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\title{\sc An infinite dimensional version of the Schur convexity property and applications
\thanks{Correspondence address:
Vicen\c tiu R\u adulescu, Department of Mathematics, University of Craiova,
200585 Craiova, Romania. E-mail: {\tt vicentiu.radulescu@math.cnrs.fr}} }
\author{Claude VALL\'EE$\,^a$ and
Vicen\c tiu R\u ADULESCU$\,^b$\\
\small $^a$ Laboratoire de M\'ecanique des Solides, UMR 6610,
Universit\'e de Poitiers,\\
\small    Boulevard Marie et Pierre Curie,  BP 30179,
86962 Futuroscope Chasseneuil,  France\\ \small E-mail: {\tt
vallee@lms.univ-poitiers.fr}\\
\small $^b$ Department of Mathematics, University of Craiova,
200585 Craiova, Romania\\ \small E-mail: {\tt vicentiu.radulescu@math.cnrs.fr}}

\date{}

\begin{document}
\baselineskip16pt \maketitle
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\catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12

\begin{abstract} We extend to infinite dimensional separable Hilbert spaces the Schur convexity %%@
property
of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of %%@
linear, selfadjoint, compact operators, and that of linear selfadjoint operators that can be %%@
approximated by operators of finite rank
and having a countable family of eigenvalues. The abstract
results of the present paper are illustrated  by several examples from mechanics or quantum %%@
mechanics, including the Sturm-Liouville problem, the Schr\"odinger equation, and the harmonic %%@
oscillator.
\\
{\bf 2000 Mathematics Subject Classification}: 35E10, 35P15, 47A75, 47F05, 52A40. \\
{\bf Key words}: Schur convexity, selfadjoint operator, convex function, eigenvalue, Schr\"odinger %%@
equation.
\end{abstract}




\section{Introduction}
An important notion in the finite dimensional theory of convex functions is that of {\it Schur %%@
convexity}. 
Roughly speaking, Schur-convex functions are real-valued mappings which
are monotone with respect to the majorization ordering. A rigorous definition is stated in what %%@
follows.
Let $\RR^n_\geq$ denote the cone of vectors with nonincreasing components, that is,
$$\RR^n_\geq=\{x=(x_1,x_2,\ldots ,x_n)\in\RR^n;\ x_1\geq x_2\geq \ldots\geq x_n\}\,.$$
The dual cone of the cone $\RR^n_\geq$ is defined by
$$(\RR^n_\geq)^+=\left\{ y\in\RR^n;\  (x,y) \geq 0\mbox{ for all }x\in\RR^n_\geq \right\}\,.$$
A straightforward computation shows that
$$(\RR^n_\geq)^+=\left\{ y\in\RR^n;\  \sum_{i=1}^j y_i\geq 0\mbox{ for all }j=1,\ldots ,n-1 \mbox{ %%@
and }
 \sum_{i=1}^ny_i=0 \right\}\,.$$

We recall (see, e.g., Roberts and Varberg \cite{roberts}, Borwein and Lewis \cite{borlew}) that a %%@
function $f:\RR^n\ri\RR$ is {\it Schur convex}
if it is $(\RR^n_\geq)^+$-isotone, that is,
$$x,y\in \RR^n_\geq,\ y-x\in (\RR^n_\geq)^+\Longrightarrow f(x)\leq f(y)\,.$$

The Schur-convex functions were introduced by Schur \cite{schur23} in 1923  and they have many
important applications in analytic inequalities. Hardy, Littlewood and P\'olya \cite{hardylp}
were also interested in some
inequalities that are related to Schur-convex functions.
The notion of Schur-convexity has shown its importance in many domains. For instance, Merkle %%@
proved in \cite{merkle}
that if $I\subset\RR$ is an interval and $f:I\ri\RR$ is differentiable, then $f'$ is convex
if and only if the mapping
$$F(x,y)=\left\{
\begin{array}{lll}
&\di \frac{f(y)-f(x)}{y-x}\qquad &\di\mbox{if $y\not=x$}\\
& f'(x)\qquad &\di\mbox{if $y=x$}
\end{array}\right.
$$
is Schur convex. This property is applied in order to obtain some
inequalities for the ratio of Gamma functions. We also refer to
Hwang and  Rothblum \cite{hwa}, who study
optimization problems over partitions of a finite set and
obtain conditions that allow for simple constructions of partitions that are uniformly optimal for %%@
all Schur convex functions. Stochastic Schur convexity properties have been established by Shaked, %%@
Shanthikumar and  Tong 
\cite{shaked}. Exciting results such as
Schur's analytic criteria for Schur convexity, equivalence with Muirhead's inequality, 
majorization and stochastic matrix conditions in $\RR^n$, and Schur's majorization inequality
can be found in the excellent book by Steele \cite{steele}. Recently, Guan \cite{guan} has proved %%@
that the complete
elementary symmetric function $c_r=c_r(x)=\sum_{i_1+\ldots +i_n=r}x_1^{i_1}\ldots x_n^{i_n}$ and %%@
the function
$c_r(x)/c_{r-1}(x)$ are Schur-convex functions in $\RR^n_+=\{(x_1,\ldots ,x_n);\ x_i>0\}$, where %%@
$r$ is a positive 
integer and $i_1,\ldots ,i_n$ are nonnegative integers.

Zhang \cite{zhangx} proved that every Schur-convex function $f:D\subset\RR^n\ri\RR$ is a symmetric %%@
function,
that is, $f(x)=f\left(x_{\sigma (1)},\ldots ,x_{\sigma (n)}\right)$ 
for any permutation $\sigma\in{\cal P}_n$ and for all $x=(x_1,\ldots ,x_n)\in D$. 
The converse is not true (see, e.g.,
Roberts and Varberg \cite[p.~258]{roberts}). However, if $I$ is an open interval and $f:I^n\ri\RR$ %%@
is symmetric and 
of class $C^1$, then  $f$ is Schur-convex if and only if
$$(x_i-x_j)\left( \frac{\partial f}{\partial x_i}-\frac{\partial f}{\partial x_j}\right)\geq %%@
0\qquad\mbox{on $I^n$},$$
for all $i,j\in\{1,\ldots ,n\}$ (see Roberts and Varberg \cite[p.~259]{roberts}).


