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\begin{document}
\title{Periodic Schr\"odinger Operators with Large Gaps and
Wannier--Stark Ladders}
\date{}
\author{J.E.Avron$^1$, P.Exner$^2$, Y.Last$^1$}
\maketitle
\begin{center}
$^1$ Department of Physics, Technion -- Israel Institute of
Technology, \\ 32000 Haifa, Israel \\
$^2$ Nuclear Physics Institute, AS CR, 25068 \v Re\v z near Prague,
Czech Republic
\end{center}
\vspace{25mm}
\noindent
We describe periodic, one dimensional Schr\"odinger operators, with
the property that the widths of the forbidden gaps increase at large
energies and the gap to band ratio is not small. Such systems can
be realized by periodic arrays of geometric scatterers, {\it e.g.},
a necklace of rings. Small, multichannel scatterers lead (for low
energies) to the same band spectrum as that of a periodic array of
(singular) point interactions known as $\,\delta'$. We consider the
Wannier--Stark ladder of $\:\delta'\,$ and show that the
corresponding Schr\"odinger operator has no absolutely continuous
spectrum.
\vspace{60mm}
\noindent
PACS: 73.20Dx; 71.50+t
\pagebreak
In this letter we discuss spectral properties of some one
dimensional, one electron Schr\"odinger equations. Our purpose is
threefold: first, we want to point out interesting features in the
band--gap structure of periodic arrays of geometric scatterers, such
as the necklace shown in Fig.1; second, we show that a singular
point interaction, known as $\,\delta'$, which leads to interesting
spectral properties and has been extensively studied \cite{point}, is
a useful paradigm for the finite energy behavior of appropriate
geometric scatterers; and finally, we describe the spectral
properties of Wannier--Stark ladders for a periodic array of
$\,\delta'$ scatterers, which are quite unlike those of the
Wannier--Stark ladders of smooth potentials (see, {\it e.g.},
\cite{GMS,Ne,CFKS} and references therein).
By a classical result, the spectrum of the one electron Schr\"odinger
equation with periodic potential is in the form of bands and gaps.
Recall that for smooth periodic potentials the size of the $\,n$--th
gap is rapidly decreasing and the band widths increase linearly with
the band index $\,n\;$ (for more precise information see {\it e.g.}
\cite{WK}). A common wisdom says that the Kronig--Penney model
(made of a periodic array of Dirac delta functions) gives the slowest
decay of gap widths. In this case the gap widths approach a constant
at high energies and the gap to band ratio goes to zero like
$\,1/n\,$, with $\,n\,$ the band index. So, in general, periodic
potentials are expected to have a gap to band ratios that decrease at
high energies at least as fast as $\,1/n\,$.
Periodic Schr\"odinger operators with singular interactions may have
increasing gaps and even increasing gap to band ratios. This is the
case for a point interaction known as $\,\delta'$, which, like the
usual Dirac $\,\delta$, is concentrated on a lattice of points. More
precisely, a $\,\delta'$ point scatterer of strength $\,\lambda\;$
(measured in units of length), is characterized by the transfer
matrix $\,T_{\delta'}(\lambda) \equiv \left(\matrix{ 1 & \lambda\cr
0 & 1\cr}\right)\,$ which relates $\,\left(\matrix{ \psi \cr
\psi'\cr}\right)\,$ on the two sides of the scatterer, {\it i.e.}
the wave function has continuous first derivatives on the right and
left, and a jump proportional to the first derivative. This boundary
condition satisfies Kirchoff's law and leads to a self-adjoint
Schr\"odinger operator for any real $\,\lambda\,$~
\cite{point,Se}.
The boundary conditions embodied in $\,\delta'$ appear at first to be
unnatural for quantum mechanics, and some efforts have been made in
order to assign $\,\delta'$ a quantum mechanical interpretation
\cite{Se}. It turns out that unlike Dirac's $\,\delta$, the
$\,\delta'\,$ can not be approximated by potentials, {\it i.e.}
functions of the coordinate with small support. In particular, it has
little to do with the derivative of Dirac's $\,\delta\,$ function.
