,*;
\tag{5}
]

(R^{\alpha_0\ldots\alpha_{l-1}}) is the (3(n-l))-dimensional Euclidean space of the variables (x_i,y_i,z_i), (i\ne \alpha_k), (k=0,1,\ldots,l-1); (R^{(\alpha_0\ldots\alpha_{l-1})}) is the (3l)-dimensional Euclidean space of the variables (x_i,y_i,z_i), (i=\alpha_k), (k=0,1,\ldots,l-1). Define the spaces (\mathscr L_2(R^{\alpha_0\ldots\alpha_{l-1}})), (W_2^1(R^{\alpha_0\ldots\alpha_{l-1}})) by analogy with (\mathscr L_2(R_{3n})), (W_2^1(R_{3n})).

[
\text{* Here and below, a prime on a sum means that the terms with } i,j=\alpha_k,\ k=0,1,\ldots,l-1,\text{ are absent.}
]

Let (\psi) be an arbitrary function from (W_2^1(R^{\alpha_0\cdots \alpha_{l-1}})),

[
\begin{aligned}
L^{\alpha_0\cdots \alpha_{l-1}}[\psi]
&=
\sum_{i=1}^{n}{}' a_i
\int_{R^{\alpha_0\cdots \alpha_{l-1}}}
|\operatorname{grad}i \psi|^2\,d\Omega
\
&\quad
+2a_0
\sum'}}^{n}{
\int_{R^{\alpha_0\cdots \alpha_{l-1}}}
(\operatorname{grad}i\psi,\operatorname{grad}_j\psi)\,d\Omega
-\sum' b_i}^{n}{
\int_{R^{\alpha_0\cdots \alpha_{l-1}}}
\frac{|\psi|^2}{r_i}\,d\Omega
\
&\quad
+\sum_{\substack{i,j=1\ i<j}}^{t}{}'
c_{ij}
\int_{R^{\alpha_0\cdots \alpha_{l-1}}}
\frac{|\psi|^2}{r_{ij}}\,d\Omega;
\end{aligned}
\tag{6}
]

[
Q^{\alpha_0\cdots \alpha_{l-1}}
=
\left{\psi,\ \psi\in W_2^1(R^{\alpha_0\cdots \alpha_{l-1}}),\
|\psi|{\mathscr L_2(R^{\alpha_0\cdots \alpha=1\right};}})
]

[
\lambda_{\alpha_0\cdots \alpha_{l-1}}
=
\inf_{\psi\in Q^{\alpha_0\cdots \alpha_{l-1}}}
L^{\alpha_0\cdots \alpha_{l-1}}[\psi];
\qquad
\mu_l=\min{\lambda_{\alpha_0\cdots \alpha_{l-1}}};
]

[
l=1,2,\ldots,n-1;\qquad \mu_n=0.
]

We shall prove that (\mu_1) is the number whose existence is asserted in the theorem.

Let (\nu) be an arbitrary point of the limiting spectrum of (\widetilde H). We shall show that (\nu\geq \mu_1). Let (E_\lambda) be the spectral function of the operator (\widetilde H); let (\sigma_{\nu,\varepsilon}) be the subspace onto which the operator (E_{\nu+\varepsilon}-E_{\nu-\varepsilon}) projects functions from (\mathscr L_2(R_{3n})). For an arbitrary sequence ({\varepsilon_k}), (\varepsilon_k>0), (\varepsilon_k\underset{k\to\infty}{\longrightarrow}0), one can indicate an orthonormal sequence of functions ({\psi_{\varepsilon_k}}) such that (\psi_{\varepsilon_k}\in \sigma_{\nu,\varepsilon_k}) ((({}^{2}),\ \text{pp. }391\text{--}392)). Obviously, (\psi_{\varepsilon_k}\in D_{\widetilde H}\subset W_2^1(R_{3n})) and

[
\lim_{k\to\infty} L[\psi_{\varepsilon_k}]
=
\lim_{k\to\infty}(\widetilde H\psi_{\varepsilon_k},\psi_{\varepsilon_k})
=
\nu.
\tag{7}
]

It follows that (|\operatorname{grad}\psi_{\varepsilon_k}|<M), (k=1,2,\ldots). By virtue of the lemma and (4),

[
\lim_{k\to\infty} L[\psi_{\varepsilon_k}]\geq \mu_1.
\tag{8}
]

From (7) and (8) the required inequality (\nu\geq \mu_1) follows.