Eigenvalues of real
symmetric matrices exhibit remarkable convexity properties.
Let ${\bf S}^n$ denote the set of all 
symmetric matrices $X\in {\cal M}_{n,n}(\RR)$. 
In Borwein and Lewis \cite[p.~108]{borlew} it is stated the following elementary property of 
eigenvalues of $X\in {\bf S}^n$.

\smallskip
{\bf The Schur Convexity Property.} {\sl Let
$\lambda_1(X)\geq \lambda_2(X)
\geq\ldots\geq \lambda_n(X)$ be the eigenvalues (counted by multiplicity)
of an arbitrary matrix $X\in {\bf S}^n$. Assume that $\mu =(\mu_1,\mu_2,\ldots ,\mu_n)\in %%@
\RR^n_\geq$. 
Then the functional $\varphi (X)=\sum_{i=1}^n\mu_i\lambda_i(X)$ is sublinear.
}

\smallskip
A direct consequence of this result is that the mapping $\varphi :{\bf S}^n\ri\RR$ is convex.

In the particular case $\mu_1=\ldots =\mu_k=1$, $\mu_{k+1}=\ldots =\mu_n=0$ 
($1\leq k\leq n$), we deduce that the sum of the largest $k$  eigenvalues of a matrix $X\in {\bf %%@
S}^n$ is a convex function. 
An alternative proof is based on the observation that, for any fixed $1\leq k\leq n$,
\begin{equation}\label{01}
\lambda_1(X)+\lambda_2(X)+\ldots +\lambda_k(X)=\sup_{A\in{\cal A}}{\rm tr}\, (AA^TX)\,,
\end{equation}
where $${\cal A}=\left\{A\in{\cal M}_{n,k}(\RR);\ A^TA={\rm I}_k\right\}\,.$$
Since ${\cal A}$ is a compact set, the supremum in \eq{01} is attained in ${\cal A}$.
We deduce that the mapping ${\bf S}^n\ni X\longmapsto\lambda_1(X)+\lambda_2(X)+\ldots %%@
+\lambda_k(X)$
is convex, as a supremum of linear functions on ${\bf S}^n$.
The extreme situations $k=1$ and $k=n$ show that both the
largest eigenvalue of $X$ and the trace of $X$ are convex functions on ${\bf S}^n$. We also %%@
deduce, by taking differences, that $\sum_{j=k+1}^n\lambda_j(X)$ is a concave function, for all %%@
$1\leq k\leq n-1$. In particular, the mapping ${\bf S}^n\ni X\longmapsto\lambda_n(X)$ is concave. 

A classical result (see, e.g., Borwein and Lewis \cite{borlew}, and
Rockafellar and  Wets \cite{rock}) asserts that Schur convex functions are precisely
restrictions to $\RR^n_\geq$ of symmetric convex functions.
This result is strictly related to the class of convex functions $f:{\bf S}^n\ri\RR$
(like the functions $\sum_{j=1}^k\lambda_j(X)$) depending only on the eigenvalues of $X$.  
In fact, if we write diag$\,(\lambda)$
(where $\lambda=(\lambda_1,\ldots ,\lambda_n)\in\RR^n$) 
for the diagonal matrix with diagonal entries $\lambda_1,\ldots ,\lambda_n$, and define a function %%@
$\Phi:\RR^n\ri\RR$
by $\Phi (\lambda)=f({\rm diag}\,(\lambda))$, then $\Phi$ is convex and symmetric: $\Phi %%@
(\lambda)=
\Phi (\sigma\circ\lambda)$ for all permutation $\sigma\in{\cal P}_n$. The converse is also true: %%@
if $\Phi:\RR^n\ri\RR$
is a symmetric convex function then the function $f:{\bf S}^n\ri\RR$ defined by $f(X)=\Phi %%@
(\lambda(X))$ 
(where $\lambda (X)=(\lambda_1(X),\ldots ,\lambda_n(X))^T$) is convex and satisfies
$f(U^*XU)=f(X)$ whenever $U\in {\cal M}_{n,n}(\RR)$ is a unitary matrix. The above result is due %%@
to
Davis \cite{davis}. 

The above considerations show that it is natural to impose an adequate ``symmetry" assumption in %%@
order to obtain a
Schur convexity property for linear operators defined on arbitrary Hilbert spaces. That is why we %%@
consider throughout this paper linear selfadjoint operators defined on infinite dimensional %%@
Hilbert spaces.



\section{A Schur convexity property in Hilbert spaces}
In the first part of this section we establish an infinite dimensional version of the Schur %%@
convexity property
for linear, selfadjoint and compact operators defined on separable Hilbert spaces. Next, we extend %%@
this 
property to the class of linear selfadjoint operators that can be approximated by operators of %%@
finite rank. Several
examples from mechanics and quantum mechanics illustrate both cases.

