# Numerical approximation of the spectrum of self-adjoint continuously
invertible operators

Tomáš Gergelits Faculty of Mathematics and Physics, Charles University,
Prague, Czech Republic. Email<EMAIL_ADDRESS>This author was
supported by Lawrence Livermore National Security, LLC Subcontract Award
B639388 under Prime Contract No. DE-AC52-07NA27344. Bjørn Fredrik Nielsen
Faculty of Science and Technology, Norwegian University of Life Sciences, P.O.
Box 5003, NO-1432 Ås, Norway. Email<EMAIL_ADDRESS>Nielsen’s work
was supported by The Research Council of Norway, project number 239070. Zdeněk
Strakoš Faculty of Mathematics and Physics, Charles University, Prague, Czech
Republic. Email<EMAIL_ADDRESS>Supported by the Grant Agency of
the Czech Republic under the contract No. 17-04150J.

###### Abstract

This paper deals with the generalized spectrum of continuously invertible
linear operators defined on infinite dimensional Hilbert spaces. More
precisely, we consider two bounded, coercive, and self-adjoint operators
${\mathcal{A},B}:V\mapsto V^{\\#}$, where $V^{\\#}$ denotes the dual of $V$,
and investigate the conditions under which the whole spectrum of
${\mathcal{B}}^{-1}{\mathcal{A}}:V\mapsto V$ can be approximated to an
arbitrary accuracy by the eigenvalues of the finite dimensional discretization
${\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}$. Since
${\mathcal{B}}^{-1}{\mathcal{A}}$ is continuously invertible, such an
investigation cannot use the concept of uniform (normwise) convergence, and it
relies instead on the pointwise (strong) convergence of
${\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}$ to
${\mathcal{B}}^{-1}{\mathcal{A}}$.

The paper is motivated by operator preconditioning which is employed in the
numerical solution of boundary value problems. In this context,
${\mathcal{A}},{\mathcal{B}}:H_{0}^{1}(\Omega)\mapsto H^{-1}(\Omega)$ are the
standard integral/functional representations of the differential operators
$-\nabla\cdot(k(x)\nabla u)$ and $-\nabla\cdot(g(x)\nabla u)$, respectively,
and $k(x)$ and $g(x)$ are scalar coefficient functions. The investigated
question differs from the eigenvalue problem studied in the numerical PDE
literature which is based on the approximation of the eigenvalues within the
framework of compact operators.

This work follows the path started by the two recent papers published in [SIAM
J. Numer. Anal., 57 (2019), pp. 1369-1394 and 58 (2020), pp. 2193-2211] and
addresses one of the open questions formulated at the end of the second paper.

Keywords: Second order PDEs, bounded non-compact operators, generalized
spectrum, numerical approximation, preconditioning.

##  1 Introduction.

Extending the path of research started in [14, 7, 8], this paper will consider
the differential operators $-\nabla\cdot(k(x)\nabla u)$ and
$-\nabla\cdot(g(x)\nabla u)$ on the open and bounded Lipschitz domain
$\Omega\subset\mathbb{R}^{2}$, where the scalar functions $g(x)$ and $k(x)$
are uniformly positive and continuous throughout the closure
$\overline{\Omega}$. The associated operator representations
${\mathcal{A}},\,{\mathcal{B}}:H_{0}^{1}(\Omega)\mapsto H^{-1}(\Omega)$, are
given by

$\displaystyle\langle\mathcal{A}u,v\rangle=\int_{\Omega}k(x)\nabla
u\cdot\nabla v,\quad u,v\in H_{0}^{1}(\Omega),$ (1)
$\displaystyle\langle\mathcal{B}u,v\rangle=\int_{\Omega}g(x)\nabla
u\cdot\nabla v,\quad u,v\in H_{0}^{1}(\Omega).$ (2)

In the first part of this paper we characterize the spectrum of the
preconditioned operator

${\mathcal{B}}^{-1}{\mathcal{A}}:H_{0}^{1}(\Omega)\to H_{0}^{1}(\Omega),$ (3)

defined as the complement of the resolvent set, i.e.,

$\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A}):=\left\\{\lambda\in\mathbb{C};\,\lambda\mathcal{I}-\mathcal{B}^{-1}\mathcal{A}\mbox{
does not have a bounded inverse}\right\\}.$ (4)

More specifically, we prove that

$\mathrm{sp}(\mathcal{L}^{-1}\mathcal{A})=\left[\inf_{x\in\overline{\Omega}}\
\frac{k(x)}{g(x)},\,\sup_{x\in\overline{\Omega}}\ \frac{k(x)}{g(x)}\right].$

Consider a sequence of finite dimensional subspaces of $H_{0}^{1}(\Omega)$
defined via the nodal polynomial basis functions111As in [7], we consider
conforming FE methods using Lagrange elements. $\phi_{1},\ldots,\phi_{n}$ with
the local supports

${\mathcal{T}}_{j}=\mbox{supp}(\phi_{j}),\quad j=1,\ldots,n.$ (5)

The standard Galerkin finite element discretization of the operators
${\mathcal{A}}$ and ${\mathcal{B}}$ gives the matrix representations of the
discretised operators $\mathcal{A}_{n}$ and $\mathcal{B}_{n}$ in terms of the
basis $\phi_{1},\ldots,\phi_{n}$,

$\displaystyle[{\mathbf{A}}_{n}]_{ij}$
$\displaystyle=\int_{\Omega}k(x)\nabla\phi_{j}\cdot\nabla\phi_{i},\,\quad
i,j=1,\ldots,n.$ (6) $\displaystyle[{\mathbf{B}}_{n}]_{ij}$
$\displaystyle=\int_{\Omega}g(x)\nabla\phi_{j}\cdot\nabla\phi_{i},\,\quad
i,j=1,\ldots,n.$ (7)

Part one of this paper also contains an investigation of the approximation of
the whole spectrum of ${\mathcal{B}}^{-1}{\mathcal{A}}$ by the eigenvalues of
the preconditioned matrices ${\mathbf{B}}_{n}^{-1}{\mathbf{A}}_{n}$ as
$n\rightarrow\infty$.

In the second part, i.e., in section 4, we generalize the results obtained for
(1) and (2). More precisely, the spectrum issue is explored in terms of an
abstract setting where ${\mathcal{A}},{\mathcal{B}}:V\mapsto V^{\\#}$ only are
assumed to be bounded, coercive, and self-adjoint222Self-adjoint in the sense
that
$\langle{\mathcal{A}}u,v\rangle=\langle{\mathcal{A}}v,u\rangle,\langle{\mathcal{B}}u,v\rangle=\langle{\mathcal{B}}v,u\rangle$
for all $u,v\in V$, where $\langle\cdot,\cdot\rangle:V^{\\#}\times
V\mapsto\mathbb{R}$ is the duality pairing. Equivalently, $\tau{\mathcal{A}}$,
$\tau{\mathcal{B}}:V\rightarrow V$ are self-adjoint, where $\tau$ is the Riesz
map. linear operators. Here, $V^{\\#}$ denotes the dual of $V$ consisting of
all linear bounded functionals from the infinite dimensional Hilbert space $V$
to $\mathbb{R}$. We present a condition under which the whole spectrum of
${\mathcal{B}}^{-1}{\mathcal{A}}:V\mapsto V$, defined as the complement of the
resolvent set, is approximated as $n\rightarrow\infty$ to an arbitrary
accuracy by the eigenvalues of the finite dimensional discretizations
${\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}$. More precisely, we will concentrate
on Galerkin discretizations using a sequence $\\{V_{n}\\}$ of subspaces
$V_{n}\subset V$ satisfying the approximation property

$\lim_{n\rightarrow\infty}\inf_{v\in V_{n}}\|w-v\|=0\quad\mbox{for all}\;w\in
V\,.$ (8)

Since ${\mathcal{B}}^{-1}{\mathcal{A}}$ is continuously invertible and $V$ is
of infinite dimension, such an investigation cannot be based on the uniform
(normwise) convergence, and it relies instead upon the pointwise (strong)
convergence of ${\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}$ to
${\mathcal{B}}^{-1}{\mathcal{A}}$.

##  2 Preconditioning by Laplacian ($g(x)=1$).

Considering the case $g(x)=1$, i.e., the preconditioner equals the operator
representation of the Laplacian ${\mathcal{B}}={\mathcal{L}}$, the paper [8]
determines the spectrum of the preconditioned operator
${\mathcal{L}}^{-1}{\mathcal{A}}$ in the following way (for brevity we use a
bit stronger assumptions than in [8] and consider a uniformly positive scalar
coefficient function $k(x)$):

######  Theorem 2.1 (cf. [8], Theorem 1.1).

Consider an open and bounded Lipschitz domain $\Omega\subset\mathbb{R}^{2}$.
Assume that the scalar function $k(x)$ is uniformly positive and continuous
throughout the closure $\overline{\Omega}$. Then the spectrum of the operator
${\mathcal{L}}^{-1}{\mathcal{A}}$ equals the interval

$\mathrm{sp}(\mathcal{L}^{-1}\mathcal{A})=\left[\inf_{x\in\overline{\Omega}}\
k(x),\,\sup_{x\in\overline{\Omega}}\ k(x)\right].$ (9)

In other words, for a uniformly positive continuous function $k(x)$, the
spectrum of ${\mathcal{L}}^{-1}{\mathcal{A}}$ equals the range
$k(\overline{\Omega})$.

Eigenvalues of the discretized operator $\mathcal{L}_{n}^{-1}\mathcal{A}_{n}$,
that is represented by the matrix ${\mathbf{L}}_{n}^{-1}{\mathbf{A}}_{n}$, can
be approximated using the following theorem from [7]. (Here we again use
assumptions conforming to the setting in the current paper.)

######  Theorem 2.2 (cf. [7], Theorem 3.1).

Let $0<\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{n}$ be the eigenvalues
of ${\mathbf{L}}_{n}^{-1}{\mathbf{A}}_{n}$, where ${\mathbf{A}}_{n}$ and
${\mathbf{L}}_{n}$ are defined in (6) and (7)333With $g(x)=1$., respectively.
Let $k(x)$ be uniformly positive and continuous throughout the closure
$\overline{\Omega}$. Then there exist a (possibly non-unique) permutation
$\pi$ such that the eigenvalues $\lambda_{\pi(j)}$ of the matrix
${\mathbf{L}}_{n}^{-1}{\mathbf{A}}_{n}$ satisfy

$\lambda_{\pi(j)}\in\left[\inf_{x\in{\mathcal{T}}_{j}}k(x),\sup_{x\in{\mathcal{T}}_{j}}k(x)\right],\quad
j=1,\ldots,n,$ (10)

where ${\mathcal{T}}_{j}$ is defined in (5).

Consequently, there is a one-to-one mapping (possibly non-unique) between the
eigenvalues of ${\mathbf{L}}_{n}^{-1}{\mathbf{A}}_{n}$ and the ranges of
$k(x)$ over the supports of the individual basis functions. With an
appropriate grid refinement of the discretization, the size of the intervals
containing the individual eigenvalues of
${\mathbf{L}}_{n}^{-1}{\mathbf{A}}_{n}$ converge linearly to zero.444An
interesting application inspired by this result that uses a different approach
is presented in [12]

The paper [8] formulates a dual open question about the distribution of the
eigenvalues of the discretized operators within the interval (9). Assuming in
addition that $k(x)\in{\mathcal{C}}^{2}(\overline{\Omega})$, we will now show
that theorems 2.1 and 2.2 yield that any point in the spectrum of the infinite
dimensional operator ${\mathcal{L}}^{-1}{\mathcal{A}}$ is approximated, as
$n\rightarrow\infty$, to an arbitrary accuracy by the eigenvalues (10) of the
matrices ${\mathbf{L}}_{n}^{-1}{\mathbf{A}}_{n}$. The individual points in the
infinite dimensional spectrum can, however, be approximated with different
speed that is at least linear and uniformly bounded from zero.

