On simultaneous approximation of several eigenvalues of a semi-definite self-adjoint linear operator in a Hilbert space

TL;DR

This paper addresses the problem of approximating multiple eigenvalues of a semi-definite self-adjoint operator in a Hilbert space using finite-dimensional operators, with accuracy improving as the dimension increases.

Contribution

It provides a method for simultaneous approximation of several eigenvalues of semi-definite self-adjoint operators with increasing accuracy as the finite-dimensional space expands.

Findings

Approximation accuracy improves as the dimension s increases.

Method applies to operators with non-empty discrete spectrum.

Convergence of eigenvalue approximations is established.

Abstract

A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $\lambda_{min}=\lambda_1\leqslant\ldots\leqslant\lambda_m<\sigma_{ess}$. The problem of approximation of these eigenvalues by eigenvalues of some linear operator in a finite-dimensional space of the dimension $s$ is considered and solved. The accuracy of the approximation obtained becomes unlimitedly high as $s\to\infty$.

Equations

$$X=\underset{n\to \infty }{\lim }{X}_n\text{, where }{X}_n=\sum _{i=1}^n{x}^i{h}_i.$$

$$\exists \underset{n\to \infty }{\lim }⟨X_n|F{X}_n⟩={\lambda }_{\text{min}}\Vert X{\Vert }^2.$$

$${\lambda }_{\text{min}}=\underset{n\to \infty }{\lim }{\hat{\mu }}_1^{(n)},$$

$$X=\sum _{i=1}^{\infty }{x}^i{h}_i,$$

$$\Gamma (F)=\{(X,Y)\in H\times H:X\in D(F)\text{ and }Y=FX\}.$$

$$\underset{n\to \infty }{\lim }{X}_n=X,$$

$$D(F^{\ast })=\{X\in H:\exists Y\in H:⟨X|FZ⟩=⟨Y|Z⟩\forall Z\in D(F)\}$$

$$Q(F)=D(\sqrt{|F|})$$

$$q_F(X,Y)=\overline{q_F(Y,X)}\text{ for all }X,Y\in Q(F).$$

$$q_F(X,Y)=\underset{n\to \infty }{\lim }⟨X_n|F{Y}_n⟩.$$

$$\begin{gathered} {\mu }_n(F)=\underset{X_1,\dots ,X_{n-1}}{\sup }\inf \text{Sb}X\in D(F),\Vert X\Vert =1 \\ X\perp X_1,\dots ,X\perp X_{n-1}⟨X|FX⟩. \end{gathered}$$

$${\mu }_1(F)=\underset{X\in D(F),\Vert X\Vert =1}{\inf }⟨X|FX⟩.$$

$${\lambda }_{\text{min}}={\lambda }_1⩽\dots ⩽{\lambda }_{n-1}⩽{\lambda }_n$$

$$\begin{gathered} {\mu }_n(F)=\inf \text{Sb}X\in D(F),\Vert X\Vert =1 \\ X\perp X_1,\dots ,X\perp X_{n-1}⟨X|FX⟩. \end{gathered}$$

$${\mu }_m(F)⩽{\hat{\mu }}_m\text{, where }m=1,\dots ,n.$$

$$⟨Y|F_{V}X⟩=q_F(Y,X)\text{ for all }X,Y\in V.$$

$${\mu }_m(F)⩽{\hat{\mu }}_m\text{, where }m=1,\dots ,n.$$

$$\begin{array}{c}{\hat{\mu }}_m=\underset{X_1,\dots ,X_{m-1}\in V}{\sup }\inf \text{Sb}X\in V,\Vert X\Vert =1\\ X\perp X_1,\dots ,X\perp X_{m-1}⟨X|F_{V}X⟩=\\ =\underset{X_1,\dots ,X_{m-1}\in V}{\sup }\inf \text{Sb}X\in V,\Vert X\Vert =1\\ X\perp X_1,\dots ,X\perp X_{m-1}{q}_F(X,X).\end{array}$$