Rather, the (known) approximants involve functions of both coordinate
and momenta \cite{Se}. The absence of a good realization of
$\,\delta'$ may be the reason why it has not attracted more
attention. One of our aims is to rectify this situation, and to show
that the unique spectral properties of $\,\delta'$ are, in fact, a
paradigm for geometric scatterers.
Consider the band--gap structure of periodic Schr\"odinger
operators that come from allowing complicated geometries in one
dimension, {\it e.g.} like those of a periodic necklace of rings and
its onion--like generalizations.
%\special
Fig.1 illustrates one such a system, where onion--like scatterers
made of four wires, or channels, are connected by wires. The gap to
band ratio of such objects does not decrease at high energies, and,
in particular, both bands and gaps tend to increase. As we shall
show, in certain (limiting) cases, such as the periodic array of
small geometric scatterers with many short channels, the band--gap
structure (for low energies) approximates that of a periodic array of
$\,\delta'$.
The out of the ordinary band--gap structure of $\,\delta'$ comes
together with an out of the ordinary Stark effect: recall that for a
large class of Stark Hamiltonians in one dimension, including
Wannier--Stark Hamiltonians, under rather week differentiability
conditions on the potential, the spectrum coincides with the real
axis and is (purely) absolutely continuous~\cite{CFKS}. The
Wannier--Stark ladder {\bf is not} a ladder of eigenvalues (we
refer the reader to \cite{GMS}, and references therein, for results
on the question of existence of Wannier--Stark ladder as a ladder of
resonances, and to \cite{MB} for the experimental status). In
contrast, tight--binding models, which have only a finite number of
bands, have Wannier--Stark ladders of (discrete) eigenvalues.
Tight--binding models may be thought of as a limiting situation of
the Schr\"odinger operator with an infinitely large gap at high
energies. The question whether the Wannier--Stark ladder is a
ladder of eigenvalues or not seems, therefore, to be related to the
structure of gaps at high energies. This point of view has been
stressed by Ao who also argued that the Kronig--Penney model is a
borderline situation, which for weak electric fields has
Wannier--Stark ladder of eigenvalues, but for strong electric fields
does not \cite{Ao}. Although we have nothing to say about this
intriguing transition, our results provide some support to the
overall point of view (see also \cite{DSS} for related phenomena in
the random setting).
Let $\,\lambda\delta'_x\,$ denote a $\,\delta'\,$ point scatterer of
strength $\,\lambda$, located at $\,x\,$, and
\begin{equation}
H(a,\lambda,F) \,=\,
-\,{\hbar^2\over 2m}\, {d^2\over dx^2} \,+\,
\sum_{n\in Z}\,\lambda\delta'_{na} \,-\, eFx\,,
\end{equation}
denote the Wannier--Stark ladder operator for $\,\delta'$, with
$\,a\,$ the period, and $\,F\,$ the electric field. We choose $\,eF
>0\,$. The symbol $\,H(a,\infty,F)\,$ stands for Neumann boundary
conditions at each point of the lattice.
It is a general fact about the Wannier--Stark ladder Hamiltonians (which
follows from the unitary translation by lattice spacing) that the
spectral properties are periodic with period $\,eFa\,$. For example,
\begin{equation}
{\rm Spec}\bigl( H(a,\lambda,F)\bigr) \,=\,
{\rm Spec}\bigl(H(a,\lambda,F)\bigr) \,+\,eFa\,.
\end{equation}
We now state our results on Wannier--Stark ladders for $\,\delta'\,$:
\vspace{3mm}
\noindent
{\bf Theorem.} {\it For $\,F,\lambda\neq 0\,$, the absolutely
continuous spectrum of $\,H(a,\lambda,F)\,$ is empty.}
\vspace{3mm}
\begin{description}
\item{\hspace{-3mm} {\it Remarks:} 1.} The proof of the Theorem
is rather technical and, therefore, shall be given elsewhere
\cite{AEL}. Below we shall sketch the ideas and the basic
intuition behind the proof.