Consider an arbitrary point $\mu$ in the spectrum of the operator
${\mathcal{L}}^{-1}{\mathcal{A}}$. It should be noted that $\mu$ may not be an
eigenvalue, and that our investigation differs from the eigenvalue problem
studied in the numerical PDE literature which is based on approximations of
the eigenvalues within the framework of infinite dimensional compact solution
operators.

Using theorem 2.1, $\mu$ is the image under $k(x)$ of some point
$y\in\overline{\Omega}$, i.e., $\mu=k(y)$. We first consider the case
$y\in{\Omega}$. The case $y\in{\partial\Omega}$ will be resolved later by a
simple limiting argument. Let $\varepsilon>0$ be an arbitrarily small positive
constant, and let

$\delta=\frac{\varepsilon}{2\sup_{x\in\Omega}\|\nabla k(x)\|},$

provided that $\sup_{x\in\Omega}\|\nabla k(x)\|>0$. (The case
$\sup_{x\in\Omega}\|\nabla k(x)\|=0$ is uninteresting because then
${\mathcal{A}}=c{\mathcal{L}}$ for some constant $c$.) Consider further a
Galerkin discretization such that the support of at least one of the nodal
basis functions555Supports of all discretization functions are contained in
$\overline{\Omega}$. that contains the point $y$ is itself contained in the
disc with center $y$ and radius $\delta$. Denote this support
${\mathcal{T}}_{j}$ and the associated eigenvalue of the discretized operator
given by theorem 2.2 as $\lambda_{\pi(j)}$. Using Corollary 3.2 in [7] (with
$y=\hat{x}_{j}$)

$\displaystyle|\lambda_{\pi(j)}-k(y)|$
$\displaystyle\leq\sup_{x\in{\mathcal{T}}_{j}}\|k(x)-k(y)\|$
$\displaystyle\leq\delta\,\|\nabla
k(y)\|+\frac{1}{2}\delta^{2}\sup_{x\in{\mathcal{T}}_{j}}\|D^{2}k(x)\|,$ (11)

where $D^{2}k(x)$ denotes the second order derivative of the function
$k(x)$666See [4, Section 1.2] for the definition of the second order
derivative.. For $\varepsilon$ sufficiently small we thus get, after a simple
manipulation,

$|\lambda_{\pi(j)}-\mu|\leq\varepsilon.$ (12)

If $\mu=k(y)$ and $y\in\partial\Omega$, the same conclusion can be obtained
using the previous derivation and the continuity of $k(x)$ throughout
$\overline{\Omega}.$ Summing up, this proves the following theorem:

######  Theorem 2.3 (Approximation of the spectrum by matrix eigenvalues).

Let $k(x)$ be twice continuously differentiable and uniformly positive
throughout the closure $\overline{\Omega}$. Let the maximal diameter of the
local supports of the basis functions used in the Galerkin discretization
(5)-(7) vanishes as $n\rightarrow\infty$. Then any point in the spectrum of
the operator ${\mathcal{L}}^{-1}{\mathcal{A}}$ is for $n\rightarrow\infty$
approximated to an arbitrary accuracy by the eigenvalues of the matrices
${\mathbf{L}}_{n}^{-1}{\mathbf{A}}_{n}$ representing the discretized
preconditioned operators.

The linear part of the upper bound (11) for the approximation error is for the
individual spectral points $\mu$ proportional to the size of the gradient
$\|\nabla k(y)\|$, where $k(y)=\mu$. Since the size of this gradient is
uniformly bounded from above throughout $\overline{\Omega}$, the speed of
convergence towards the individual spectral points, as the h-refinement
proceeds, is uniformly bounded from zero throughout the whole spectrum of
${\mathcal{L}}^{-1}{\mathcal{A}}$. It can however differ for different
spectral points.

##  3 Generalization to (piecewise) continuous and uniformly positive $g(x)$.

The purpose of this section is to generalize the results presented above to
preconditioners in the form (2). We first present the theorems and a
corollary, and thereafter their proofs are discussed.

The content of the present section is motivated by the desire to increase our
knowledge about second order differential operators and preconditioning
issues. In particular, to obtain a better understanding of the benefits of
applying piecewise constant preconditioners.

######  Theorem 3.1 (Spectrum of the infinite dimensional preconditioned
operator).

Consider an open and bounded Lipschitz domain $\Omega\subset\mathbb{R}^{2}$.
Assume that the scalar functions $g(x)$ and $k(x)$ are uniformly positive and
continuous throughout the closure $\overline{\Omega}$. Then the spectrum of
the operator ${\mathcal{B}}^{-1}{\mathcal{A}}$, defined in (4), equals

$\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A})=\left[\inf_{x\in\overline{\Omega}}\
\frac{k(x)}{g(x)},\,\sup_{x\in\overline{\Omega}}\ \frac{k(x)}{g(x)}\right].$
(13)

The next theorem deals with the localization of the eigenvalues of the
preconditioned matrix arising from the discretization. It does not consider
the approximation of the spectrum of the infinite dimensional operator
${\mathcal{B}}^{-1}{\mathcal{A}}$. Analogously to [7], we can therefore relax
the assumptions about the continuity of the coefficient functions $k(x)$ and
$g(x)$.

######  Theorem 3.2 (Eigenvalues of the preconditioned matrix).

Let $0<\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{n}$ be the eigenvalues
of ${\mathbf{B}}_{n}^{-1}{\mathbf{A}}_{n}$, where ${\mathbf{A}}_{n}$ and
${\mathbf{B}}_{n}$ are defined in (6) and (7), respectively. Let $g(x)$ and
$k(x)$ be bounded, uniformly positive and piecewise continuous functions. Then
there exists a (possibly non-unique) permutation $\pi$ such that the
eigenvalues of the matrix ${\mathbf{B}}_{n}^{-1}{\mathbf{A}}_{n}$ satisfy

$\lambda_{\pi(j)}\in\left[{\inf_{x\in{\mathcal{T}}_{j}}}\frac{k(x)}{g(x)},{\sup_{x\in{\mathcal{T}}_{j}}}\frac{k(x)}{g(x)}\right],\quad
j=1,\ldots,n,$ (14)

where ${\mathcal{T}}_{j}$ is defined in (5).

######  Corollary 3.3 (Pairing the eigenvalues and the nodal values).

Using the notation and the assumptions of Theorem 3.2, consider any point
${\hat{x}_{j}}\in{{\mathcal{T}}_{j}}$. Then the associated eigenvalue
$\lambda_{\pi(j)}$ of the matrix ${\mathbf{B}}^{-1}{\mathbf{A}}$ satisfies

$|\lambda_{\pi(j)}-r({\hat{x}_{j}})|\leq{\sup_{x\in{\mathcal{T}}_{j}}}|r(x)-r({\hat{x}_{j}})|,\quad{j=1,\ldots,n,}$
(15)

where

$r(x)\equiv\frac{k(x)}{g(x)}.$

If, in addition, $k(x)$ and $g(x)\in{\mathcal{C}}^{2}({{\mathcal{T}}_{j}})$,
then

$\displaystyle|\lambda_{\pi(j)}-r({\hat{x}_{j}})|$
$\displaystyle\leq{\sup_{x\in{\mathcal{T}}_{j}}}|r(x)-r({\hat{x}_{j}})|$
$\displaystyle\leq\hat{h}\|\nabla
r({\hat{x}_{j}})\|+\tfrac{1}{2}\hat{h}^{2}{\sup_{x\in{\mathcal{T}}_{j}}}\|D^{2}r(x)\|,\quad{j=1,\ldots,n,}$
(16)

where $\hat{h}={\mathrm{diam}({\mathcal{T}}_{j})}$ and $D^{2}r(x)$ denotes the
second order derivative of $r(x)$. In particular, (15) and (16) hold for any
discretization mesh node ${\hat{x}_{j}}\in{{\mathcal{T}}_{j}}$.

While the proofs of Theorem 3.1 and Theorem 3.2 are presented below, Corollary
3.3 follows immediately by applying [7, Corollary 3.2] to the ratio function
$r(x)$.

### Proof of Theorem 3.1.

Recall the definition (2) of the operator ${\mathcal{B}}$, and let us
introduce the following notation for the inner product and norm induced by
${\mathcal{B}}$,

$\displaystyle(u,v)_{{\mathcal{B}}}$
$\displaystyle=\langle\mathcal{B}u,v\rangle,\quad u,v\in H_{0}^{1}(\Omega),$
$\displaystyle\|u\|_{{\mathcal{B}}}^{2}$
$\displaystyle=(u,u)_{{\mathcal{B}}},\quad u\in H_{0}^{1}(\Omega).$

  1. 1.

The proof of the fact that

$\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A})\subset\left[\inf_{x\in\overline{\Omega}}\
\frac{k(x)}{g(x)},\,\sup_{x\in\overline{\Omega}}\ \frac{k(x)}{g(x)}\right]$

is analogous to the proof in [8, Section 3], employing the inner product
induced by ${\mathcal{B}}$ instead of that induced by the Laplacian
${\mathcal{L}}$. More precisely, using the self-adjointness of the operator
${\mathcal{B}}^{-1}{\mathcal{A}}$ with respect to the inner product
$(\cdot,\cdot)_{{\mathcal{B}}}$, the spectrum of
${\mathcal{B}}^{-1}{\mathcal{A}}$ is real and it is contained in the interval

$\displaystyle\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A})$
$\displaystyle\subset\left[\inf_{u\in
H_{0}^{1}(\Omega)}\frac{({\mathcal{B}}^{-1}{\mathcal{A}}u,u)_{{\mathcal{B}}}}{(u,u)_{{\mathcal{B}}}},\
\sup_{u\in
H_{0}^{1}(\Omega)}\frac{({\mathcal{B}}^{-1}{\mathcal{A}}u,u)_{{\mathcal{B}}}}{(u,u)_{{\mathcal{B}}}}\right]$
$\displaystyle=\left[\inf_{u\in
H_{0}^{1}(\Omega)}\frac{\langle{\mathcal{A}}u,u\rangle}{\langle{\mathcal{B}}u,u\rangle},\
\sup_{u\in
H_{0}^{1}(\Omega)}\frac{\langle{\mathcal{A}}u,u\rangle}{\langle{\mathcal{B}}u,u\rangle}\right].$
(17)

Moreover, the endpoints of this interval are contained in the spectrum. It
remains to bound

$\frac{\langle{\mathcal{A}}u,u\rangle}{\langle{\mathcal{B}}u,u\rangle}$ (18)

in terms of the values of the scalar functions $g(x)$ and $k(x)$. Let
$\|\cdot\|$ denote the standard Euclidean norm. Then,

$\displaystyle\sup_{u\in
H_{0}^{1}(\Omega)}\frac{\langle{\mathcal{A}}u,u\rangle}{\langle{\mathcal{B}}u,u\rangle}$
$\displaystyle=\sup_{u\in H_{0}^{1}(\Omega)}\frac{\int_{\Omega}k(x)\|\nabla
u\|^{2}}{\int_{\Omega}g(x)\|\nabla u\|^{2}}=\sup_{u\in
H_{0}^{1}(\Omega)}\frac{\int_{\Omega}\frac{k(x)}{g(x)}g(x)\|\nabla
u\|^{2}}{\int_{\Omega}g(x)\|\nabla u\|^{2}}$
$\displaystyle\leq\sup_{x\in\Omega}\frac{k(x)}{g(x)},$ (19)

where we have used the assumption that $k(x)$ and $g(x)$ are uniformly
positive and continuous on $\overline{\Omega}$. Similarly,

$\displaystyle\inf_{u\in
H_{0}^{1}(\Omega)}\frac{\langle{\mathcal{A}}u,u\rangle}{\langle{\mathcal{L}}u,u\rangle}$
$\displaystyle\geq\inf_{x\in\Omega}\frac{k(x)}{g(x)}.$ (20)

  2. 2.