$$\begin{array}{c}{\hat{\mu }}_m=\underset{X_1,\dots ,X_{m-1}\in H}{\sup }\inf \text{Sb}X\in V,\Vert X\Vert =1\\ X\perp P{X}_1,\dots ,X\perp P{X}_{m-1}{q}_F(X,X)=\\ =\underset{X_1,\dots ,X_{m-1}\in H}{\sup }\inf \text{Sb}X\in V,\Vert X\Vert =1\\ X\perp X_1,\dots ,X\perp X_{m-1}{q}_F(X,X)⩾\\ ⩾\underset{X_1,\dots ,X_{m-1}\in H}{\sup }\inf \text{Sb}X\in Q(F),\Vert X\Vert =1\\ X\perp X_1,\dots ,X\perp X_{m-1}{q}_F(X,X)={\mu }_m(F).\end{array}$$

$$X_k=\underset{n\to \infty }{\lim }{X}_{kn}\text{ and }\underset{n\to \infty }{\lim }{q}_F(X_{kn},X_{qn}⟩=⟨X_k|F{X}_q⟩,$$

$${\lambda }_k=\underset{n\to \infty }{\lim }{\hat{\mu }}_k^{(n)}\text{ for all }k=1,\dots ,m,$$

$$V_n=\mathrm{Span}(\{X_{ks}\text{, where }1⩽k⩽m,1⩽s⩽n\})\subset Q(F)$$

$$\text{Cal}V=\mathrm{Span}(X_1,\dots ,X_m)\subset D(F)\subset Q(F)$$

$$F_{\text{Cal}V}=\text{Cal}P\text{raise}1.0pt\circ F\text{raise}1.0pt\circ \text{Cal}P\text{vrule}{height=10.0pt,depth=8.0pt,width=0.5pt}_{\text{Cal}V}.$$

$$\text{Cal}F=\Vert \begin{array}{ccc}{\lambda }_1& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \dots & {\lambda }_m\end{array}\Vert .$$

$$\text{Cal}H=G\text{Cal}F,$$

$$\text{Cal}{H}_{sk}=q_F(X_s,X_k)=⟨X_s|F{X}_k⟩,$$

$$\text{Cal}{V}_n=\mathrm{Span}(X_{1n},\dots ,X_{mn})\subset Q(F).$$

$$\dim \text{Cal}{V}_n=\dim \text{Cal}V=m.$$

$$\text{Cal}{H}^{(n)}=G^{(n)}\text{Cal}{F}^{(n)},$$

Full text

1. Introduction.

The interest to Hilbert spaces and to linear operators in them is mainly due to their applications in quantum mechanics where the spaces of square integrable complex functions $L_{2}(\mathbb{C},\mathbb{R}^{3N})$ and differential operators in them are considered. The most important of them is the energy operator which is called the Hamiltonian in general case and is called the Schrödinger operator in the case of particles in potential fields. In many cases the Schrödinger operator appears to be self-adjoint and semi-definite. With the quantum mechanical applications in view, Reed and Simon in their book [1] give the following theorem (see § 2 in Chapter XIII of [1]).

\special{html:<a

name="thr1.1">}Theorem 1.1\special{html:</a>}

Let $F$ be a lower semi-definite self-adjoint linear operator with the domain $D(F)$ in a Hilbert space $H$ such that its discrete spectrum is not empty and $\lambda_{\text{min}}$ is ts minimal eigenvalue111 It is implicitly assumed that $\lambda_{\text{min}}$ is below the lower limit of the essential spectrum $\sigma_{\text{ess}}(F)$ of the operator $F$.. Let $X$ be an eigenvector of $F$ associated with the eigenvalue $\lambda_{\text{min}}$. Assume that the eigenvector $X$ is expanded in some orthonormal basis $\{h_{i}\}_{i=1,\,\ldots,\,\infty}$ of the Hilbert space $H$:

[TABLE]

Assume that $h_{i}\in D(F)$ for all $i=1,\,\ldots,\,\infty$ and suppose that

[TABLE]

Under these assumptions we have the equality

[TABLE]

where $\hat{\mu}^{(n)}_{1}$ is the minimal eigenvalue of the Hermitian $n\times n$ matrix $\Phi^{(n)}$ whose elements are $\Phi_{ij}=\langle h_{i}|Fh_{j}\rangle$.