\subsection{Schur convexity property for selfadjoint, compact operators}
Let $H$ be a separable Hilbert space and assume that $S:H\ri H$ is a linear, selfadjoint and %%@
compact operator. 
Since $S$ is compact then, by the Riesz-Schauder theorem (Theorem VI.15 in Reed and Simon 
\cite{reed}), the spectrum $\sigma (S)$
of $S$ is a discrete set having no limit points except perhaps the origin. Moreover, any %%@
$\lambda\in\sigma (S)\setminus
\{0\}$ is an eigenvalue of finite multiplicity. Next, the classical spectral theory of compact %%@
selfadjoint operators
(see, e.g., Brezis \cite[Proposition VI.9]{brezis}) ensures that $\sigma (S)\subset [m,M]$ and $m, %%@
M\in\sigma (S)$,
where $m=\inf\{ (Su,u);\ u\in H,\ \|u\|=1\}$ and $M=\sup\{ (Su,u);\ u\in H,\ \|u\|=1\}$. In %%@
conclusion, the spectrum of $S$ is discrete and it consists of a countable family of eigenvalues %%@
$\left(\lambda_n(S)\right)_{n\geq 1}$ with the additional
property that $\lambda_n(S)\ri 0$ as $n\ri\infty$. At this stage,
the Hilbert-Schmidt theorem (Theorem VI.16 in Reed and Simon \cite{reed}) 
implies that there is a complete orthonormal
basis $(e_n)_{n\geq 1}$ of $H$ such that $Se_n=\lambda_ne_n$ for all $n\geq 1$, where %%@
$\lambda_n=\lambda_n(S)$. So,
$Sx=\sum_{n=1}^\infty \lambda_n(x,e_n)e_n$, for all $x\in H$.

We observe that for any fixed positive integer $n$, the set 
$$\left\{\lambda\in\sigma (S);\ |\lambda|\geq\frac 1n\right\}$$
is either empty or finite. Thus, we can rearrange the eigenvalues of $S$ such that 
\begin{equation}\label{laspectr}
\lambda_1(S)\geq \lambda_2(S)\geq\ldots\geq \lambda_n(S)\geq\ldots >0>\ldots %%@
\geq\lambda_{-n}(S)\geq\ldots
\geq\lambda_{-2}(S)\geq\lambda_{-1}(S)\,.\end{equation}
Moreover, the unique limit point of the sequence $\left(\lambda_n(S)\right)_{n\in\ZZ}$ is 0.  
If $S$ has a finite number of negative eigenvalues (say, $n$), we denote them by
$\lambda_{-1}(S)\leq \ldots\leq \lambda_{-n}(S)$ and we set $\lambda_{-k}(S)$ for all $k\leq n+1$. 
We make a similar convention if $S$ has finitely many positive eigenvalues.
If $0$ is an eigenvalue of $S$, we denote $\lambda_0 (S)=0$.

Denote by $\kplus$ the vector space of linear, selfadjoint and compact operators $S:H\ri H$.

We prove the following infinite dimensional variant of  Schur's convexity property.

\begin{theorem}\label{th1}
Let $H$ be a separable Hilbert space and assume that $S:H\ri H$ is an arbitrary compact %%@
selfadjoint operator. 
Assume that the eigenvalues of $S$ are arranged as in \eq{laspectr} and let 
$(\mu_n)_{n\in \ZZ}$ be real numbers such that $\mu_1\geq\mu_2\geq\ldots\geq\mu_n\geq\ldots\geq %%@
\mu_{-n}\geq\ldots
\geq\mu_{-2}\geq\mu_{-1}$ and $\sum_{n=-\infty}^\infty
\mu_n$ is an absolutely convergent series.

Then the functional $\psi :\kplus\ri\RR$ defined by $\psi %%@
(S)=\sum_{n=-\infty}^\infty\mu_n\lambda_n(S)$ is convex
and lower semicontinuous.
\end{theorem}

\proof 
We first observe that since $S\in\kplus$ is not assumed to be a nuclear operator, then the series %%@
$\sum_{n\in\ZZ}
\lambda_n(S)$ is not necessarily convergent. However, our hypothesis that the series %%@
$\sum_{n=-\infty}^\infty
\mu_n$ is absolutely convergent implies that the series $\sum_{n=-\infty}^\infty\mu_n\lambda_n(S)$ %%@
is absolutely convergent, too,
so the mapping $\psi$ is well-defined. Indeed, for all $S\in\kplus$,
$$|\psi (S)|\leq \sum_{n=-\infty}^\infty |\mu_n|\cdot |\lambda_n(S)|\leq
\max\left\{-\lambda_{-1}(S),\lambda_1(S)\right\}\sum_{n=-\infty}^\infty |\mu_n|<\infty\,.$$

Any operator $S\in\kplus$ is the norm limit of a sequence of operators of finite rank. Indeed, if %%@
$(e_n)_{n\in\ZZ}$
is a complete orthonormal basis of $H$ so that $Se_n=\lambda_n(S)e_n$ for all $n\in\ZZ$, with %%@
$\lambda_n(S)$
arranged as in \eq{laspectr}, then 
$Sx=\sum_{n=-\infty}^\infty \lambda_n(S)(x,e_n)e_n$, for all $x\in H$. Set, for any $m\geq 1$,
$S_mx=\sum_{j=-m}^m \lambda_j(S)(x,e_j)e_j$, for all $x\in H$. Then $S_m\ri S$ in $L(H)$ as %%@
$m\ri\infty$ and the (nontrivial) eigenvalues of $S_m$ are 
$\lambda_1(S)\geq \ldots\geq \lambda_m(S)>0>\lambda_{-m}(S)\geq\ldots\geq\lambda_{-1}(S)$. 
So, by the finite-dimensional Schur convexity property,
the mapping
$$\psi_m :\kplus\ri\RR\,,\qquad\qquad \psi_m (S)=\sum_{j=-m}^m\mu_j\lambda_j(S)$$ is sublinear. 
So, for any $S,\, T\in\kplus$ and all $\alpha\in\RR$,
\begin{equation}\label{1}
\psi_m(S+T)\leq\psi_m(S)+\psi_m(T)\qquad\mbox{and}\qquad\psi_m(\alpha S)=|\alpha|\,\psi_m(S)\,.
\end{equation}

On the other hand,
$$\left| \psi (S)-\psi_m(S)\right|=\left| \sum_{|j|\geq m+1}
\mu_j\lambda_j(S) \right|\leq
\max\left\{-\lambda_{-m-1}(S),\lambda_{m+1}(S)\right\}\sum_{|j|\geq m+1} |\mu_j|\,.$$
Therefore 
\begin{equation}\label{2}\psi_m(S)\ri\psi (S)\qquad \mbox{as $m\ri\infty$}.\end{equation}
Thus, by \eq{1} and \eq{2}, $\psi$ is a sublinear functional. In particular, $\psi$ is convex.