The proof of the converse inclusion

$\left[\inf_{x\in\overline{\Omega}}\
\frac{k(x)}{g(x)},\,\sup_{x\in\overline{\Omega}}\
\frac{k(x)}{g(x)}\right]\subset\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A})$

is similar to the proof of [14, Theorem 3.1].

     * •

For an arbitrary $x_{0}\in\Omega$, consider $\lambda_{0}=k(x_{0})/g(x_{0})$.

     * •

Let $\\{v_{r}\\}_{r\in\mathbb{R}_{+}}$ be a set of functions satisfying777Note
that no limit of $v_{r}$, as $r\rightarrow 0$, is needed in this proof. Only
the existence of a set of functions satisfying (21) and (22) is required.

$\displaystyle{\rm supp}(v_{r})\subset x_{0}+U_{r},$ (21)
$\displaystyle\|v_{r}\|_{{\mathcal{B}}}=1,$ (22)

where $U_{r}=\\{z\in\mathbb{R}^{2}|\;\|z\|\leq r\\}$.

     * •

Next, let

$u_{r}=(\lambda_{0}{\mathcal{I}}-{\mathcal{B}}^{-1}{\mathcal{A}})v_{r},$ (23)

and observe that

$\displaystyle{\mathcal{B}}u_{r}=(\lambda_{0}{\mathcal{B}}-{\mathcal{A}})v_{r}.$

Consequently (see (2)),

$\displaystyle\|u_{r}\|_{{\mathcal{B}}}^{2}$
$\displaystyle=\langle(\lambda_{0}{\mathcal{B}}-{\mathcal{A}})v_{r},u_{r}\rangle$
$\displaystyle=\int_{x_{0}+U_{r}}\left(g(x)\lambda_{0}-k(x)\right)\nabla
v_{r}\cdot\nabla u_{r}$ $\displaystyle\leq\sup_{x\in
x_{0}+U_{r}}\left|g(x)\frac{k(x_{0})}{g(x_{0})}-k(x)\right|\,g_{\mathrm{min}}^{-1}\,|(v_{r},u_{r})_{{\mathcal{B}}}|$
$\displaystyle\leq\sup_{x\in
x_{0}+U_{r}}\left|g(x)\frac{k(x_{0})}{g(x_{0})}-k(x)\right|\,g_{\mathrm{min}}^{-1}\,\|v_{r}\|_{{\mathcal{B}}}\|u_{r}\|_{{\mathcal{B}}},$

where

${\color[rgb]{0,0,0}g_{\mathrm{min}}=\min_{x\in x_{0}+U_{r}}g(x).}$

Employing (22),

$\displaystyle\|u_{r}\|_{{\mathcal{B}}}\leq\sup_{x\in
x_{0}+U_{r}}\left|g(x)\frac{k(x_{0})}{g(x_{0})}-k(x)\right|\,g_{\mathrm{min}}^{-1},$

and from the continuity of $g(x)$ and $k(x)$ we conclude that

$\lim_{r\to 0}\|u_{r}\|_{{\mathcal{B}}}=0.$ (24)

     * •

Assume that $\lambda_{0}\notin\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A})$. Then
$\lambda_{0}{\mathcal{I}}-{\mathcal{B}}^{-1}{\mathcal{A}}$ has a bounded
inverse, and (23) and (24) imply that

$\|v_{r}\|_{{\mathcal{B}}}\leq\|(\lambda_{0}{\mathcal{I}}-{\mathcal{B}}^{-1}{\mathcal{A}})^{-1}\|_{{\mathcal{B}}}\,\|u_{r}\|_{{\mathcal{B}}}\longrightarrow
0$

as $r\rightarrow 0$, which contradicts (22). We conclude that

$\lambda_{0}=\frac{k(x_{0})}{g(x_{0})}\in\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A}).$

     * •

Since $x_{0}\in\Omega$ was arbitrary in this argument, and $g$ and $k$ are
continuous and uniformly positive on $\overline{\Omega}$, it follows that

$\left(\inf_{x\in\overline{\Omega}}\
\frac{k(x)}{g(x)},\,\sup_{x\in\overline{\Omega}}\
\frac{k(x)}{g(x)}\right)\subset\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A}).$ (25)

As mentioned above, according to the general result for self-adjoint
operators, the endpoints of the interval in (17) are contained in the spectrum
of ${\mathcal{B}}^{-1}{\mathcal{A}}$. Combining this with (19) and (20) yield
that also the endpoints of the interval (25) belong to
$\mathrm{sp}(\mathcal{B}^{-1}\mathcal{A})$.

### Proof Theorem 3.2.

The proof of Theorem 3.2 is analogous to the proof of Theorem 3.1 in[7]. As
was explained in detail in [9, Section 3.2], due to the use of the Hall’s
theorem for bipartite graphs (see, e.g., [2, Theorem 5.2]), it is sufficient
to prove the statement formulated in the following lemma (cf. [7, Lemma 3.3]).

######  Lemma 3.4.

Assume that $k(x)$ and $g(x)$ are uniformly positive, bounded and piecewise
continuous functions, and let the matrix
${\mathbf{B}}_{n}^{-1}{\mathbf{A}}_{n}$ be given by (6) and (7). Let,
moreover, ${\mathcal{J}}\subset\\{1,\ldots,n\\}$ and let
${\mathcal{T}}_{\mathcal{J}}:=\bigcup_{j\in{\mathcal{J}}}{\mathcal{T}}_{j}$ be
the union of the supports of the basis functions $\phi_{j}$,
$j\in{\mathcal{J}}$. Then there are at least $|{\mathcal{J}}|$ eigenvalues of
the matrix ${\mathbf{B}}_{n}^{-1}{\mathbf{A}}_{n}$ located in the interval

$\left[{\inf_{x\in{\mathcal{T}}_{\mathcal{J}}}}\frac{k(x)}{g(x)},{\sup_{x\in{\mathcal{T}}_{\mathcal{J}}}}\frac{k(x)}{g(x)}\right].$
(26)

###### Proof.

Following the proof of Lemma 3.3 in [7], consider, for any set of indices
${\mathcal{J}}\subset\\{1,\ldots,n\\}$, the (local) perturbation
$\tilde{k}_{\mathcal{J}}(x)$ of the coefficient function $k(x)$,

$\displaystyle\tilde{k}_{\mathcal{J}}(x)$ $\displaystyle=\begin{cases}K\cdot
g(x)\qquad\mbox{for}\quad x\in{\mathcal{T}}_{{\mathcal{J}}},\\\
k(x)\quad\mbox{elsewhere},\end{cases}$ (27)

where $K$ is a positive scalar. Analogously to (6), the matrix
$\tilde{{\mathbf{A}}}_{{\mathcal{J}},n}$ obtained by the discretization of the
associated perturbed operator $\tilde{{\mathcal{A}}}_{{\mathcal{J}}}$ is given
by

$\displaystyle[\tilde{{\mathbf{A}}}_{{\mathcal{J}},n}]_{lj}$
$\displaystyle=\left\langle\tilde{{\mathcal{A}}}_{{\mathcal{J}},h}\phi_{j},\phi_{l}\right\rangle=\int_{\Omega}\tilde{k}_{{\mathcal{J}}}(x)\nabla\phi_{j}\cdot\nabla\phi_{l}\,.$

The simple observation

$\tilde{{\mathbf{A}}}_{{\mathcal{J}},n}{\mathbf{e}}_{j}=K\,{\mathbf{B}}_{n}{\mathbf{e}}_{j},\quad
j\in{{\mathcal{J}}},$

shows that $K$ is an eigenvalue of the matrix
${\mathbf{B}}_{n}^{-1}\tilde{{\mathbf{A}}}_{{\mathcal{J}},n}$ with
multiplicity of at least $|{\mathcal{J}}|$.

By similarity transformations, the spectrum of
${\mathbf{B}}_{n}^{-1}{\mathbf{A}}_{n}$ equals the spectrum of
${\mathbf{B}}_{n}^{-1/2}{\mathbf{A}}_{n}{\mathbf{B}}_{n}^{-1/2}$, and the
spectrum of ${\mathbf{B}}_{n}^{-1}\tilde{{\mathbf{A}}}_{{\mathcal{J}},n}$ is
equal to the spectrum of
${\mathbf{B}}_{n}^{-1/2}\tilde{{\mathbf{A}}}_{{\mathcal{J}},n}{\mathbf{B}}_{n}^{-1/2}$.
Using the standard perturbation result for symmetric matrices (see, e.g., [15,
Corollary 4.9, p. 203]), there are at least $|{\mathcal{J}}|$ eigenvalues of
${\mathbf{B}}_{n}^{-1}{\mathbf{A}}_{n}$ in the interval

$[K+\theta_{min},K+\theta_{max}]\subset[K-\Theta,K+\Theta],$ (28)

where $\Theta=\max\\{|\theta_{min}|,|\theta_{max}|\\}$ and $\theta_{min}$ and
$\theta_{max}$ denote the smallest and largest eigenvalues, respectively, of
the perturbation matrix
${\mathbf{B}}_{n}^{-1}({\mathbf{A}}_{n}-\tilde{{\mathbf{A}}}_{{\mathcal{J}},n})$.

The Rayleigh quotient for an eigenvalue-eigenvector pair
$(\theta,{\mathbf{q}})$ of the perturbation matrix, with the associated
eigenfunction
$q=\sum_{j=1}^{N}\nu_{j}\phi_{j},\,{\mathbf{q}}^{T}=[\nu_{1},\ldots,\nu_{N}]$,
satisfies

$\displaystyle\theta$
$\displaystyle=\frac{{\mathbf{q}}^{T}({\mathbf{A}}_{n}-\tilde{{\mathbf{A}}}_{{\mathcal{J}},n}){\mathbf{q}}}{{\mathbf{q}}^{T}{\mathbf{B}}_{n}{\mathbf{q}}}=\frac{\langle({\mathcal{A}}-\tilde{{\mathcal{A}}}_{{\mathcal{J}}})q,q\rangle}{\langle{\mathcal{B}}q,q\rangle}$
$\displaystyle=\frac{\int_{\Omega}(k(x)-\tilde{k}_{{\mathcal{J}}}(x))\nabla
q\cdot\nabla q\,dx}{\int_{\Omega}g(x)\|\nabla
q\|^{2}\,dx}=\frac{\int_{{\mathcal{T}}_{{\mathcal{J}}}}(k(x)-Kg(x))\|\nabla
q\|^{2}\,dx}{\int_{\Omega}g(x)\|\nabla q\|^{2}\,dx}$
$\displaystyle=\frac{\int_{{\mathcal{T}}_{{\mathcal{J}}}}\left(\frac{k(x)}{g(x)}-K\right)g(x)\|\nabla
q\|^{2}\,dx}{\int_{\Omega}g(x)\|\nabla q\|^{2}\,dx},$

where we used the fact that $\tilde{k}_{{\mathcal{J}}}(x)=k(x)$ for
$x\in\Omega\setminus{\mathcal{T}}_{{\mathcal{J}}}$; see (27). Using the
uniform positivity of $g(x)$,

$|\theta|\leq{\sup_{x\in{\mathcal{T}}_{{\mathcal{J}}}}}\left|\frac{k(x)}{g(x)}-K\right|.$
(29)

Substituting (29) into (28) yields the existence of at least $|{\mathcal{J}}|$
eigenvalues of
${\mathbf{B}}_{{\color[rgb]{0,0,0}n}}^{-1}{\mathbf{A}}_{{\color[rgb]{0,0,0}n}}$
in the interval