The angular brackets $\langle\bullet|\bullet\rangle$ in Theorem 1.1 mean the standard sesquilinear scalar product of the Hilbert space $H$. The formula 1.1 in Theorem 1.1 means convergence of the series

[TABLE]

with respect to the norm in $H$, while the coefficients $x^{i}$ in 1.1 and 1.2 are denoted according to the Einstein’s tensorial notation (see § 20 of Chapter I in [2]), i. e. using the upper index $i$ for the coordinates of the vector $X$.

Theorem 1.1 is in background of amost all numerical methods of quantum chemistry, though it is used without recognition and without references to it. The goal of this paper is

2. Standard definitions and prerequisites.

A Hilbert space $H$ is a complex (generally speaking infinite-dimensional) linear vector space with some fixed positive sesquilinear form $\langle X|Y\rangle=\overline{\langle Y|X\rangle}$ that defines the norm $\|X\|=\sqrt{\langle X|X\rangle}$ and thus defines the topology of a complete metric space in $H$. With the quantum mechanical applications in view, all Hilbert spaces in this paper are implicitly assumed to be separable.

The form $\langle X|Y\rangle$ in $H$ is assumed to be linear in its second argument $Y$ and to be conjugate linear with respect to the first argument $X$. This convention is used in quantum mechanics (see. [3, 4]). The form $\langle X|Y\rangle$ is called the standard scalar product in $H$.

A linear operator $F$ in a Hilbert space $H$ is usually given along with its domain $D(F)$ which is assumed to be a dense subspace of $H$. The graph of a linear operator $F$ is the following subset of the Cartesian product $H\times H$:

[TABLE]

Definition Definition 2.1

A linear operator $F$ in a Hilbert space $H$ is called closed, if its graph $\Gamma(F)$ is closed in $H\times H$.

An extension of an operator $F$ is another operator with the domain bigger than $D(F)$ which upon restricting to $D(F)$ coincides with $F$.

Definition Definition 2.2

A linear operator $F$ in a Hilbert space $H$ is called closable if it has at least one closed extension in $H$. The minimal closed extension of a closable operator $F$ is its closed extension with the minimal domain. It is denoted through $\bar{F}$ and is shortly called the closure of $F$.

The minimal closed extension $\bar{F}$ of a closable operator $F$ is constructed by closing its graph: $\Gamma(\bar{F})=\overline{\Gamma(F)}$. This yields the folloiwing theorem.

\special{html:<a

name="thr2.1">}Theorem 2.1\special{html:</a>}

For any closable operator $F$ in a Hilbert space $H$ a vector $X$ belongs to the domain $D(\bar{F})$ if and only if there is a sequence of vectors $X_{n}\in D(F)$ such that the following relationships hold:

[TABLE]

Limits here are in the sense of convergence with respect to the norm in $H$, while the vector $Y$ is the value of the operator $\bar{F}$ applied to the vector $X$, i. e. $\bar{F}X=Y$.

Definition Definition 2.3

Let $F$ be a linear operator in a Hilbert space $H$. The linear operator $F^{*}$ with the domain

[TABLE]

given by the formula $F^{*}X=Y$, where the element $Y\in H$ is uniquely fixed by the condition $\langle X|FZ\rangle=\langle Y|Z\rangle\ \forall\,Z\in D(F)$ from 2.1, is called the conjugate operator for the operator $F$.

In the book [5] the following theorem is given (see § 1 of Chapter VIII in [5]).

\special{html:<a

name="thr2.2">}Theorem 2.2\special{html:</a>}

Let $F$ be a linear operator in a Hilbert space $H$. Then

Definition Definition 2.4

A linear operator $F$ in a Hilbert space $H$ is called symmetric if the conjugate operator $F^{*}$ is an extension of $F$.

Practically the definition 2.4 can be replaced with a more simple and equivalent definition.

Definition Definition 2.5

A linear operator $F$ in a Hilbert space $H$ is called symmetric if $\langle X|FY\rangle=\langle FX|Y\rangle$ for all $X,Y\in D(F)$.

\special{html:<a

name="thr2.3">}Theorem 2.3\special{html:</a>}

Each symmetric operator $F$ in a Hilbert space $H$ is closable. Its closure $\bar{F}$ itself is a symmetric operator in $H$.

Definition Definition 2.6

A linear operator $F$ in a Hilbert space $H$ is called self-adjoint if $F^{*}=F$, i. e. if $F$ is symmetric and $D(F)=D(F^{*})$.