It remains to argue that $\psi$ is lower semicontinuous, that is, $\psi %%@
(S)\leq\liminf_{n\ri\infty}
\psi (S_n)$ for all $S\in\kplus$, provided $S_n\in\kplus$ and $\|S_n-S\|\ri 0$ as $n\ri\infty$.
The key ingredient is Theorem 4.2 in Gohberg-Krein \cite{gohkre}, which asserts that
$\lambda_j(S)=\lim_{n\ri\infty}\lambda_j(S_n)$. Fix an integer $m\geq 1$ and choose arbitrarily
$0<\ep <\max\left\{-\lambda_{-m}(S),\lambda_m(S)\right\}$. 
It follows that there exists $N_0=N_0(\ep)\in\NN$ such that,
for all $n\geq N_0$,
$$\begin{array}{ll}\di \psi_m(S)&\di=\sum_{j=-m}^m\mu_j\lambda_j(S)=
\sum_{j=-m}^m\mu_j^+\lambda_j(S)-\sum_{j=-m}^m\mu_j^-\lambda_j(S)\\
&\di\leq \sum_{j=-m}^m\mu_j^+\left(\lambda_j(S_n)+\ep\right)-
\sum_{j=-m}^m\mu_j^-\left(\lambda_j(S_n)-\ep\right)\\ &\di %%@
=\sum_{j=-m}^m\mu_j\lambda_j(S_n)+\ep\sum_{j=-m}^m|\mu_j|=
\psi_m(S_n)+\ep \sum_{j=-m}^m|\mu_j|\,.\end{array}$$
Taking $\ep\ri 0$ we obtain $\psi_m(S)\leq \psi_m(S_n)$, for all positive integers $m$ and $n$.
So, for all $n\geq 1$,
$$\psi (S)=\lim_{m\ri\infty} \psi_m(S_n)\leq\lim_{m\ri\infty} \psi_m(S_n)=\psi (S_n)\,.$$
We deduce that $\psi (S)\leq\liminf_{n\ri\infty}
\psi (S_n)$ and the proof is concluded. \qed

\medskip
{\bf Examples.} 1. {\it Sturm-Liouville differential operators}. Many eigenvalue problems in %%@
quantum mechanics as well as classical physics are described by the Sturm-Liouville problem
\begin{equation}\label{sturm}
\left\{
\begin{array}{ll}
&\di -\frac{d}{dx}\left(p(x)\frac{dy}{dx}\right)+q(x)y=\Lambda y\qquad \mbox{in $(0,L)$}\\
&\di y(0)=y(L)=0\,,
\end{array}\right.
\end{equation}
where $y(x)$ is the quantum mechanical wave function or other physical quantity, while $p\in %%@
C^1[0,L]$
($p>0$ in $[0,L]$) and $q\in C[0,L]$ are given functions that are determined by the nature of the %%@
system of interest.
We can assume, without loss of generality, that $q\geq 0$ in $[0,L]$. Indeed, if not, we choose %%@
$C\in\RR$
sufficiently large such that $q+C\geq 0$ in $[0,L]$ (in such a case, $\Lambda$ is replaced by %%@
$\Lambda +C$ in
\eq{sturm}). Fix $f\in L^2(0,L)$. Thus, by the Lax-Milgram lemma, there exists a unique
$u\in H^2(0,L)\cap H^1_0(0,L)$ such that
$$
\left\{
\begin{array}{ll}
&\di -\frac{d}{dx}\left(p(x)\frac{dy}{dx}\right)+q(x)y=f\qquad \mbox{in $(0,L)$}\\
&\di y(0)=y(L)=0\,.
\end{array}\right.
$$
Let $S:L^2(0,L)\ri L^2(0,L)$ be the operator defined by $Sf=u$. Then, by Theorem VIII.20 in Brezis %%@
\cite{brezis}, 
$S$ is linear, selfadjoint, compact, and
nonnegative. Let $\lambda_1(S)\geq\lambda_2(S)\geq\ldots\geq \lambda_n(S)\geq\ldots >0$ denote the %%@
eigenvalues of
$S$. Then $\Lambda_n(S)=1/\lambda_n(S)$ is an eigenvalue corresponding to the Sturm-Liouville %%@
problem \eq{sturm}.
In the particular case $p\equiv 1$ and $q\equiv 0$, a straightforward computation shows that %%@
$\lambda_n(S)=L^2(n^2\pi^2)^{-1}$.

Let $\mu_n$ ($n\geq 1$) be real numbers such that $\mu_i\geq \mu_j$ if $i<j$
and such that the series $\sum_{n=1}^\infty\mu_n$ converges absolutely. So, by Theorem \ref{th1}, %%@
the mapping
$$S\longmapsto\sum_{n=1}^\infty
\mu_n\lambda_n(S)$$
is convex and lower semicontinuous.