$\left[\,K-{\sup_{x\in{\mathcal{T}}_{{\mathcal{J}}}}}\left|\frac{k(x)}{g(x)}-K\right|,\,K+{\sup_{x\in{\mathcal{T}}_{{\mathcal{J}}}}}\left|\frac{k(x)}{g(x)}-K\right|\,\right].$
(30)

Setting
$K=\tfrac{1}{2}\left(\inf_{x\in{\mathcal{T}}_{{\mathcal{J}}}}\frac{k(x)}{g(x)}+\sup_{x\in{\mathcal{T}}_{{\mathcal{J}}}}\frac{k(x)}{g(x)}\right)$
finishes the proof. ∎

##  4 Abstract setting.

This section investigates numerical approximations of the spectrum of
preconditioned linear operators within an abstract Hilbert space setting; see,
e.g., [13, 10]. Let $V$ be an infinite dimensional real Hilbert space with the
inner product

$(\cdot,\cdot):V\times V\mapsto\mathbb{R}.$ (31)

Throughout this text, $V^{\\#}$ denotes the dual of $V$ consisting of all
linear bounded functionals from $V$ to $\mathbb{R}$, with the associated
duality pairing

$\langle\cdot,\cdot\rangle:V^{\\#}\times V\mapsto\mathbb{R},$

and the Riesz map

$\langle\cdot,\cdot\rangle=:(\tau\cdot,\cdot),\quad\tau:V^{\\#}\mapsto V.$

Consider two bounded and coercive linear operators
${\mathcal{A}},{\mathcal{B}}:V\mapsto V^{\\#}$ that are self-adjoint with
respect to the duality pairing. We will investigate whether all the points in
the spectrum of the preconditioned operator
${\mathcal{B}}^{-1}{\mathcal{A}}:V\mapsto V$ are approximated to an arbitrary
accuracy by the eigenvalues of the finite dimensional operators in a sequence
$\\{{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\\}$ determined via the Galerkin
discretization.

For the problems considered in sections 2 and 3, we obtained concrete
expressions for the approximations of the spectrum of
${\mathcal{B}}^{-1}{\mathcal{A}}$ in terms of the coefficient functions $g(x)$
and $k(x)$. Such detailed information is, of course, not obtainable for the
present abstract problem setting. We must be content with analyzing whether,
or in what sense, the set of eigenvalues of the discretized mapping converges
toward the spectrum of the corresponding infinite dimensional operator.

Since ${\mathcal{B}}^{-1}{\mathcal{A}}$ is self-adjoint with respect to the
inner product

$(\cdot,\cdot)_{{\mathcal{B}}}:=\langle{\mathcal{B}}\,\cdot,\cdot\rangle:V\times
V\mapsto\mathbb{R},$ (32)

it is convenient to use this inner product instead of (31) whenever
appropriate888Since
${\mathcal{B}}^{-1}{\mathcal{A}}=(\tau{\mathcal{B}})^{-1}(\tau{\mathcal{A}})$,
one can as an alternative investigate approximations of the spectrum of the
symmetrized operator
$(\tau{\mathcal{B}})^{-1/2}\tau{\mathcal{A}}(\tau{\mathcal{B}})^{-1/2}$ that
is self-adjoint with respect to the inner product (31).. The associated norm
$\|\cdot\|_{{\mathcal{B}}}$ is equivalent to the norm $\|\cdot\|$ defined by
the inner product (31), and the Riesz map $\tau_{{\mathcal{B}}}$ representing
the operator preconditioning is determined by

$\langle\cdot,\cdot\rangle=:(\tau_{{\mathcal{B}}}\,\cdot,\cdot)_{{\mathcal{B}}}=\langle{\mathcal{B}}\,\tau_{{\mathcal{B}}}\,\cdot,\cdot\rangle,\quad\mbox{i.e.,}\;\tau_{{\mathcal{B}}}={\mathcal{B}}^{-1}.$

The investigated preconditioned operator ${\mathcal{B}}^{-1}{\mathcal{A}}$ is
continuously invertible on the Hilbert space $V$ of infinite dimension.
Therefore its finite dimensional approximations can not converge to it in norm
(uniformly). We will instead use Theorem 4.1 (below) that assumes the
pointwise (strong) convergence. Its statement reformulates a theorem presented
in [11, chapter VIII, § 1.2, Theorem 1.14, p. 431], which is reproduced also
in [3, section 5.4, Theorem 5.12, pp. 239-240]. The second monograph also
provides several references to related results of J. Descloux and
collaborators published earlier; see, in particular, [6]. In terms of the
spectral representation of self-adjoint operators, a bit stronger statements
were proved in the context of the problem of moments in [16, section III.3,
Theorem IX, p. 61] and, more generally, in [11, chapter VIII, § 1.2, Theorem
1.15, p. 432]. The formulations in [11] and [3] require a careful study of
parts of the books. We will therefore, for the sake of convenience, include a
proof of the following theorem in the Appendix.

######  Theorem 4.1 (Approximation of the spectrum of self-adjoint operators).

Let ${\mathcal{Z}}$ be a linear self-adjoint operator on a Hilbert space $V$
and let $\\{{\mathcal{Z}}_{n}\\}$ be a sequence of linear self-adjoint
operators on $V$ converging to ${\mathcal{Z}}$ pointwise (strongly). Then, for
any point $\lambda\in\mathrm{sp}({\mathcal{Z}})$ in the spectrum of
${\mathcal{Z}}$, and for any of its neighbourhoods, there exists
${\mathcal{Z}}_{j}$ such that the intersection of its spectrum
$\mathrm{sp}({\mathcal{Z}}_{j})$ with this neighbourhood is nonempty.

Using this theorem and the Hilbert space $V$ equipped with the inner product
(32), it remains, within our setting, to prove that the self-adjoint operators
${\mathcal{Z}}_{n}$, which arise from
${\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}$ by extending it to the whole space
$V$, converge pointwise to the original self-adjoint operator

${\mathcal{Z}}:={\mathcal{B}}^{-1}{\mathcal{A}}.$

The discretization will be based on a sequence of subspaces
$\\{V_{n}\\},\,V_{n}\subset V,$ satisfying the approximation property (8)999In
the limit equal to zero, i.e., it does not matter which of the equivalent
norms, $\|\cdot\|_{{\mathcal{B}}}$ or $\|\cdot\|$, we use.

$\lim_{n\rightarrow\infty}\inf_{v\in V_{n}}\|w-v\|=0\quad\mbox{for all}\;w\in
V,$ (33)

see, e.g., [1, relation (8)]. Note that (33) typically yields that Galerkin
discretizations of boundary value problems are consistent; see also [13,
chapter 9, relation (9.8)].

Consider a basis $\Phi_{n}=(\phi_{1}^{(n)},\dots,\phi_{n}^{(n)})$ of the
$n$-dimensional subspace $V_{n}\subset V$. Then the Galerkin discretizations
${\mathcal{A}}_{n}$ and ${\mathcal{B}}_{n}$ of the operators ${\mathcal{A}}$
and ${\mathcal{B}}$ are determined by (see [10, section 4.1] and [13, chapter
6]),

$\langle\mathcal{A}_{n}w,v\rangle:=\langle\mathcal{A}w,v\rangle\quad\mbox{and}\quad\langle\mathcal{B}_{n}w,v\rangle:=\langle\mathcal{B}w,v\rangle,\quad\mbox{for
all}\;w,v\in V_{n}.$ (34)

Their matrix representations are given by

$\mathbf{A}_{n}=\left(\langle\mathcal{A}\phi_{j}^{(n)},\phi_{i}^{(n)}\rangle\right)_{i,j=1,\dots,n},$
(35)

and

$\mathbf{B}_{n}=\left(\langle\mathcal{B}\phi_{j}^{(n)},\phi_{i}^{(n)}\rangle\right)_{i,j=1,\dots,n}.$
(36)

The spectrum of the discretized operator
${\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}:V_{n}\mapsto V_{n}$ is given by the
eigenvalues of its matrix representation $\mathbf{B}_{n}^{-1}\mathbf{A}_{n}$.
The operator ${\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}$ is self-adjoint with
respect to the inner product (32), and the matrix
$\mathbf{B}_{n}^{-1}\mathbf{A}_{n}$ is self-adjoint with respect to the
algebraic inner product
$(\mathbf{x},\mathbf{y})_{\mathbf{B}_{n}}:=\mathbf{y}^{*}\mathbf{B}_{n}\mathbf{x}$.

Using the orthogonal projection

$\Pi_{{\mathcal{B}}}^{n}:V\mapsto V_{n},$

where the orthogonality is determined by the inner product (32),
${\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}$ is extended to the whole space $V$,

${\mathcal{Z}}_{n}:={\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}:V\mapsto
V_{n}\subset V.$ (37)

We need to show that ${\mathcal{Z}}_{n}$ is self-adjoint with respect to the
inner product (32). Using, for any $u,w\in V$, the associated orthogonal
decompositions $u=\Pi_{{\mathcal{B}}}^{n}u+u^{\perp}$ and
$w=\Pi_{{\mathcal{B}}}^{n}w+w^{\perp}$, we can write

$\displaystyle\langle{\mathcal{B}}{\mathcal{Z}}_{n}u,w\rangle$
$\displaystyle=\langle{\mathcal{B}}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}u,w\rangle=\langle{\mathcal{B}}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}u,\Pi_{{\mathcal{B}}}^{n}w\rangle+\langle{\mathcal{B}}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}u,w^{\perp}\rangle$
$\displaystyle=\langle{\mathcal{B}}_{n}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}u,\Pi_{{\mathcal{B}}}^{n}w\rangle=\langle{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}u,\Pi_{{\mathcal{B}}}^{n}w\rangle=\langle{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}w,\Pi_{{\mathcal{B}}}^{n}u\rangle$
$\displaystyle=\langle{\mathcal{B}}_{n}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}w,\Pi_{{\mathcal{B}}}^{n}u\rangle=\langle{\mathcal{B}}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}w,\Pi_{{\mathcal{B}}}^{n}u\rangle$
$\displaystyle=\langle{\mathcal{B}}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}w,\Pi_{{\mathcal{B}}}^{n}u\rangle+\langle{\mathcal{B}}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}w,u^{\perp}\rangle$
$\displaystyle=\langle{\mathcal{B}}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}w,u\rangle=\langle{\mathcal{B}}{\mathcal{Z}}_{n}w,u\rangle,$

which gives the self-adjointness.

Summarizing, the sequence of subspaces $\\{V_{n}\\}$ determines a sequence of
self-adjoint operators $\\{{\mathcal{Z}}_{n}\\}$ defined on the whole space
$V$. The dimension of the ranges of these operators is finite, but increases
as $n$ increases. It remains to prove that $\\{{\mathcal{Z}}_{n}\\}$ converges
pointwise to ${\mathcal{Z}}$.