Definition Definition 2.7

A symmetric operator $F$ in a Hilbert space $H$ is called essentially self-adjoint if its closure $\bar{F}$ is a self-adjoint opertator in $H$.

Each linear operator $F$ in a Hilbert space $H$ is associated with the sesquilinear form $q_{F}(X,Y)=\langle X|FY\rangle$ which is defined for all $X$ and $Y$ in $D(F)$. In the case of a symmetric operator $F$ the form $q_{F}$ is symmetric, i. e. the following relationship holds: $q_{F}(X,Y)=\overline{q_{F}(Y,X)}\text{\ \ for all \ }X,Y\in D(F)$.

Definition Definition 2.8

A symmetric linear operator $F$ in a Hilbert space $H$ is called lower semi-definite if its form $q_{F}$ is lower semi-definite, i. e. if there is a real constant $C$ such that $q_{F}(X,X)=\langle X|FX\rangle\geqslant C\,\|X\|^{2}\text{\ \ for all \ }X\in D(F)$. If $C=0$, such an operator $F$ is called non-negative.

In the case of a self-adjoint operator $F$, using the spectral theorem, the form $q_{F}$ is extended from $D(F)$ to the bigger set

[TABLE]

(see § 6 of Chapter VIII in [5] and § 53 of Part XV in [6]), preserving its symmetry:

[TABLE]

In the case of a lower semi-definite self-adjoint linear operator $F$ the set 2.2 and the values of the form $q_{F}$ in it admit a more constructive description.

\special{html:<a

name="thr2.4">}Theorem 2.4\special{html:</a>}

Let $F$ be a lower semi-definite self-adjoint linear operator in a Hilbert space $H$. Then a vector $X$ belongs to the domain $Q(F)$ of the corresponding sesquilinear form $q_{F}$ if and only if there is a sequence of vectors $X_{n}$ in the domain $D(F)$ of the operator $F$ such that

\special{html:<a

name="thr2.5">}Theorem 2.5\special{html:</a>}

Under the assumptions of Theorem 2.4 let $X$ and $Y$ be two vectors from $Q(F)$ and let $X_{n}$ and $Y_{n}$ be two sequences of vectors from $D(F)$ approximating $X$ and $Y$ in the sense of Theorem 2.4. Then

[TABLE]

3. Minimax principle.

\special{html:<a

name="thr3.1">}Theorem 3.1\special{html:</a>}

Let $F$ be a lower semi-definite self-adjoint operator with the domain $D(F)$ in a Hilbert space $H$. For some positive integer $n$ denote

[TABLE]

Then exactly one of the two options holds:

Theorem 3.1 expresses the minimax principle for linear operators in Hilbert spaces. Its statement and its proof are given in the book [1] (see § 1 of Chapter XIII).

Remark. In [1] it is said that in Theorem 3.1 the domain $D(F)$ of the operator $F$ can be replaced with the domain $Q(F)$ of its sesquilinear form $q_{F}$ (see 2.2). At the same time the quantity $\langle X|FX\rangle$ is replaced by the value of the form $q_{F}(X,X)$.

The case $n=1$ in Theorem 3.1 is somewhat different from others. In this case the formula 3.1 simplifies and takes the form

[TABLE]

The next theorem is a corollary of Theorem 3.1.

\special{html:<a

name="thr3.2">}Theorem 3.2\special{html:</a>}

If a lower semi-definite self-adjoint operator $F$ with the domain $D(F)$ in a Hilbert space $H$ has a non-empty discrete spectrum below $\sigma_{\text{ess}}(F)$, then its minimal eigenvalue $\lambda_{\text{min}}=\mu_{1}(F)$ is given by the formula 3.2.

Remark. The domain $D(F)$ of the operator $F$ in the formula 3.2 can be replaced by the domain $Q(F)$ of its sesquilinear form $q_{F}$ (see 2.2). At the same time the quantity $\langle X|FX\rangle$ is replaced by the value of the form $q_{F}(X,X)$.

4. Minimum principle.

In the case $n=1$ the minimax principle is expressed by the formula 3.2. It transforms into minimum principle. A similar minimum principle can be formulated for the case $n>1$.