\smallskip
2. {\it The electron atom model}. On the Hilbert space $H=L^2(\RR^3)$, let $x$, $y$, $z$ be the %%@
components of
the momentum of the electron and denote by $r=(x,y,z)$ its position. Consider on $H$ the %%@
selfadjoint operator
$$S=\Delta +\frac{\alpha}{|r|}\,,\qquad  \qquad |r|=\sqrt{x^2+y^2+z^2}\,.$$
Notice that the potential $V(|r|)=\alpha /|r|$ is the energy of the electric field surrounding the %%@
electron, $\alpha$
depends on the electron's charge, and $|r|$ is its distance from the atom's nucleus.
As established in Reed and Simon \cite{reed4}, $S$ has no eigenvalues for any $\alpha<0$ and, if %%@
$\alpha>0$,
then all eigenvalues of $S$ are
$$\lambda_n(S)=\frac{\alpha}{4n^2}\,,\qquad \qquad n=1,2,\ldots$$

Let $(\mu_n)_{n\geq 1}$ be a sequence of real numbers such that $\mu_1\geq %%@
\mu_2\geq\ldots\geq\mu_n\geq\ldots$
and the series $\sum_{n=1}^\infty\mu_n$ converges absolutely. So, by Theorem \ref{th1}, the %%@
mapping
$$S\longmapsto\sum_{n=1}^\infty
\mu_n\lambda_n(S)$$
is convex and lower semicontinuous. 

\smallskip
3. {\it Nonrelativistic model for $2$-electron atom}.
Set $H=L^2(\RR^6)$ and define on $H$ the selfadjoint operator
$$S=\Delta_1 +\frac{\alpha}{|r_1|}+\Delta_2 +\frac{\beta}{|r_2|}\,,$$
where $\alpha, \beta>0$, $r_k=(x_k,y_k,z_k)$,  
and 
$$\Delta_k=\frac{\partial^2}{\partial x_k^2}+\frac{\partial^2}{\partial y_k^2}+
\frac{\partial^2}{\partial z_k^2}\,,\qquad \mbox{for all $k=1,2$}.$$

Cf. Reed and Simon \cite{reed4}, the eigenvalues of $S$ are precisely
$$\lambda_{n,m}(S)=\frac{\alpha}{4n^2}+\frac{\beta}{4m^2}\,,\qquad\qquad n,m=1,2,\ldots$$

The countable family of positive numbers $(\lambda_{n,m}(S))_{n,m\geq 1}$ can be rearranged in a %%@
sequence
$(\gamma_p(S))_{p\geq 1}$ such that $\gamma_i(S)\geq\gamma_j(S)$ provided $i<j$.
Let $(\mu_p)_{p\geq 1}$ be a sequence of real numbers such that $\mu_1\geq %%@
\mu_2\geq\ldots\geq\mu_p\geq\ldots$
and the series $\sum_{p=1}^\infty\mu_p$ converges absolutely. 
Thus, by Theorem \ref{th1}, the mapping
$$S\longmapsto\sum_{p=1}^\infty
\mu_p\gamma_p(S)$$
is convex and lower semicontinuous.

\smallskip
4. {\it Schr\"odinger operators with periodic potential}. 
The basic equation of quantum mechanics is the Schr\"odinger equation
\begin{equation}\label{erwin1}
i\hbar\psi_t=-\frac{\hbar^2}{2m}\,\Delta\psi+V(x)\psi\,.\end{equation}
Schr\"odinger \cite{schre} studied the stationary equation
\begin{equation}\label{erwin2}
\lambda\varphi=-\frac{\hbar^2}{2m}\,\Delta\varphi+V(x)\varphi\,,\end{equation}
which follows from \eq{erwin1} through $\psi(x,t)=\varphi (x)e^{-i\lambda t/\hbar}$. From %%@
\eq{erwin2}
Schr\"odinger derived the spectrum of the hydrogen atom. In this case, $V$ is the potential of
the electrostatic attracting force of the atomic nucleus, while from the eigenvalues $\lambda$
of \eq{erwin2} one obtains the energy levels of the electron of the hydrogen atom. 

Solutions of Schr\"odinger's equation have to fulfill strict conditions to be useful in describing %%@
the electron. Some of the solutions are associated with special values of the electron's energy %%@
level, known as eigenvalues.
We consider in what follows the class of piecewise continuous potential
functions $V:\RR\ri\RR$ which are periodic of period $2\pi$. Let $S$ denote 
the one dimensional Schr\"odinger operator associated to $V$ defined on $L^2_{\rm per}(\RR)$ 
 with $2\pi$-periodic conditions. 
This operator is defined as follows: for any $f\in L^2_{\rm per}(\RR)$ periodic of period $2\pi$, %%@
let $u\in H^1_{\rm per}(\RR)$ be the unique solution of the problem 
$$
\left\{
\begin{array}{ll}
&\di -u''+V(x)u=f\qquad \mbox{in $(0,2\pi)$}\\
&\di u(0)=u(2\pi),\ u'(0)=u'(2\pi)\,.
\end{array}\right.
$$
Then $S$ is defined by $L^2_{\rm per}(\RR)\ni f\longmapsto u=Sf\in L^2_{\rm per}(\RR)$.
 According to Theorem XIII.89 in Reed and Simon \cite{reed4}, $S$ has a countable family of %%@
eigenvalues
$\lambda_1(S)>\lambda_2(S)>\ldots >\lambda_n(S)>\ldots$ and $\lambda_n(S)\ri 0$ as $n\ri\infty$.
Assume that $\mu_n$ ($n\geq 1$) are real numbers such that $\mu_i\geq \mu_j$ if $i<j$
and such that the series $\sum_{n=1}^\infty\mu_n$ converges absolutely. So, by Theorem \ref{th1}, %%@
the mapping
$$S\longmapsto\sum_{n=1}^\infty\mu_n\lambda_n(S)$$
is convex and lower semicontinuous.