######  Theorem 4.2 (Pointwise convergence).

Let ${\mathcal{Z}}={\mathcal{B}}^{-1}{\mathcal{A}}$ be a linear self-adjoint
operator on a Hilbert space $V$, where ${\mathcal{A}},{\mathcal{B}}:V\mapsto
V^{\\#}$ are bounded and coercive linear operators that are self-adjoint with
respect to the duality pairing, and let $\\{{\mathcal{Z}}_{n}\\}$ be the
sequence of linear self-adjoint operators defined in (37). Assume that the
sequence of subspaces $\\{V_{n}\\}$ satisfy the approximation property (33).
Then the sequence $\\{{\mathcal{Z}}_{n}\\}$ converges pointwise (strongly) to
${\mathcal{Z}}$, i.e., for all $w\in V$

$\lim_{n\rightarrow\infty}\|{\mathcal{Z}}w-{\mathcal{Z}}_{n}w\|=0.$

###### Proof.

Let $w\in V$ be an arbitrary fixed element and define

$f:={\mathcal{Z}}w.$

Consider a finite dimensional subspace $V_{n}\subset V$ and the Galerkin
discretization of the equation ${\mathcal{Z}}w=f$, using the innerproduct
$(\cdot,\cdot)_{{\mathcal{B}}}$: Find $w_{n}\in V_{n}$ such that

$({\mathcal{Z}}w_{n},v)_{{\mathcal{B}}}=(f,v)_{{\mathcal{B}}}\quad\mbox{for
all }v\in V_{n}.$

This gives, for all $v\in V_{n}$,

$\displaystyle 0$
$\displaystyle=(f-{\mathcal{Z}}w_{n},v)_{{\mathcal{B}}}=(f_{n}-{\mathcal{Z}}w_{n},v)_{{\mathcal{B}}}+(f-f_{n},v)_{{\mathcal{B}}}$
$\displaystyle=(f_{n}-\widehat{{\mathcal{Z}}_{n}}w_{n},v)_{{\mathcal{B}}},$
(38)

where we used the definition $f_{n}=\Pi_{{\mathcal{B}}}^{n}f$ for the
discretized right hand side, i.e.,

$(f_{n}-f,v)_{{\mathcal{B}}}=0\quad\mbox{for all}\;v\in V_{n},$

giving

$\|f-f_{n}\|_{{\mathcal{B}}}=\inf_{g\in V_{n}}\|f-g\|_{{\mathcal{B}}}.$

The equivalence of the norms induced by the innerproducts (31) and (32) and
the approximation property (33) then assure that

$\lim_{n\rightarrow\infty}\|f-f_{n}\|=0\,.$ (39)

The discretized operator $\widehat{{\mathcal{Z}}_{n}}$ is determined by

$(\widehat{{\mathcal{Z}}_{n}}u,v)_{{\mathcal{B}}}:=({\mathcal{Z}}u,v)_{{\mathcal{B}}}\quad\mbox{for
all}\;u,v\in V_{n},$

and it is easy to verify that
$\widehat{{\mathcal{Z}}_{n}}={\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}$. Indeed,
for all $u,v\in V_{n}$,

$\displaystyle(\widehat{{\mathcal{Z}}_{n}}u,v)_{{\mathcal{B}}}$
$\displaystyle=({\mathcal{B}}^{-1}{\mathcal{A}}u,v)_{{\mathcal{B}}}$
$\displaystyle=\langle{\mathcal{A}}u,v\rangle$
$\displaystyle=\langle{\mathcal{A}}_{n}u,v\rangle$
$\displaystyle=\langle{\mathcal{B}}_{n}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}u,v\rangle$
$\displaystyle=\langle{\mathcal{B}}{\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}u,v\rangle$
$\displaystyle=({\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}u,v)_{{\mathcal{B}}}.$

The previous considerations remain valid when replacing
$\widehat{{\mathcal{Z}}_{n}}$ by its extension ${\mathcal{Z}}_{n}$ to the
whole space $V$ given in (37) because

$\displaystyle({\mathcal{Z}}_{n}u,v)_{{\mathcal{B}}}$
$\displaystyle=({\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}\,\Pi_{{\mathcal{B}}}^{n}u,v)_{{\mathcal{B}}}$
$\displaystyle=({\mathcal{B}}_{n}^{-1}{\mathcal{A}}_{n}u,v)_{{\mathcal{B}}}$
$\displaystyle=(\widehat{{\mathcal{Z}}_{n}}u,v)_{{\mathcal{B}}}\quad\mbox{for
all}\;u,v\in V_{n}.$

Consequently,

${\mathcal{Z}}_{n}w_{n}=f_{n},$

see (38).

With the previous considerations we can write

$\displaystyle{\mathcal{Z}}w-{\mathcal{Z}}_{n}w$
$\displaystyle=({\mathcal{Z}}w-f)+(f-f_{n})+(f_{n}-{\mathcal{Z}}_{n}w_{n})+{\mathcal{Z}}_{n}(w_{n}-w)$
$\displaystyle=(f-f_{n})+{\mathcal{Z}}_{n}(w_{n}-w).$ (40)

Using (39), the first term vanishes as $n\rightarrow\infty$. As for the second
term, ${\mathcal{Z}}_{n}$ results from the Galerkin discretization of
${\mathcal{Z}}$ and therefore its norm is bounded independently of $n$. It
remains to examine the size of the difference $w_{n}-w$. Following the
standard derivation of the Céa’s lemma (cf., e.g., [13, chapter 9, derivation
of the relation (9.8)]), we have for any $v\in V_{n}$

$\displaystyle\alpha_{{\mathcal{A}}}\|w-w_{n}\|^{2}$
$\displaystyle\leq\langle{\mathcal{A}}(w-w_{n}),w-w_{n}\rangle$
$\displaystyle=\langle{\mathcal{B}}({\mathcal{B}}^{-1}{\mathcal{A}})(w-w_{n}),w-w_{n}\rangle$
$\displaystyle=(({\mathcal{B}}^{-1}{\mathcal{A}})(w-w_{n}),w-w_{n})_{{\mathcal{B}}}$
$\displaystyle=(({\mathcal{B}}^{-1}{\mathcal{A}})(w-w_{n}),w-v)_{{\mathcal{B}}}+(({\mathcal{B}}^{-1}{\mathcal{A}})(w-w_{n}),v-w_{n})_{{\mathcal{B}}}$
$\displaystyle=(({\mathcal{B}}^{-1}{\mathcal{A}})(w-w_{n}),w-v)_{{\mathcal{B}}}+((f-{\mathcal{Z}}w_{n}),v-w_{n})_{{\mathcal{B}}}$
$\displaystyle=\langle{\mathcal{A}}(w-w_{n}),w-v\rangle\leq
C_{{\mathcal{A}}}\|w-w_{n}\|\|w-v\|.$

Here, $\alpha_{{\mathcal{A}}}$ and $C_{{\mathcal{A}}}$ are the coercivity and
boundedness constants associated with ${\mathcal{A}}$, and in the derivation
we used the Galerkin orthogonality (38), i.e.,
$((f-{\mathcal{Z}}w_{n}),v-w_{n})_{{\mathcal{B}}}=0$. Consequently, denoting
$\kappa({\mathcal{A}}):=C_{{\mathcal{A}}}/\alpha_{{\mathcal{A}}}$,

$\|w-w_{n}\|\leq\kappa({\mathcal{A}})\,\inf_{v\in V_{n}}\|w-v\|\,.$

Using again the approximation property (33), it follows that the second term
in (4) also vanishes as $n\rightarrow\infty$, and the proof is finished. ∎

Theorems 4.1 and 4.2 immediately give the final corollary.

######  Corollary 4.3 (Spectral approximation).

Consider an infinite dimensional Hilbert space $V$, its dual $V^{\\#}$, and
bounded and coercive linear operators ${\mathcal{A}},{\mathcal{B}}:V\mapsto
V^{\\#}$ that are self-adjoint with respect to the duality pairing. Consider
further a sequence of subspaces $\\{V_{n}\\}$ of $V$ satisfying the
approximation property (33).

Let the sequences of matrices $\\{\mathbf{A}_{n}\\}$ and
$\\{\mathbf{B}_{n}\\}$ be defined by (34) - (36). Then all points in the
spectrum of the preconditioned operator

${\mathcal{B}}^{-1}{\mathcal{A}}:V\mapsto V$

are approximated to an arbitrary accuracy by the eigenvalues of the
preconditioned matrices in the sequence
$\\{\mathbf{B}_{n}^{-1}\mathbf{A}_{n}\\}.$ That is, for any point
$\lambda\in\mathrm{sp}({\mathcal{{\mathcal{missing}}}{B}^{-1}{\mathcal{A}}})$
and any $\epsilon>0$, there exists $n^{*}$ such that
$\mathbf{B}_{n^{*}}^{-1}\mathbf{A}_{n^{*}}$ has an eigenvalue $\lambda^{*}$
satisfying $|\lambda-\lambda^{*}|<\epsilon$.

##  5 Numerical experiments

We used Matlab’s PDE-Toolbox to compute scalars $\lambda$ satisfying

${\mathbf{A}}_{n}{\mathbf{v}}=\lambda{\mathbf{B}}_{n}{\mathbf{v}},$ (41)

where ${\mathbf{A}}_{n}$ and ${\mathbf{B}}_{n}$ denote the stiffness matrices
defined in (6) and (7), respectively. Whereas our theoretical study concerns
problems with homogeneous Dirichlet boundary conditions, we employed
homogeneous Neumann boundary conditions in the numerical experiments below.
This was done for the sake of completeness: One can show, in a straightforward
manner, that the results presented in sections 2 and 3 also hold in the case
of Neumann boundary conditions.

Due to the homogeneous Neumann boundary condition,
${\mathbf{0}}={\mathbf{A}}_{n}{\mathbf{c}}=\lambda{\mathbf{B}}_{n}{\mathbf{c}}={\mathbf{0}}$
for any constant vector ${\mathbf{c}}$ and any scalar $\lambda$. Matlab
handles this matter, i.e., (41) is solved subject to the constraint that
${\mathbf{v}}$ must not belong to the intersection of the nullspaces of
${\mathbf{A}}_{n}$ and ${\mathbf{B}}_{n}$.

The generalized eigenvalues and the nodal values of the function
$r(x,y)=k(x,y)/g(x,y)$ are sorted in increasing order in the plots below,
$\Omega=(-1,1)^{2}$, and

$\displaystyle k(x,y)$
$\displaystyle=(1+50\exp(-5(x^{2}+y^{2})))(2+sin(x+y)),$ $\displaystyle
g(x,y)$ $\displaystyle=1+50\exp(-5(x^{2}+y^{2})),$ $\displaystyle r(x,y)$
$\displaystyle=2+\sin(x+y).$

Figure 1 shows the generalized eigenvalues (41) and the nodal values of
$r(x,y)$ computed with the mesh depicted in Figure 2. The results obtained
with computations performed on a finer grid is visualized in figures 3 and 4.
The outcome of these experiments is as one could have anticipated from Theorem
3.1, Theorem 3.2 and Corollary 3.3. As expected, we can also observe spreading
of the computed eigenvalues over the whole spectral interval.

Figure 1:  Generalized eigenvalues (blue circles) and nodal values of $r(x,y)$
(red asterisks) computed with the coarse mesh displayed in Fig. 2. Figure 2:
Matlab’s grid, coarse case. Figure 3:  Generalized eigenvalues (blue circles)
and nodal values of $r(x,y)$ (red asterisks) computed with the “fine” mesh
displayed in Fig. 4. Figure 4:  Matlab’s grid, “fine” case.

##  6 Conclusions and further work.

We have not only extended our earlier results [14, 7, 8], addressing Laplacian
preconditioning, to preconditioners defined in terms of more general elliptic
differential operators, but we have also proved that the entire spectrum of
any operator in the form ${\mathcal{B}}^{-1}{\mathcal{A}}$ can be approximated
with arbitrary accuracy by the eigenvalues of the associated discretized
mappings. Here, ${\mathcal{A},B}:V\mapsto V^{\\#}$ are linear, bounded,
coercive and self-adoint operators defined on an infinite dimensional Hilbert
space $V$, and $V^{\\#}$ denotes the dual space. Our analysis differs
significantly from the classical investigations of the point spectrum of
second order differential operators, which is typically done within the
framework of compact (solution) operators.