\special{html:<a

name="thr4.1">}Theorem 4.1\special{html:</a>}

Let $F$ be a lower semi-definite self-adjoint linear operator with the domain $D(F)$ in a Hilbert space $H$ whose discrete spectrum below $\sigma_{\text{ess}}(F)$ comprises at least $n$ eigenvalues arranged in the non-decreasing order

[TABLE]

so that each eigenvalue in the sequence is repeated according to its multiplicity, except for maybe the last one. Let $X_{1},\,\ldots,\,X_{n-1}$ be linerly independent eigenvectors for the initial $n-1$ eigenvalues in this non-decreasing sequence. Then the quantity $\mu_{n}(F)=\lambda_{n}$ is given by the formula

[TABLE]

The minimum principle similar to Theorem 4.1 for symmetric operators in a finite-dimensional Euclidean space can be found in the book [7]. For the Laplace operator it is formulated in [8]. The proof of Theorem 4.1 can be found in [9]. In this paper Theorem 4.1 is given for the sake of completeness of our preliminary review. It is not used in what follows.

5. The Rayleigh-Ritz method.

\special{html:<a

name="thr5.1">}Theorem 5.1\special{html:</a>}

Let $F$ be a lower semi-definite self-adjoint linear operator with the domain $D(F)$ in a Hilbert space $H$, let $V\subset D(F)$ be a finite-dimensional subspace of the dimension $n$ in $D(F)$, and let $P$ be the orthogonal projector onto $V$. The composite operator P\,\raise 1.0pt\hbox{\scriptscriptstyle\circ}\,F\,\raise 1.0pt\hbox{\scriptscriptstyle\circ}\,P is lower semi-definite and self-adjoint. Its restriction $F_{V}$ to the subspace $V$ has exactly $n$ eigenvalues $\hat{\mu}_{1},\,\ldots,\,\hat{\mu}_{n}$ that can be arranged in the non-decreasing order $\hat{\mu}_{1}\leqslant\ldots\leqslant\hat{\mu}_{n}$ where each eigenvalue is repeated according to its multiplicity. Under these assumptions the following inequalities hold:

[TABLE]

The quantities $\mu_{m}(F)$ in the left hand side of the inequalities 5.1 are given by the formula 3.1. They do not depend on the choice of the subspace $V$ in Theorem 5.1. The statement and the proof of Theorem 5.1 are given in the book [1] (see § 2 of Chapter XIII).

Remark. As we already noted above, the domain $D(F)$ of the operator $F$ when computing $\mu_{m}(F)$ in the formula 3.1 can be replaced by the domain $Q(F)$ of its sesquilinear form $q_{F}$ (see 2.2). At the same time the quantity $\langle X|FX\rangle$ is replaced by the value of the form $q_{F}(X,X)$. In Theorem 5.1 the domain $D(F)$ can also be replaced by the domain $Q(F)$. Although then we should define the operator $F_{V}$ in a different way through the restriction of the form $q_{F}$ to the subspace $V\subset Q(F)$.

\special{html:<a

name="thr5.2">}Theorem 5.2\special{html:</a>}

Let $F$ be a lower semi-definite self-adjoint linear operator in a Hilbert space $H$ whose associated sesquilinear form is $q_{F}$ and the domain of $q_{F}$ is $Q(F)$. Let $V\subset Q(F)$ be a finite-dimensional subspace of the dimension $n$ in $Q(F)$ and let $F_{V}$ be a linear operator in the subspace $V$ defined by the restriction of the form $q_{F}$ to $V$ according to the formula

[TABLE]

The operator $F_{V}$ has exactly $n$ eigenvalues $\hat{\mu}_{1},\,\ldots,\,\hat{\mu}_{n}$ that can be arranged in the non-decreasing order $\hat{\mu}_{1}\leqslant\ldots\leqslant\hat{\mu}_{n}$ where each eigenvalue is repeated according to its multiplicity. Under these assumptions the following inequalities hold:

[TABLE]

Demonstration Proof

Applying the minimax principle expressed by Theorem 3.1 to the operator $F_{V}$ in the finite-dimensional space $V$, from 3.1 we derive

[TABLE]

Let $P$ be the orthogonal projector onto the subspace $V$. Them 5.3 implies

[TABLE]

The inequality in 5.4 arises since we replace $V$ in the left hand side of the inequality by a larger subspace $Q(F)$ in the right hand side of this inequality. The last equality in 5.4 is due to the remark to Theorem 3.1 on page 5. ∎

Theorems 5.1 and 1.1 constitute a base for the Rayleigh-Ritz method. Its application to the experimental confirmation of the Lamb shift is described in [1] (see § 2 of Chapter XIII therein). See also [10--17].