\item{2.} The assertion of the theorem generalizes to
non-identical scatterers, {\it i.e.}, to situations where $\,\lambda\,$
is replaced by a sequence $\,\{\lambda_n\}\,$ which may be position
dependent, provided $\,|\lambda_n| \ge \lambda >0\,$.
\item{3.} The essential spectrum \cite{RS1} of $\,H(a,\infty,F)\,$ is
given explicitly by the accumulation points of the set
\begin{equation}
\Big\{ eFa\big(\gamma n^2 -k -1/2\big)\,\big|\;\; n,k\in Z \;\Big\}\,,
\end{equation}
where $\,\gamma \equiv\ h^2/8meFa^3$. Thus, for $\,\gamma =p/q\,$
rational, the (Neumann) essential spectrum is contained in the set
$\,(eFa)\, \{\ell/q -1/2\, |\; \ell \in Z\,\}\,$ and has gaps (in
fact, it is a nowhere dense, countable set). For $\,\gamma\,$
irrational it is the real axis. Our results say nothing on what is
the essential spectrum (as a set) for finite $\,\lambda\,$ nor if the
spectrum is pure point or singular continuous.
\item{4.} The Wannier--Stark problem in one dimension and the
Zeeman problem with periodic potentials in two dimensions have a
dimensionless parameter which characterizes commensuration of
periods. In the Zeeman problem it is the number of quantum flux units
per unit cell \cite{Ne}. In the Wannier--Stark problem it is
$\,\gamma\,$. Aspects of this fact, in the tight binding setting,
have been stressed in \cite{Niu}. In the Zeeman problem it is known
that the spectrum, as a set, is sensitive to number theoretic
properties the of the flux per unit cell. There is, of course, no
corresponding sensitivity of the spectrum for Wannier-Stark
Hamiltonians with smooth periodic potentials. The sensitivity of the
spectrum for the $\,\delta'\,$ Wannier--Stark Hamiltonians as
function of $\,\gamma\,$, is open.
\end{description}
A basic intuition to the results discussed above comes from
considering first the scattering properties of a single scatterer.
Let us start by contrasting $\,\delta'\,$ of strength $\,\lambda\,$,
with the Dirac $\,\delta\,$ interaction of strength $\,1/\lambda\,$.
The Dirac delta interaction has transfer matrix $\,T_{\delta
}(1/\lambda) \equiv\, \left(\matrix{ 1 & 0\cr 1/\lambda &
1\cr}\right)\,$. For a single delta scatterer, the reflection
amplitude for wave number $\,k\,$ is \cite{lr} $\;r=
-1/(1\!-\!2ik\lambda)\,$ which goes to $\,-1\,$ as $\,k\lambda \to
0\,$. (Here $\,k\equiv \sqrt{ 2m E/\hbar^2}\ge 0\,$ is the wave
number associated with an energy $\,E\,$). Total reflection can be
interpreted as a decoupling of the two sides of the scatterer which
occurs here through a Dirichlet boundary condition. Decoupling is,
therefore, a low energy phenomenon for Dirac's $\,\delta$. On the
other hand, for a single $\,\delta'$ the reflection amplitude is
$\,r=\,ik\lambda /(ik\lambda\!-\!2)$ which approaches $\,+1\,$ in
the limit $\,k\lambda \to \infty\,$. Now the total reflection can be
interpreted as decoupling the two sides of the scatterer through an
approach to a Neumann boundary condition. The remarkable fact about
$\,\delta '\,$ is that the decoupling is a {\bf high} energy
phenomenon! The effective decoupling at large wavenumbers is at the
heart of many of the unique properties we shall discuss below.
Consider now the onion--like scatterers with $\,N\,$ channels, {\it
i.e.} replace the $\,N=4\,$ channels of Fig.1 by a general $\,N\,$.