In our opinion, these results yield a new perspective on the continuous
spectrum of preconditioned elliptic differential operators, and there are
several unanswered questions: For example, are the results presented in
section 3 also valid if the coefficient functions $k(x)$ and $g(x)$ are
replaced by symmetric and uniformly positive definite conductivity tensors
$K(x)$ and $G(x)$, respectively? Also, and perhaps even more interesting, do
Theorem 4.2 and Corollary 4.3 also hold for more general continuously
invertible operators ${\mathcal{A}}:V\mapsto V^{\\#}$, e.g., for saddle point
problems?

## Acknowledgments.

The authors thank David Krejčiřík, Josef Málek and Ivana Pultarová for
stimulating discussions during the work on this paper.

##  Appendix A Approximations of the spectrum of self-adjoint operators.

Using [3, Chapter 3] and [11, Chapter 8], we first recall several results
concerning the convergence of linear self-adjoint operators defined on
infinite dimensional Hilbert spaces. By the Hellinger-Toeplitz theorem (see
[5, Theorem 5.7.2, p. 260]), any linear self-adjoint operator
${\mathcal{Z}}:V\mapsto V$ on a Hilbert space $V$ is closed and, according to
the Banach closed-graph theorem, bounded (continuous).

Consider a bounded linear operator ${\mathcal{G}}:V\mapsto V$ (not necessarily
self-adjoint, therefore we for the moment change the notation) and a sequence
of its bounded linear approximations
$\\{{\mathcal{G}}_{n}\\},{\mathcal{G}}_{n}:V\mapsto V$, that can converge to
${\mathcal{G}}$ in different ways:

  * •

pointwise (strongly), i.e., ${\mathcal{G}}_{n}\stackrel{{\scriptstyle
p}}{{\rightarrow}}{\mathcal{G}}$

$\mbox{iff for all}\;x\in
V,\quad\lim_{n\rightarrow\infty}\|{\mathcal{G}}x-{\mathcal{G}}_{n}x\|=0\,;$

  * •

uniformly (in norm), iff
$\lim_{n\rightarrow\infty}\|{\mathcal{G}}-{\mathcal{G}}_{n}\|=0\,;$

  * •

stably, i.e. ${\mathcal{G}}_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}{\mathcal{G}}$ iff

    * –

${\mathcal{G}}_{n}\stackrel{{\scriptstyle p}}{{\rightarrow}}{\mathcal{G}}$,
and

    * –

the inverse operators $\\{{\mathcal{G}}_{n}^{-1}\\}$ are uniformly bounded,
i.e., for some $C>0$, $\|{\mathcal{G}}_{n}^{-1}\|\leq C$ for all $n$.

Clearly, uniform convergence implies pointwise convergence, but the converse
implication does not hold. Since the class of compact operators is closed with
respect to uniform convergence, the uniform convergence concept can not be
used to investigate the convergence of compact to non-compact operators, such
as to bounded continuously invertible operators defined on infinite
dimensional Hilbert spaces.

The spectral theory for bounded linear operators is based on the concept of
the operator resolvent

${\mathcal{R}}(\mu):=(\mu\mathcal{I}-\mathcal{G})^{-1}$

and on the resolvent set

$\rho({\mathcal{G}}):=\left\\{\mu\in\mathbb{C};\,\mu\mathcal{I}-\mathcal{G}\;\mbox{has
a bounded inverse}\right\\}.$ (42)

It is interesting to notice that, for any $\mu\in\rho({\mathcal{G}})$, a
sequence of shifted operators $\\{\mu\mathcal{I}-\mathcal{G}_{n}\\}$ converge
to $\mu\mathcal{I}-\mathcal{G}$ stably if, and only if,
$\\{\mathcal{G}_{n}\\}$ and the resolvents
$\\{{\mathcal{R}}_{n}(\mu)\\},{\mathcal{R}}_{n}(\mu):=(\mu\mathcal{I}-\mathcal{G}_{n})^{-1}$,
converge to $\mathcal{G}$ and ${\mathcal{R}}(\mu)$ pointwise, respectively,
i.e.,101010See [3, Lemma 3.16].

$\mu\mathcal{I}-\mathcal{G}_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\mu\mathcal{I}-\mathcal{G}\quad\mbox{iff}\quad\mathcal{G}_{n}\stackrel{{\scriptstyle
p}}{{\rightarrow}}\mathcal{G}\;\mbox{and}\;{\mathcal{R}}_{n}(\mu)\stackrel{{\scriptstyle
p}}{{\rightarrow}}{\mathcal{R}}(\mu).$ (43)

Indeed, using the resolvent identity

${\mathcal{R}}_{n}(\mu)-{\mathcal{R}}(\mu)={\mathcal{R}}_{n}(\mu)\,({\mathcal{G}}_{n}-{\mathcal{G}})\,{\mathcal{R}}(\mu),$

the right implication follows immediately from the definition of stable
convergence. Conversely, from the pointwise convergence of
${\mathcal{R}}_{n}(\mu)$ and the uniform boundedness principle
(Banach–Steinhaus theorem) we conclude that
$\\{{\mathcal{R}}_{n}(\mu)\\}=\\{(\mu\mathcal{I}-\mathcal{G}_{n})^{-1}\\}$ is
uniformly bounded and the result follows.

We will now present the proof of Theorem 4.1; cf. also [11, chapter VIII, §
1.2, Theorem 1.14, p. 431] and [3, chapter 5, section 4, Theorem 5.12, p.
239-240].

######  Theorem A.1 (Approximation of the spectrum of self-adjoint operators).

Let ${\mathcal{Z}}$ be a linear self-adjoint operator on a Hilbert space $V$
and let $\\{{\mathcal{Z}}_{n}\\}$ be a sequence of linear self-adjoint
operators on $V$ converging to ${\mathcal{Z}}$ pointwise (strongly). Then, for
any point $\lambda\in\mathrm{sp}({\mathcal{Z}})$ in the spectrum of
${\mathcal{Z}}$, and for any of its neighbourhoods, there exists
${\mathcal{Z}}_{j}$ such that the intersection of its spectrum
$\mathrm{sp}({\mathcal{Z}}_{j})$ with this neighbourhood is nonempty.

###### Proof.

Consider any point $\lambda\in\mathrm{sp}({\mathcal{Z}})\subset\mathbb{R}$ in
the spectrum of ${\mathcal{Z}}$. Then, for any $\varepsilon>0$, the point
$\mu:=\lambda+\iota\varepsilon$, where $\iota$ is the complex unit, belongs to
the resolvent set $\rho({\mathcal{Z}})$ (because ${\mathcal{Z}}$ is self-
adjoint). For self-adjoint operators, the norm of the resolvent at any point
in the resolvent set is equal to the inverse of the distance of the given
point to the spectrum (see, e.g., [3, Proposition 2.32]). Therefore, for all
$n$,

$\displaystyle\|{\mathcal{R}}(\mu)\|$
$\displaystyle:=\|(\mu\mathcal{I}-\mathcal{Z})^{-1}\|=\frac{1}{\mathrm{dist}(\mu,\mathrm{sp}({\mathcal{Z}}))}=\frac{1}{\varepsilon}\,,$
(44) $\displaystyle\|{\mathcal{R}}_{n}(\mu)\|$
$\displaystyle:=\|(\mu\mathcal{I}-\mathcal{Z}_{n})^{-1}\|=\frac{1}{\mathrm{dist}(\mu,\mathrm{sp}({\mathcal{Z}}_{n}))}\leq\frac{1}{\varepsilon}\,.$
(45)

The inequality in (45) follows from the assumption that
$\\{{\mathcal{Z}}_{n}\\}$ is a sequence of self-adjoint operators, i.e.,
$\mathrm{sp}({\mathcal{Z}}_{n})\subset\mathbb{R}$. Note that inequality (45)
provides the uniform boundedness of $\\{{\mathcal{R}}_{n}(\mu)\\}$, which,
together with the pointwise convergence of $\\{{\mathcal{Z}}_{n}\\}$, yields
that

$\mu{\mathcal{I}}-{\mathcal{Z}}_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\mu{\mathcal{I}}-{\mathcal{Z}}.$

Using (43), we thus have the pointwise convergence of
$\\{{\mathcal{R}}_{n}(\mu)\\}$, i.e., for any $x\in V,||x\|=1,$

$\|{\mathcal{R}}(\mu)x\|=\lim_{n\rightarrow\infty}\|{\mathcal{R}}_{n}(\mu)x\|.$

Consider a fixed $x\in V,||x\|=1,$ such that

$\frac{1}{2\varepsilon}\leq\|{\mathcal{R}}(\mu)x\|\leq\frac{1}{\varepsilon}.$

(The existence of such a $x\in V$ follows from (44).) Then, from the pointwise
convergence, there must exist $n$ such that

$\|{\mathcal{R}}_{n}(\mu)\|\geq\|{\mathcal{R}}_{n}(\mu)x\|\geq\frac{1}{3\varepsilon}.$

Recall that
$\|{\mathcal{R}}_{n}(\mu)\|=\mathrm{[dist}(\mu,\mathrm{sp}({\mathcal{Z}}_{n}))]^{-1}$.
Therefore there exists a point $\lambda_{n}\in\mathrm{sp}({\mathcal{Z}}_{n})$
such that

$|\lambda_{n}-\mu|\leq 3\varepsilon\quad\mbox{and,
consequently,}\quad|\lambda_{n}-\lambda|\leq 4\varepsilon,$

which finishes the proof. ∎

## References

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# All-Optical Ultrafast Valley Switching in Two-Dimensional Materials

Navdeep Rana  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076, India  Gopal Dixit<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

###### Abstract

Electrons in two-dimensional materials possess an additional quantum
attribute, the valley pseudospin, labelled as $\mathbf{K}$ and
$\mathbf{K}^{\prime}$ – analogous to the spin up and spin down. The majority
of research to achieve valley-selective excitations in valleytronics depends
on resonant circularly-polarised light with a given helicity. Not only
acquiring valley-selective electron excitation but also switching the
excitation from one valley to another is quintessential for bringing
valleytronics-based technologies in reality. Present work introduces a
coherent control protocol to initiate valley-selective excitation, de-
excitation, and switch the excitation from one valley to another on the fly
within tens of femtoseconds – a timescale faster than any valley decoherence
time. Our protocol is equally applicable to both gapped and gapless two-
dimensional materials. Monolayer graphene and molybdenum disulfide are used to
test the universality. Moreover, the protocol is robust as it is insensitive
to significant parameters of the protocol, such as dephasing times,
wavelengths, and time delays of the laser pulses. Present work goes beyond the
existing paradigm of valleytronics, and opens a new realm of valley switch at
PetaHertz rate.

##  I Introduction

The successful synthesis of monolayer graphene has led to the proliferation in
synthesizing of other two-dimensional (2D) materials, such as hexagonal boron
nitride and transition-metal dichalcogenides 1; 2; 3. These 2D materials
exhibit attractive transport and optoelectronic properties, which hold promise
for upcoming technologies 4. One of the fascinating features of these 2D
materials is the electron’s additional quantum attribute, the valley
pseudospin – analogous to the electron’s spin. The valley pseudospin is
associated with the valleys in the energy landscape of these materials. The
hexagonal 2D materials are endowed with two degenerate valleys situated at the
corners of the Brillouin zone 5; 6. The flexible control over these valley
pseudospins offers a platform to write, process, and store quantum information
7; 8. Moreover, coherent switching of electron excitation from one valley to
another, on a timescale faster than the valley decoherence, is quintessential
for valleytronics-based emerging quantum technologies at ambient conditions 9;
10.