\smallskip
5. {\it Indefinite weight elliptic problems on the whole space}. Consider the class of measurable %%@
functions
$V:\RR^N\ri\RR$ ($N\geq 3$) such that $V^+\in L^{N/2}(\RR^N)$, where $V=V^+-V^-$. We observe that %%@
this class contains
potentials $V$ satisfying $V^+(x)\leq C(1+|x|^2)^{-\alpha}$ for all $x\in\RR^N$, where $\alpha >1$ %%@
and $C$
is a positive constant. For some fixed $\lambda>0$, let $E$ be the completion of %%@
$C^\infty_0(\RR^N)$ with
respect to the norm
$$\|u\|^2=\int_{\RR^N}\left[|\nabla u|^2+\max \left(\lambda V^-,\omega\right)u^2\right]dx\,,$$
where $\omega (x)=K(1+|x|^2)^{-1}$ with $K>0$ sufficiently small. Then, by Lemma~0 in Allegretto %%@
\cite{alegr},
the operator $S:E\ri E^*\hookrightarrow E$ defined by $S\varphi =V^+\varphi$ is compact and %%@
selfadjoint. Next, by Theorem~1 in Allegretto \cite{alegr}, there exist infinitely many %%@
eigenvalues $\lambda_1(S)>\lambda_2(S)\geq\ldots
\geq \lambda_n(S)\geq \ldots\geq 0$ of $S$ with $\lambda_n(S)\ri 0$ as $n\ri\infty$. So, if
$\mu_n$ ($n\geq 1$) are real numbers such that $\mu_i\geq \mu_j$ if $i<j$
and $\sum_{n=1}^\infty |\mu_n|<\infty$ then, by Theorem \ref{th1}, the mapping
$S\longmapsto\sum_{n=1}^\infty
\mu_n\lambda_n(S)$
is convex and lower semicontinuous.



\subsection{A more general framework}
Consider the class $\kdoi$ of linear selfadjoint operators $S:H\ri H$ having a countable family of %%@
eigenvalues and such that $S$ can be approximated by operators of finite rank. For any operator %%@
$S\in\kdoi$, passing eventually at a 
rearrangement, let $\lambda_1(S)\geq \lambda_2(S)\geq\ldots \geq\lambda_n(S)\geq\ldots$ denote the %%@
eigenvalues of $S$.

Fix a family $\mu=(\mu_1,\mu_2,\ldots ,\mu_n,\ldots)$ of real numbers such that $\mu_i\geq\mu_j$ %%@
if $i<j$. 
Consider the class $\kdoimu$ of operators $S\in\kdoi$ such that the series
$\sum_{n=1}^\infty\mu_n\lambda_n(S)$ converges.

Under these hypotheses, we establish the following infinite dimensional version of the Schur %%@
convexity property.

\begin{theorem}\label{th2}
The functional $\psi :\kdoimu\ri\RR$ defined by $\psi (S)=\sum_{n=1}^\infty\mu_n\lambda_n(S)$ is %%@
convex
and lower semicontinuous.
\end{theorem}

\proof By the definition of $\kdoimu$, for any operator belonging to this class there exists
a sequence $(S_n)_{n\geq 1}$ of operators of finite rank such that $\| S_n-S\|\ri 0$ as %%@
$n\ri\infty$.
So, by Theorem 4.2 in Gohberg-Krein \cite{gohkre}, we have
$\lim_{n\ri\infty}\lambda_j(S_n)=\lambda_j(S)$, for all positive integer $j$. Define, for all %%@
$m\geq 1$,
$$\psi_m :\kdoimu\ri\RR\,,\qquad\qquad \psi_m (S)=\sum_{j=1}^m\mu_j\lambda_j(S)\,.$$
Therefore
\begin{equation}\label{bre1}
\lim_{n\ri\infty}\sum_{j=1}^m\mu_j\lambda_j(S_n)=\sum_{j=1}^m\mu_j\lambda_j(S)=\psi_m(S)\,.
\end{equation}
On the other hand, since $S\in\kdoimu$,
\begin{equation}\label{bre2}
\lim_{m\ri\infty}\psi_m(S)=\sum_{j=1}^\infty\mu_j\lambda_j(S)=\psi(S)\,.
\end{equation}

Let $S,T\in\kdoimu$ and assume that $S_n,T_n$ are operators of finite rank such that $\|S_n-S\|\ri %%@
0$ and
$\|T_n-T\|\ri 0$ as $n\ri\infty$. Applying the Schur convexity property we obtain
$$\psi_m(S_n+T_n)\leq\psi_m(S_n)+\psi_m(T_n)\,,\qquad\mbox{for all $m,n\geq 1$}\,.$$
Taking $n\ri\infty$ and using \eq{bre1} we find
$$\psi_m(S+T)\leq\psi_m(S)+\psi_m(T)\,,\qquad\mbox{for all $m\geq 1$}\,.$$
Next, by \eq{bre2}, we deduce that 
$$\psi(S+T)\leq\psi(S)+\psi(T)\,,\qquad\mbox{for all $S,T\in\kdoimu$}\,.$$
A similar argument shows that $\psi$ is positive homogeneous.

The lower semicontinuity of $\psi$ follows with the same arguments as in the proof of Theorem %%@
\ref{th1}. \qed

\medskip
{\bf Examples.} 1. {\it Schr\"odinger operators with arbitrary potential}. 
Let $H_0$ denote the differential operator $d^2/dx^2$ on $L^2(0,1)$ with the boundary conditions
$u(0)=u(1)=0$ and assume that $V\in L^\infty (0,1)$ is an arbitrary potential. Let $\lambda_n(S)$ %%@
be the $n$th 
eigenvalue of the operator $S=H_0+V$. Then, by Theorem XIII.82.5 in Reed and Simon \cite{reed4},
\begin{equation}\label{erwin4}
\lambda_n(S)=-n^2\pi^2+\int_0^1V(x)dx+o(1)\qquad\mbox{as $n\ri\infty$}.\end{equation}

Fix the real numbers $\mu_n$ ($n\geq 1$) such that $\mu_i\geq \mu_j$ if $i<j$ and the series %%@
$\sum_{n=1}^\infty\mu_n\lambda_n(S)$ converges. 
Using the asymptotic estimate \eq{erwin4}, we deduce that, for the last purpose, it is enough to %%@
choose $\mu_n$ so that $\mu_n=O\left(n^{-p}\right)$, for some $p>3$. 
Then, by Theorem \ref{th2}, the mapping
$$S\longmapsto\sum_{n=1}^\infty\mu_n\lambda_n(S)$$
is convex and lower semicontinuous.