6. Approximation of several eigenvalues.

\special{html:<a

name="thr6.1">}Theorem 6.1\special{html:</a>}

Let $F$ be a lower semi-definite self-adjoint linear operator with the domain $D(F)$ in a Hilbert space $H$ whose discrete spectrum below $\sigma_{\text{ess}}(F)$ is not empty and comprises at least $m$ eigenvalues arranged in the non-decreasing order $\lambda_{\text{min}}=\lambda_{1}\leqslant\ldots\leqslant\lambda_{m}$ so that each eigenvalue is repeated according to its multiplicity, except for maybe the last one. Let $X_{1},\,\ldots,\,X_{m}$ be linearly independent eigenvectors corresponding to the eigenvalues $\lambda_{\text{min}}=\lambda_{1}\leqslant\ldots\leqslant\lambda_{m}$ and assume that for each eigenvector $X_{k}$ a sequence of vectors $X_{kn}$ from the domain $Q(F)$ of the associated with $F$ sesquilinear form $q_{F}$ is given so that

[TABLE]

where $1\leqslant k,q\leqslant m$. Under these assumptions

[TABLE]

where $\hat{\mu}^{(n)}_{\text{min}}=\hat{\mu}^{(n)}_{1}\leqslant\ldots\leqslant\hat{\mu}^{(n)}_{m}$ are initial $m$ eigenvalues of the operator $F_{V_{n}}$ generated by the restriction of the form $q_{F}$ to the finite-dimensional subspace

[TABLE]

taken in the non-decreasing order so that each eigenvalue is repeated according to its multiplicity, except for maybe the last one.

Demonstration Proof

The eigenvectors $X_{1},\,\ldots,\,X_{m}$ belong to the domain $D(F)$. They are linearly independent. Their span

[TABLE]

is an $m$-dimensional subspace in $D(F)$ and in $Q(F)$. Let $\Cal{P}$ be the orthogonal projector onto the subspace 6.4. We set

[TABLE]

In other words, we denote through $F_{\Cal{V}}$ the restriction of the composite operator \Cal{P}\,\raise 1.0pt\hbox{\scriptscriptstyle\circ}\,F\,\raise 1.0pt\hbox{\scriptscriptstyle\circ}\,\Cal{P} to the subspace 6.4. The eigenvectors $X_{1},\,\ldots,\,X_{m}$ of the operator $F$ are eigenvectors of the operator 6.5, while the corresponding eigenvalues coincide with $\lambda_{\text{min}}=\lambda_{1}\leqslant\ldots\leqslant\lambda_{m}$. Therefore the matrix of the operator 6.5 in the basis of the eigenvectors $X_{1},\,\ldots,\,X_{m}$ is

[TABLE]

The operator 6.5 is generated by the restriction of the form $q_{F}$ to the subspace 6.4, therefore the matrix 6.6 obeys the relationships

[TABLE]

where $\Cal{H}$ and $G$ are two Hermitian matrices with the components

[TABLE]

The matrix $G$ is the Gram matrix of the basis of vectors $X_{1},\,\ldots,\,X_{m}$. Since these vectors are linearly independent, $\det G\neq 0$ and hence the matrix relationships 6.7 are consistent.