In order to prove that every fixed number (\nu), (\nu\geq \mu_1), is a point of the limiting spectrum of the operator (\widetilde H), it is enough ((({}^{2}),\ \text{pp. }392\text{--}393)) to indicate a sequence of functions ({F_m}) such that

[
\begin{array}{ll}
\text{a)} & F_m\in D_{\widetilde H};\[2mm]
\text{b)} & |F_m|=1;\[2mm]
\text{c)} & F_m \xrightarrow{\text{weakly}} 0 \text{ in } \mathscr L_2(R_{3n});\[2mm]
\text{d)} & |\widetilde H F_m-\nu F_m|\underset{m\to\infty}{\longrightarrow}0.
\end{array}
\tag{9}
]

We shall construct the sequence ({F_m}) for the operator (\widetilde H) of the form (1) with (a_0=0).

It can be shown that (\mu_n>\mu_{n-1}\geq \cdots \geq \mu_1). Let (\mu_s) be the smallest of the numbers (\mu_k) ((1\leq k\leq n-1)) for which

[
\mu_k<\mu_{k+1}
\tag{10}
]

holds, and (\mu_s=\lambda_{i_0\cdots i_{s-1}}), where (i_0,i_1,\ldots,i_{s-1}) are some fixed numbers. It follows from (10) that (\lambda_{i_0\cdots i_{s-1}}<\lambda_{i_0\cdots i_{s-1}i_s}) for any (i_s) ((1\leq i_s\leq n,\ i_s\ne i_\alpha,\ \alpha=0,1,\ldots,s-1)). Hence, using the lemma of the present note and general—

applying Lemma 2 from (1), we obtain that (\lambda_{i_0\ldots i_{s-1}}) is the least eigenvalue of the operator (\widetilde H^{\,i_0\ldots i_{s-1}}); let (\theta(x_i,y_i,z_i)), (i=1,2,\ldots,n), (i\ne i_\alpha), (\alpha=0,1,\ldots,s-1), be the corresponding eigenfunction. Consider the operator
[
H_3=-a_{i_0}\Delta_{i_0}-b_{i_0}\frac{1}{r_{i_0}}.
]
It has limiting spectrum filling the whole ray ([0,+\infty)) (5). Therefore (4) implies that for each number (\nu), (\nu\geqslant\mu_1), there exists a completely spreading sequence of functions (f_m), for which:
[
\text{a) } f_m\in C_2\bigl(R^{(i_0)}\bigr)*;
]
[
\text{b) } |f_m|{\mathscr L_2(R^{(i_0)})}=1;
]
[
\text{c) }|H_3f_m-(\nu-\mu_1)f_m|0.})}\xrightarrow[m\to\infty]{
]

Let (g(x_p,y_p,z_p)), (p=i_1,i_2,\ldots,i_{s-1}), be an arbitrary finite twice continuously differentiable function in (R^{(i_1,\ldots,i_{s-1})}), (|g|{\mathscr L_2(R^{(i_1,\ldots,i=1),})})
[
g_m(x_p,y_p,z_p)=m^{-3/2(s-1)}g(m^{-1}x_p,m^{-1}y_p,m^{-1}z_p),\quad m=1,2,\ldots
]
Put (F_m=\theta f_m g_m,\ m=1,2,\ldots). Using the properties of (\theta), ({f_m}), and ({g_m}), one can show that the functions (F_m) satisfy the relations (9). The first part of the theorem is proved.

The proof of the second part of the theorem is carried out, using the lemma, in the same way as the proof of the main theorem in (1).

The author expresses his gratitude to Prof. A. G. Sigalov, who supervised the execution of this work, for numerous pieces of advice and guidance.

Gorky State University
named after N. I. Lobachevsky

Received
16 V 1958

CITED LITERATURE

¹ G. M. Zhislin, DAN, 117, 931 (1957). ² F. Riesz, B. Sz.-Nagy, Lectures on Functional Analysis, Moscow, 1954. ³ R. Courant, D. Hilbert, Methods of Mathematical Physics, 1, Moscow—Leningrad, 1951. ⁴ E. E. Shnol’, Mat. sbornik, 42, 273 (1957). ⁵ I. M. Glazman, DAN, 80, 153 (1951).

* (\widetilde H^{\,i_0\ldots i_{s-1}}) is a self-adjoint extension of the operator (H^{i_0\ldots i_{s-1}}).

** (C_2(R^{(i_0)})) is the space of twice continuously differentiable functions in (R^{(i_0)}) belonging to (\mathscr L_2(R^{(i_0)})) together with all their derivatives up to order 2 inclusive.