\smallskip
2. {\it Wave functions on infinite depth wells}. Fix arbitrarily the positive numbers $a$ and $b$. %%@
Define the
following discontinuous potential energy of a particle in the force field
$$V(x)=\left\{
\begin{array}{lll}
&\di -\infty\qquad &\mbox{if $x<-b$}\\
&\di 0\qquad &\mbox{if $-b<x<a$}\\
&\di -\infty\qquad &\mbox{if $x>a$}\,.
\end{array}\right.$$
Consider the Schr\"odinger equation 
$$\left\{
\begin{array}{ll}
&\di \frac{\hbar^2}{2m}\psi ''+V(x)\psi =\lambda\psi \\
&\di \psi (-b)=\psi (a)=0\,,
\end{array}\right.$$
where $m$ is the mass of the particle and $\hbar$ is Dirac's constant (reduced Planck's constant). 
Cf. Pluvinger \cite[p.~102]{pluvi}, the definition
of $V$ forces $\psi=0$ outside $(-b,a)$.
A straightforward computation shows that the eigenvalues of the associated operator $S$ are given %%@
by
$$\lambda_n(S)=-\frac{\hbar^2\pi^2}{2m(a+b)^2}\, n^2\,.$$

Fix the real numbers $\mu_n$ ($n\geq 1$) such that $\mu_i\geq \mu_j$ if $i<j$ and the series %%@
$\sum_{n=1}^\infty\mu_n\lambda_n(S)$ converges. 
The above expression of eigenvalues shows that it is enough to choose $\mu_n$ so that 
$\mu_n=O\left(n^{-p}\right)$, for some $p>3$. 
Applying Theorem \ref{th2}, we deduce that the mapping
$$S\longmapsto\sum_{n=1}^\infty\mu_n\lambda_n(S)$$
is convex and lower semicontinuous.

\smallskip
3. {\it Linear harmonic oscillator}. Consider the Schr\"odinger equation on the whole real axis
\begin{equation}\label{dirac7}\left\{
\begin{array}{ll}
&\di \frac{\hbar^2}{2m}\psi ''+V(x)\psi =\lambda\psi \\
&\di \lim_{|x|\ri\infty}\psi (x)=0=0\,.
\end{array}\right.\end{equation}
In the particular case where $V(x)=-m\omega^2x^2/2$ the above problem describes the linear %%@
harmonic oscillator. Cf. 
Pluvinger \cite[p.~74]{pluvi} the energy levels of the corresponding linear operator $S$ are given %%@
by
$\lambda_n(S)=-\hbar\omega(n+1/2)$. So, letting $(\mu_n)_{geq 1}$ so that $\mu_i\geq \mu_j$ if %%@
$i<j$ and such that the series $\sum_{n=1}^\infty\mu_n\lambda_n(S)$ converges, Theorem \ref{th2} %%@
implies that the mapping
$S\longmapsto\sum_{n=1}^\infty\mu_n\lambda_n(S)$
is convex and lower semicontinuous. 

We point out that in the case of Morse potentials $V(x)=V_0\left(e^{-2x/a}-2e^{-x/a}\right)$ the %%@
number of eigenvalues of the
problem \eq{dirac7} is finite.

\smallskip
4. {\it Periodic standing waves of Schr\"odinger's equation}.
In his Ph.D. thesis defended in 1923, de Broglie showed that an electron, or any other particle, %%@
has a wave associated with it. The second equation established by de Broglie establishes that the %%@
kinetic energy of a particle is directly proportional to its angular frequency. De Broglie's work %%@
resulted in the equation $\lambda=\hbar\omega$, where $\lambda$ is the kinetic energy of the %%@
associated wave and $\omega$ is the angular frequency of the particle. With the same notations as %%@
in the previous example, we consider the Schr\"odinger equation with periodic boundary conditions
$$\left\{
\begin{array}{ll}
&\di \frac{\hbar^2}{2m}\psi ''+V(x)\psi =\lambda\psi \qquad\mbox{in $(-b,a)$}\\
&\di \psi (-b)=\psi (a)\\ 
&\di \psi' (-b)=\psi ' (a) \,.
\end{array}\right.$$
Outside the fundamental segment of length $L=a+b$ the standing wave $\psi$ is prolonged by %%@
periodicity such that
$\psi (x+L)=\psi (x)$, for all $x\in\RR$. In Pluvinger \cite[p.~108]{pluvi}
it is provided a class of potentials $V$ for which the associated bound state energies to the %%@
above problem are given by 
$$\lambda_n(S)=-\frac{2\hbar\pi}{L}\, n \,.$$
Thus, by Theorem \ref{th2}, the mapping $S\longmapsto\sum_{n=1}^\infty\mu_n\lambda_n(S)$
is convex and lower semicontinuous, provided
$(\mu_n)_{n\geq 1}$ are chosen so that $\mu_i\geq \mu_j$ if $i<j$ and the series %%@
$\sum_{n=1}^\infty\mu_n\lambda_n(S)$ converges.