Let’s consider the vectors $X_{1n},\,\ldots,\,X_{mn}$. According to the first condition in 6.1 these vectors approximate $X_{1},\,\ldots,\,X_{m}$ in the sense of convergence with respect to the norm in the Hilbert space $H$. This means that for all sufficiently large $n$ the vectors $X_{1n},\,\ldots,\,X_{mn}$ are linearly independent. We set

[TABLE]

The subspace 6.9 is analogous to the subspace 6.4. For all sufficiently large $n$ the dimensions of these subspaces do coincide:

[TABLE]

Generally speaking, the subspace 6.9 is not enclosed in $D(F)$. Therefore we cannot write a formula similar to 6.5. But we can consider the operator $F_{\Cal{V}_{n}}$ generated by the restriction of the sesquilinear form $q_{F}$ to the subspace 6.9. Let $\Cal{F}^{(n)}$ be the matrix of such operator in the basis $X_{1n},\,\ldots,\,X_{mn}$. It is similar to the matrix 6.6, though it is not diagonal. The matrix $\Cal{F}^{(n)}$ obeys the relationships which are similar to the relationships 6.7:

[TABLE]

In 6.11 we see two Hermitian matrices $\Cal{H}^{(n)}$ and $G^{(n)}$. Their components are defined by the formulas similar to 6.8:

[TABLE]

The matrix $G^{(n)}$ in 6.11 is the Gram matrix for the basis composed by the vectors $X_{1n},\,\ldots,\,X_{mn}$. For all sufficiently large $n$ the matrix $G^{(n)}$ is non-degenerate, which is in agreement with 6.10. Hence the relationships 6.11 are consistent.

Now we consider the conditions 6.1. Applying them to 6.8 and 6.12, we get

[TABLE]

In the matrix form the relationships 6.13 are written as follows:

[TABLE]

The first relationship 6.14 yields $\lim\limits_{n\to\infty}\det G^{(n)}=\det G\neq 0$. Hence

[TABLE]

Let’s combine 6.15 with the second relationship 6.14 and then take into account 6.11. This yields the following relationship

[TABLE]

The eigenvalues of the matrix $\Cal{F}$ are the eigenvalues of the operator 6.5, coinciding with $\lambda_{\text{min}}=\lambda_{1}\leqslant\ldots\leqslant\lambda_{m}$. They are presented explicitly in 6.6. The eigenvalues of the matrix $\Cal{F}^{(n)}$ in 6.16 are the eigenvalues of the self-adjoint operator $F_{\Cal{V}_{n}}$ in $m$-dimensional space $\Cal{V}_{n}$. We denote them

[TABLE]

From 6.16 for the eigenvalues 6.17 we derive

[TABLE]

In the next step we consider the subspace 6.3. The sesquilinear form $q_{F}$ produces the self-adjoint operator $F_{V_{n}}$ whose initial $m$ eigenvalues in Theorem 6.1 are denoted through $\hat{\mu}^{(n)}_{\text{min}}=\hat{\mu}^{(n)}_{1}\leqslant\ldots\leqslant\hat{\mu}^{(n)}_{m}$. We can apply Theorem 5.2 to the subspace 6.3. Applying this theorem, we get

[TABLE]

Moreover, in our case the operator $F$ is such that for all $\mu_{k}(F)$ from 6.19 the first of the two alternative options from Theorem 3.1 is realized. This means that

[TABLE]

Comparing 6.19 and 6.20, we derive

[TABLE]

In order to complete the proof of Theorem 6.1 we compare the subspaces 6.3 and 6.9 along with the self-adjoint operators $F_{V_{n}}$ and $F_{\Cal{V}_{n}}$ in them. It is easy to see that $\Cal{V}_{n}\subset V_{n}$. If we denote through $\Cal{P}_{n}$ the orthogonal projector onto the smaller subspace $\Cal{V}_{n}$, we can easily derive the relationship

[TABLE]

Due to the inclusion $\Cal{V}_{n}\subset V_{n}$ and the relationship 6.22 we can apply Theorem 5.1 to the subspace $\Cal{V}_{n}$ and to the operator $F_{V_{n}}$ in the enclosing subspace $V_{n}$. Applying this theorem, we derive the inequalities

[TABLE]

From 6.21 and 6.23 we derive double inequalities:

[TABLE]

Now, applying 6.18 to 6.24, we get the required result expressed by the formula 6.2. Theorem 6.1 is proved. ∎

7. Conclusions

The main result of this paper is expressed by Theorem 6.1. As a mathematical result it has independent value. As for its application to quantum mechanics and quantum chemistry, it means that the Rayleigh-Ritz method can be used not only for computing the ground energy level of atoms and molecules, but for several consequtive excited energy levels along with the ground level. A way of applying Theorem 6.1 to quantum chemistry using polylinear splines is described in [9].