Figure 1: Schematic of an all-optical ultrafast valley switch in a two-
dimensional material. First panel: Brillouin zone of a 2D material along with
high-symmetry points in momentum space. Two degenerate valleys are located at
the corners of the Brillouin zone, i.e., at $\mathbf{K}$ and
$\mathbf{K}^{\prime}$. Second panel: valley excitation at
$\mathbf{K}^{\prime}$ (shown in yellow) is favored over $\mathbf{K}$ (shown in
green) after the end of the first linear pulse. Third panel: the favored
valley excitation at $\mathbf{K}^{\prime}$ is counterbalanced by the second
linear pulse subsequently as shown in blue. Fourth panel: reversal in the
preference of valley excitation at $\mathbf{K}$ (shown in green) over
$\mathbf{K}^{\prime}$ (shown in yellow) by the action of the third linear
pulse. All three few-cycle pulses have well-defined carrier-envelope phase and
are linearly polarised along $\Gamma-\mathbf{K}$ direction.  Figure 2:  All-
optical valley switch in graphene. (a) Energy band structure of graphene along
high-symmetry points in the Brillouin zone. Two inequivalent and degenerate
valleys located at the corners of the Brillouin zone with crystal momenta
$\mathbf{K}$ and $\mathbf{K}^{\prime}$. The first Brillouin zone is shown in
the inset. The momentum-resolved residual electronic populations in the first
Brillouin zone of the conduction band are shown in (b) - (d) after the end of
the first, second, and third linear pulses, respectively. (b) The first pulse
with CEP = 0∘ results in asymmetric electronic population, leading to
preferred excitation around $\mathbf{K}^{\prime}$ valley. (c) After the second
pulse with CEP = 180∘, symmetric residual population in the Brillouin zone,
i.e., $\mathbf{K}$ and $\mathbf{K}^{\prime}$ valleys have equal population.
(d) Asymmetric population distribution with more in the vicinity of
$\mathbf{K}$ valley. Thus, switch of excitation from $\mathbf{K}^{\prime}$ to
$\mathbf{K}$ valley after the end of the third pulse with CEP = 180∘. Red and
blue represent the residual populations around $\mathbf{K}^{\prime}$ and
$\mathbf{K}$ valleys, respectively.

Owing to the sign of the Berry curvature and resultant selection rules, left-
or right-handed circularly polarized light, resonant to the material’s band
gap, has been employed to achieve valley-selectivity excitation in gapped 2D
materials 11; 12; 13; 14; 15; 16; 17. In recent years, nonresonant laser
pulses with tailored waveforms have been applied to obtain desired valley
excitation in gapped 2D materials 18; 19; 20; 21. Similar significant success
has been reported in gapless 2D materials 22; 23; 24; 25; 26; 27; 28. Huber
and co-workers have experimentally achieved a milestone in coherent valley
switching by using a combination of a pump pulse, resonant with the band gap
in tungsten diselenide, and a controlled terahertz pulse 29. Recently, a
scheme consisting of a pair of two time-separated orthogonally polarized
linear pulses has been proposed for valley excitation in gapped 2D materials
30. Moreover, it has been shown that the desired control over valley
excitation can be obtained by controlling the delay between the two pulses and
their polarizations 30. Ivanov and co-workers have proposed a control protocol
to initiate and switch the valley excitation in gapped 2D materials using a
sequence of four time-separated orthogonally polarized linear pulses 31.
Linear pulses, resonant with the band gap of the 2D materials, are used in
both schemes 30; 31. Thus, they are unsuitable for gapless 2D materials, such
as graphene. Moreover, both the schemes are susceptible to the relative
polarizations and delay among the linear pulses. Thus, a robust and universal
control protocol, applicable to gapped and gapless 2D materials, is lacking –
a major impediment in the realization of versatile light-based valleytronics
devices.

In this work, we provide a universal control protocol for initiating and
switching the excitation from one valley to another on timescales faster than
valley decoherence time. To demonstrate universality, we apply our protocol to
gapped and gapless 2D materials. Monolayer molybdenum disulfide (MoS2) and
graphene will be used to represent gapped and gapless 2D materials,
respectively.

The direction of electron flow is sensitive to the exact shape of the electric
field waveform of the laser, which can be steered by the carrier-envelope
phase (CEP) with attosecond precision. Recently, precise sculpting the
waveforms of one and few-cycle laser pulses via multioctave shaping has been
demonstrated experimentally 32; 33; 34; 35. Linearly polarized laser pulses
with CEP = 0∘ and = 180∘ have the electric field waveform opposite in
direction. Thus, choosing few-cycle laser pulses with well-defined CEP is
expected to facilitate switching the excitation from one direction to another,
which is the essence of our protocol.

Let us understand how few-cycle CEP-stabilized linear pulse can induce valley
polarization. It is known that the canonical momenta is conserved at each
instant of time during laser-solid interaction, i.e.,
${\mathbf{k}(t_{2})}-{\mathbf{A}(t_{2})}={\mathbf{k}(t_{1})}-{\mathbf{A}(t_{1})}$
where ${t_{1}}$ and ${t_{2}}$ are two arbitrary instances during the laser
pulse, $\mathbf{k}$ is the crystal momentum, and $\mathbf{A}(t)$ is the vector
potential of the laser. The probability of an electron excitation from valence
to conduction band is maximum at $\mathbf{K}$ and $\mathbf{K}^{\prime}$
valleys, i.e., near minimum band gap. Moreover, the likelihood of the
excitation is maximum when the electric field strength of the laser is
maximum. Thus, the maximum number of electrons are excited into the conduction
band near the maxima of the electric field. Note that there is no preference
for electron ejection between the two valleys as both valleys are
energetically equivalent.

As the laser pulse gets over, the electronic populations from the $\mathbf{K}$
and $\mathbf{K}^{\prime}$ valleys are shifted to $\mathbf{k}_{1}$ and
$\mathbf{k}_{2}$ as
$\mathbf{K}=\mathbf{k}_{1}(t\rightarrow\infty)+\mathbf{A}(t_{0})$ and
$\mathbf{K}^{\prime}=\mathbf{k}_{2}(t\rightarrow\infty)+\mathbf{A}(t_{0})$
with $t_{0}$ as the time at which the electric field is maximum. The
population distribution associated with the electron excitation is symmetric
when the maxima of the electric field coincide with the zero of the vector
potential, which is true for a longer laser pulse. Thus, “relatively” longer
pulse yields symmetric electron excitation around $\mathbf{K}$ and
$\mathbf{K}^{\prime}$ valleys and, therefore, no valley polarization. This
situation changes drastically as the pulse becomes shorter and sensitive to
the CEP. In the case of the few-cycle CEP-sensitive pulse, the maxima of the
electric field does not coincide with the zero of the vector potential. The
value of the vector potential is sensitive to the value of the CEP. Thus, the
nonzero value of the vector potential at $t_{0}$ creates asymmetric population
distribution around $\mathbf{K}$ and $\mathbf{K}^{\prime}$ valleys 36. The
direction of the electron excitation is dictated by the direction of the field
amplitude. For CEP value of 0∘, the magnitude of the vector potential is
positive at the maximum of electric field, which favors $\mathbf{K}^{\prime}$
valley over $\mathbf{K}$. On the other hand, as the CEP changes to 180∘, the
magnitude of the vector potential becomes negative and thus $\mathbf{K}$
valley is favored.

The key idea of our valley-switching protocol by an all-optical means in a 2D
material, with two degenerate valleys at $\mathbf{K}$ and
$\mathbf{K}^{\prime}$, is illustrated in Fig. 1. Coherent switching of an
electron excitation from one valley to another in a 2D material can be
envisioned as a three-step process: in the first step, a linear laser pulse
induces a preferable valley excitation at $\mathbf{K}^{\prime}$ over
$\mathbf{K}$. An application of a second linear pulse during the second step
could counterbalance the preferred valley excitation at $\mathbf{K}^{\prime}$
by reversing the direction of the excitation. The final step involves the
action of a third linear pulse, similar to the one used in the second step,
which would manage the excitation away from $\mathbf{K}^{\prime}$ valley as it
favors the excitation to $\mathbf{K}$ valley. Thus, switching of the valley
excitation from one to another can be potentially realized by the action of
three time-separated CEP-controlled linear pulses polarized along the same
direction. Our approach is simpler than the one discussed in Ref. 31 where a
sequence of four orthogonally polarized linear pulses, resonant with the band
gap of a 2D material, is used. Furthermore, as we demonstrate below, our
protocol is robust against several limitations in the experimental setup as it
is insensitive to the wavelengths of the laser pulses, time delay among the
pulses, and dephasing time. The protocol discussed in Ref. 31 is susceptible
to the experimental setup.

Figure 3:  All-optical valley switch in MoS2. (a) Energy band structure of
MoS2 displays two valleys along high-symmetry points at $\mathbf{K}$ and
$\mathbf{K}^{\prime}$. Energy band gaps at $\mathbf{K}$ and
$\mathbf{K}^{\prime}$ are 1.58 eV. The momentum-resolved residual electronic
populations in the first Brillouin zone of the conduction band after the end
of the first, second, and third laser pulses are shown in (b) - (d),
respectively. Red and blue represent the residual populations around
$\mathbf{K}^{\prime}$ and $\mathbf{K}$ valleys, respectively. The details of
the linear pulses and the residual populations in (b) - (d) are the same as in
Fig. 2.

##  II Computational Details

The nearest-neighbour tight-binding approximation is used to obtain the
electronic ground state of gapped and gapless graphene. The corresponding
Hamiltonian is expressed as

$\mathcal{{H}}_{\textbf{k}}=-\gamma_{0}f^{*}(k)\hat{a}_{\textbf{k}}^{\dagger}\hat{b}_{\textbf{k}}-\dfrac{\Delta}{2}\hat{a}_{\textbf{k}}^{\dagger}\hat{a}_{\textbf{k}}+\dfrac{\Delta}{2}\hat{b}_{\textbf{k}}^{\dagger}\hat{b}_{\textbf{k}}+\text{H.c.}$
(1)

Here, $\gamma_{0}$ is the nearest-neighbor hopping energy,
$\hat{a}_{k}^{\dagger}(\hat{b}_{k})$ is creation (annihilation) operator
corresponding to the A (B) type of the atom in the unit cell. $f(k)$ is
defined as $\sum_{i\in nn}~{}e^{i\textbf{k}\cdot\textbf{d}_{i}}$ with
$\textbf{d}_{i}$ as the separation vector between an atom with its nearest
neighbor 37; 38; 39. The band structure of the 2D material is obtained by
solving Eq. (1). To obtain the minimum band gap at $\mathbf{K}$ and
$\mathbf{K}^{\prime}$ points, we use $\Delta=$ 0 and 1.58 eV for graphene and
MoS2, respectively.