\smallskip
5. {\it Generalized model of the helium atom}. Let $S$ be the differential operator on %%@
$L^2(\RR^{3n})$ given by
$$S=\sum_{i=1}^{3n}\left(-\frac{\Delta_i}{2m_i}-\frac{n}{m_i}\right)+\sum_{i<j}\left(\frac{\nabla_%%@
%%@
i\cdot\nabla_j}{M}+\frac{1}{|r_i-r_j|}\right)\,,$$
where $M$ and $m_i$ ($1\leq i\leq n$) are arbitrary positive numbers. Cf. Reed and Simon %%@
\cite{reed4}, the above operator has been introduced by Zhislin and $S$ can be viewed as the %%@
Hamiltonian of a system consisting of a
nucleus of mass $M$ and $n$ electrons of masses $m_1,\ldots ,m_n$, after the center of the mass %%@
motion has been removed.
This model generalizes the elementary model of the helium atom which is described by the operator %%@
$S$ on
$L^2(\RR^6)$ given by
$$S=-\Delta_1-\Delta_2-\frac{2}{|r_1|}-\frac{2}{|r_2|}+\frac{1}{|r_1-r_2|}\,.$$
In both cases (see Kato's Theorem and Theorem XIII.7 in Reed and Simon \cite[p.~89]{reed4}) the %%@
operator $S$ has a countable family of eigenvalues which can be supposed to be arranged so that %%@
$\lambda_i(S)\geq\lambda_j(S)$ if $i<j$
(notice that $\lambda_1(S)<-1$ in the case of the elementary model of the helium atom).
Fix the real numbers $\mu_n$ ($n\geq 1$) such that $\mu_i\geq \mu_j$ if $i<j$ and the series %%@
$\sum_{n=1}^\infty\mu_n\lambda_n(S)$ converges. 
Thus, by Theorem \ref{th2}, the mapping
$S\longmapsto\sum_{n=1}^\infty\mu_n\lambda_n(S)$
is convex and lower semicontinuous. 

\smallskip
6. {\it Schr\"odinger operators with unbounded potential}. Let $V\in L^1_{\rm loc}(\RR^N)$ %%@
belonging to the class
of operators which are bounded from above and such that $V(x)\ri -\infty$ as $|x|\ri\infty$. Then, %%@
by Theorem XIII.67
in Reed and Simon \cite{reed4}, the Schr\"odinger operator $S=-\Delta +V$ has a countable family %%@
of eigenvalues such that
$$\lambda_1(S)\geq\ldots\geq\lambda_n(S)\geq\ldots\qquad\mbox{and $\lambda_n(S)\ri-\infty$ as %%@
$n\ri\infty$}\,.$$
Consider the real numbers $\mu_n$ ($n\geq 1$) such that $\mu_i\geq \mu_j$ if $i<j$ and the series %%@
$\sum_{n=1}^\infty\mu_n\lambda_n(S)$ converges. 
Applying Theorem \ref{th2}, we deduce that the mapping
$S\longmapsto\sum_{n=1}^\infty\mu_n\lambda_n(S)$
is convex and lower semicontinuous.

\smallskip
7. {\it Quasilinear anisotropic Sturm-Liouville problems}. Let $\alpha\geq 0$, $p>1$, and $0\leq %%@
a<b<\infty$. Assume that
$q, s\in L^\infty (a,b)$ and ${\rm ess}\,\inf_{x\in (a,b)}s(x)>0$. Consider the quasilinear %%@
anisotropic eigenvalue problem
\begin{equation}\label{sturmli}
\left\{\begin{array}{ll}
&\di r^{-\alpha}\left(r^\alpha |u'|^{p-2}u'\right)'+q(r)|u|^{p-2}u=\lambda s(r)|u|^{p-2}u\qquad
\di\mbox{in $(a,b)$}\\
&\di \gamma_1\left(|u|^{p-2}u\right)(a)+ \gamma_2\left(r^\alpha |u'|^{p-2}u'\right)(a)=0\\
&\di \gamma_3\left(|u|^{p-2}u\right)(b)+ \gamma_4\left(r^\alpha |u'|^{p-2}u'\right)(b)=0\,,
\end{array}\right.
\end{equation}
where $\gamma_i\in\RR$ ($i=1,\ldots ,4$) such that $\gamma_1^2+\gamma_2^2>0$ and %%@
$\gamma_3^2+\gamma_4^2>0$. 

We distinguish two cases: the regular case where $a>0$ or $a=0$ and $0\leq\alpha <p-1$, and the %%@
singular case defined by $a=0$, $\alpha\geq p-1$. In the singular case the boundary condition at %%@
the origin is $u'(0)=0$. In both cases Walter
\cite{walter} proved that problem \eq{sturmli} has a countable number of simple eigenvalues %%@
$\lambda_1(S)>\ldots >
\lambda_n(S)>\ldots$, $\lim_{n\ri\infty}\lambda_n(S)=-\infty$ and the corresponding eigenfunction %%@
$u_n$ has $n-1$ simple zeroes in $(a,b)$. Consider the real numbers $\mu_n$ ($n\geq 1$) such that %%@
$\mu_i\geq \mu_j$ if $i<j$ and the series $\sum_{n=1}^\infty\mu_n\lambda_n(S)$ converges. 
So, by Theorem \ref{th2}, the mapping
$S\longmapsto\sum_{n=1}^\infty\mu_n\lambda_n(S)$
is convex and lower semicontinuous.

\medskip
{\bf Conclusions.} In this paper we have extended the Schur convexity property of the eigenvalues
of a symmetric matrix with real entries in the framework of infinite dimensional Hilbert spaces.
First, we have considered the case of linear, selfadjoint, and compact operators. Next, we have
established a corresponding version of the Schur convexity property for linear selfadjoint %%@
operators that can be approximated by operators of finite rank
and having a countable family of eigenvalues. Our abstract results have been illustrated by %%@
various examples, including Sturm-Liouville problems, Schr\"odinger operators with variable %%@
potential, the electron atom model, the linear harmonic oscillator, the generalized model of the %%@
helium atom, and 
wave functions on infinite depth wells. We have been concerned with linear operators with 
discrete spectrum and our results do not cover the case of operators with a continuous spectrum.

\medskip
{\bf Acknowledgments.} This paper has been written while V. R\u adulescu was visiting
the Laboratoire de M\'ecanique des Solides, Universit\'e de Poitiers, in June 2006. 
V.~R\u adulescu was also partially supported by grants CEx 05-D11-36/2005,
CNCSIS 308/2006 and GAR 12/2006.


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\end{thebibliography}




\end{document}