Semiconductor-Bloch equations in Houston basis are used to simulate the
electron dynamics in the presence of few-cycle laser pulses 40 and defined as

$\displaystyle\frac{d}{dt}\rho_{vv}^{\textbf{k}}$
$\displaystyle=i\textbf{E}(t)\cdot\textbf{d}_{vc}(\textbf{k}_{t})\rho_{cv}^{\textbf{k}}+\textrm{c.c.}$
(2a) $\displaystyle\frac{d}{dt}\rho_{cv}^{\textbf{k}}$
$\displaystyle=-\left[i\varepsilon_{cv}(\textbf{k}_{t})+\frac{1}{\textrm{T}_{2}}\right]\rho_{cv}^{\textbf{k}}+i\textbf{E}(t)\cdot\textbf{d}_{cv}(\textbf{k}_{t})\left[\rho_{vv}^{\textbf{k}}-\rho_{cc}^{\textbf{k}}\right].$
(2b)

Here, $\textbf{k}_{t}$ stands for $\textbf{k}+\textbf{A}(t)$, $\textbf{E}(t)$
is the electric field of the laser and $\textrm{T}_{2}$ represents a
phenomenological dephasing term to incorporate the decoherence between
electrons and holes. $\textbf{d}_{cv}(\textbf{k})=i\langle
c,\textbf{k}|\nabla_{\textbf{k}}|v,\textbf{k}\rangle$, and
$\varepsilon_{cv}(\textbf{k})$ are the dipole matrix element and band-gap
energy between conduction and valence bands at k, respectively. Note that
$\rho_{cc}^{\textbf{k}}(t)=1-\rho_{vv}^{\textbf{k}}(t)$, and
$\rho^{\textbf{k}}_{vc}(t)=\rho^{\textbf{k}*}_{cv}(t)$. The electronic
population of the conduction band is estimated using
$n^{\textbf{k}}(t)=\rho^{\textbf{k}}_{cc}(t).$

To demonstrate the robustness of our idea, the parameters of the three laser
pulses are similar for graphene and MoS2. Realistic two-cycle pulses with
well-defined CEP are used in our control protocol. All three pulses have a
wavelength of 2.0 $\mu$m with cosine squared envelope, and are linearly
polarised along $k_{x}$ direction, i.e., the zigzag ($\Gamma-\mathbf{K}$)
direction. The CEP of the first, second, and third pulses are 0∘, 180∘, and
180∘, respectively. For graphene, the peak intensities of the first, second,
and third pulses are 3.8$\times 10^{12}$, 3.8$\times 10^{12}$, and 4.4$\times
10^{12}$ W/cm2, respectively. The peak intensities of the three pulses are
slightly tuned to optimize valley switching in MoS2. In this case, the peak
intensities of the first, second, and third pulses are 2.6$\times 10^{12}$,
2.6$\times 10^{12}$, and 3.0$\times 10^{12}$ W/cm2, respectively. In both
cases, the intensity of the third pulse is slightly higher than the second one
to favor the electron excitation to another valley. However, the ratios of the
three pulses are kept constant in graphene and MoS2. Note that the intensities
of the laser pulses are close to the damage threshold of monolayer graphene
41. By using an appropriate substrate, damage of the sample can be further
avoided.

##  III Results and Discussion

Let us first discuss monolayer graphene’s numerical results to confirm our
valley switching protocol. There are two equivalent valleys at $\mathbf{K}$
and $\mathbf{K}^{\prime}$ as evident from the energy band structure of
graphene along high-symmetry points in the first Brillouin zone, see Fig.
2(a). The band dispersion is linear, and energy gaps between valence and
conduction bands at $\mathbf{K}$ and $\mathbf{K}^{\prime}$ valleys are zero.
Thus, schemes based on resonant laser pulses for valley-selective excitation
are not suited for graphene.

The residual electronic populations in the conduction band after the first,
second, and third pulses are presented in Figs. 2(b) - 2(d), respectively. The
populations in the vicinity of $\mathbf{K}$ and $\mathbf{K}^{\prime}$ are
shown by red and blue, respectively. After the first pulse, the residual
population is asymmetric. Moreover, the population in $\mathbf{K}^{\prime}$
valley is more in comparison to the $\mathbf{K}$ valley as reflected from Fig.
2(b).

Let us uncover the underlying mechanism for the preferential population in
$\mathbf{K}^{\prime}$ valley by the first laser pulse. As discussed earlier,
the exact shape of the waveform of an intense laser dictates the direction of
the electron’s motion, either positive or negative, along the $x$ axis. Also,
a sufficiently intense laser not only excites electrons in the valley but also
its vicinity. As the time starts, electrons are driven towards a positive
direction as the vector potential of the first laser has a negative slope and
increases in the same direction. At some point in time, electrons change their
direction of motion from positive to negative due to a change in the slope of
the vector potential from negative to positive as the potential peaks twice
during each half cycle of the waveform. The two paths of the electron motion
lead an interference, which results in interference fringes in the residual
population as evident from Fig. 2(b) [see S1 within the Supplemental
Material]. Due to zero CEP of the pulse, the vector potential is asymmetric
and has a net positive component, which leads the $\mathbf{K}^{\prime}$ valley
to be more populated over $\mathbf{K}$ valley. Thus, the asymmetry of the
laser waveform gets imprinted in the asymmetric residual populations.

The preference for excitation from one valley to another can be altered by
changing the CEP from 0∘ to 180∘. To counterbalance the preferred population
in $\mathbf{K}^{\prime}$ valley caused by the first pulse, let us employ a
second pulse with CEP = 180∘ that favors the population in $\mathbf{K}$
valley. Thus, the combined effect of both pulses leads to a symmetric residual
population around both valleys as reflected from Fig. 2(c).

To break the equal population distribution around $\mathbf{K}$ and
$\mathbf{K}^{\prime}$, a third pulse with CEP = 180∘ is applied, which induces
asymmetric residual population. However, this time $\mathbf{K}$ valley has a
more residual population over the $\mathbf{K}^{\prime}$ valley, see Fig. 2(d).
Thus, the action of three time-separated linear pulses switches the population
from one valley to another completely 42. Moreover, the switch of valley
excitation from $\mathbf{K}^{\prime}$ to $\mathbf{K}$ valley occurs within a
few tens of femtoseconds – an order of magnitude faster than the valley
decoherence time 9; 10. Note that two-cycle laser pulses with CEP = 90∘ and
270∘ have zero net components and do not prefer any valley-selective
excitation.

After a successful illustration of valley switching in gapless graphene, let
us explore the universality of our control protocol. For this purpose, we
apply our protocol to MoS2, which belongs to another class of 2D materials,
i.e., gapped graphene. As evident from Fig. 3(a), there are two valleys at
$\mathbf{K}$ and $\mathbf{K}^{\prime}$ with a band gap of 1.58 eV, which makes
MoS2 a semiconductor in nature. To claim the universality, the sequence of the
three time-separated laser pulses are the same for MoS2 as for graphene.

The residual population in the conduction band after the end of the first
pulse is shown in Fig. 3(b). As evident from the figure, the first pulse with
CEP = 0∘ favors electron excitation in $\mathbf{K}^{\prime}$ valley. The
second pulse with CEP = 180∘ dispenses the population equally to both
$\mathbf{K}$ and $\mathbf{K}^{\prime}$ valleys [see Fig. 3(c)]. The valley
excitation is completely switched to $\mathbf{K}$ valley by the action of the
third pulse with CEP = 180∘ as reflected from Fig. 3(d). The results presented
in Fig. 3 unequivocally establish that our protocol with the same setup, as in
graphene, is equally applicable to MoS2 for a complete switch of valley
excitation from one valley to another. Thus, our control protocol of valley
switch is equally applicable to gapped and gapless 2D materials, unlike the
protocol discussed in Ref. 31.

Figure 4: Robustness of our protocol with respect to laser parameters and
dephasing time. The ratio of residual population in the conduction band around
$\mathbf{K}$ and $\mathbf{K}^{\prime}$ valleys
($\textrm{n}_{\textrm{K}}/\textrm{n}_{\textrm{K}^{\prime}}$) with respect to
(a) laser’s wavelength $\lambda$, (b) dephasing time $\textrm{T}_{2}$, time
delay between (c) the first and second laser pulses $\tau_{12}$, and (d) the
second and third laser pulses $\tau_{23}$. Time delay between the second and
third laser is constant in (c), and the delay between the first and second
laser is constant in (d). The wavelength of all three pulses is kept the same
in (a). The residual populations after the first, second, and third pulses are
shown by blue, black, and red, respectively.

After establishing valley switching in graphene and MoS2, let us explore the
robustness of our protocol concerning the various parameters in an
experimental setup. To this end, we analyze the ratio of the residual
population in graphene after the end of the first, second, and third laser
pulses, as shown by blue, black, and red lines [See Fig. 4]. As illustrated
earlier, the residual population in the $\mathbf{K}^{\prime}$ valley is
greater than the $\mathbf{K}$ valley, so the ratio is less than one. However,
the ratio becomes one after the end of the second pulse as it counterbalances
the population in both the valleys. The action of the third pulse favors the
residual population in the $\mathbf{K}$ valley, and hence the ratio becomes
more than one. It is expected that the change in the wavelength of the laser
pulses does not affect the outcome of the valley switch from
$\mathbf{K}^{\prime}$ to $\mathbf{K}$ as the underlying mechanism relies on
tuning the CEP of the pulses. The results in Fig. 4(a) confirm our
expectations as our protocol of the valley switch is insensitive with respect
to the change in the wavelength from 2.0 to 6.0 $\mu$m for all the three
pulses. Intensity ratios of the three pulses are kept constant while varying
the wavelengths.

It has been anticipated that the coherence between electron and hole at their
respective energy bands is a crucial factor in the functioning of light-driven
valleytronics devices 9. Thus, it is essential to check how our protocol
depends on the electron-hole coherence. The residual populations after the end
of the three pulses are presented for different dephasing time
$\textrm{T}_{2}$, which determines the decoherence between electron and hole.
As evident from Fig. 4(b), the ratios of the populations remain unaffected for
different values of $\textrm{T}_{2}$, ranging from 5 to 50 fs. Invariance of
the ratios with $\textrm{T}_{2}$ also indicates that the electron excitation
by CEP-controlled laser pulses is dominated by the intraband motion 43.

It is natural to envisage that our control protocol is robust against the time
delay among the three pulses as the populations ratio do not change with
$\textrm{T}_{2}$. To verify this hypothesis, we present the population ratios
for the different delays between the first and second pulses ($\tau_{12}$)
while keeping the delay between the second and third pulses constant [see Fig.
4(c)]. As evident from the figure, the residual populations after the end of
the three pulses remain the same for different values of $\tau_{12}$. The same
is true when the delay between the second and third pulses ($\tau_{23}$) is
changed, while $\tau_{12}$ is kept constant [see Fig. 4(d)]. Thus, the
analysis of Figs. 4(c) and 4(d) demonstrate the protocol of valley switching
is independent of the time delay among the three pulses, i.e., changes in
$\tau_{12}$ and $\tau_{23}$. The other parameters are kept identical as in
Fig. 2, while varying $\tau_{12}$ and $\tau_{23}$.

##  IV Conclusion

In summary, we illustrate a coherent control protocol to switch the valley
excitation from one valley to another in 2D materials on a timescale faster
than any valley decoherence time. Our protocol consists of three time-
separated two-cycle linear pulses, polarized along the same direction. For
efficient valley switching, the electric field waveform of the employed pulses
are tailored by tuning the CEP. We successfully illustrate the valley switch
in graphene and MoS2. Thus, our protocol is versatile as it relies on CEP-
controlled nonresonant pulses and equally applicable to hexagonal gapped and
gapless 2D materials. Moreover, our protocol is resilient to significant
parameters in an experimental setup as it is oblivious to the wavelength of
the pulses, time delay among pulses, and the dephasing time. The laser pulses
used in our protocol have been recently employed to explore electron dynamics
in solids 33; 34; 44. Therefore, our protocol of valley switching is within
reach of experimental feasibility. Additionally, the present work can be
extended to realize logical operations using valley pseudospins – similar to
the recent experimental work in graphene using real and virtual charge
carriers 45. We also test our protocol with CEP-controlled single-cycle
pulses, and our findings remain qualitatively the same. However, valley
polarization reduces drastically as the number of cycles in a laser pulse
increases 36. To end, high-harmonic spectroscopy and time-resolved angle-
resolved photoemission spectroscopy could be employed to read the outcomes of
the valley switching 46; 47; 48; 49.

##  V Acknowledgement

We acknowledge fruitful discussions with Misha Ivanov (MBI Berlin), Alvaro
Jimenez-Galan (MBI Berlin), Mandar Deshmukh (TIFR India), M. S. Mrudul
(Uppsala University, Sweden) and Sumiran Pujari (IIT Bombay). G. D.
acknowledges support from SERB India (Project No. MTR/2021/000138).

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