For simplicity sake, suppose that all the wires are ideal identical
conductors of length $\,L\,$, and that the boundary conditions at the
two vertices are that the wave function has a unique limit at the
vertices, and that $\,\sum\psi' \Big|_{\rm Vertex} =0\;$ (all legs
are considered outgoing from the vertex). The transfer matrix across
the scatterer is
\begin{equation}
T_{N,L}(k) \,=\, \left(\matrix{ \cos (kL) & \sin (kL)/Nk \cr
-Nk \sin(kL) & \cos(kL) \cr} \right)
\end{equation}
(in the special case $\,N=1\,$ it corresponds to an ideal wire of
length $\,L\,$). The reflection amplitude from one such scatterer is
\begin{equation} \label{reflection}
r(kL;N) \,=\, {-N^2+1 \over N^2 + 2iN \cot (kL)+1}\,.
\end{equation}
The reflection is periodic in $\,k\,$, something one expects for a
geometric structure; there is no limit at high energies.
Note first that in the $\,k\to 0\,$ limit, $\;T_{N,\ell}(k)\to
T_{\delta'}(L/N)\,$, expressing the fact that in the
long--wavelength limit a geometric scatterer looks point--like. This,
however, does not yet say that geometric scatterers are like $\,\delta'$,
because, from our perspective, the crucial point of $\,\delta'$ is the
high energy decoupling. In fact, as Eq.~(\ref{reflection}) shows,
the reflection has no limit as $\,k\to\infty\,$. However, the reflection
from certain geometric scatterers can mimic the reflection from a
$\,\delta'\,$ in the following sense: consider the limit of a small
scatterer with many channels: $\;L \to 0,\quad NL =\beta\,$,
\begin{equation} \label{reflection limit}
r(kL; N) \,\to\, -(1+2i/\beta k)^{-1}\,.
\end{equation}
When $\,k\to \infty\;$ (but still $\,kL\ll 1\,$) the reflection
amplitude goes to $\,-1\,$, which gives the requisite decoupling at
high energies (albeit through Dirichlet). One should therefore expect
that such geometric scatterers will share with $\,\delta'\,$ some of
its remarkable features. We shall now discuss evidence for this.
By stringing geometric scatterers or point scatterers with wires,
or with a second type of geometric scatterers, to a periodic necklace as
{\it e.g.\ } in Fig.1, we get a one dimensional system with spectrum
made of bands and gaps. Let $\,T_{\rm Period}(k)\,$ be the transfer
matrix for one period, for wave number $\,k\,$. The discriminant is
$\,\Delta (k) \equiv {\rm Tr}\, \left( T_{\rm Period}\right)\,$. By
Floquet theory, the bands are given by the condition $\,-2 \le \Delta
(k) \le 2\,$. It is a straightforward exercise to show that for the
Kronig--Penney $\,\delta'\,$ model \cite{point}
\begin{equation} \label{discriminant}
\Delta_{KP \delta'} (k) = 2 \cos (ka) -\lambda k \sin (ka)\,.
\end{equation}
For a necklace made of a pair of interlacing geometric scatterers
\begin{eqnarray} \label{necklace discriminant}
\lefteqn{\Delta_{\rm Necklace} (k) \,=\, \left( 1 +{N_1\over 2N_2}
+{N_2\over 2N_1}\right) \cos \big(k(L_1\!+L_2)\big)} \nonumber \\ \\
&& +\left( 1 - {N_1\over 2N_2} -{N_2\over 2N_1}\right) \cos
\big(k(L_1\!-L_2)\big) \nonumber
\end{eqnarray}
(for the necklace of Fig.1, $\,N_1=1\,$ and $\,N_2=4\,$). It is
known \cite{point}, and can easily be shown to follow from
Eq.~(\ref{discriminant}), that the band--gap structure of the
Kronig--Penney $\,\delta'\,$ model has gaps that increase linearly
with the band index $\,n\,$ while the bands at large energies
approach a constant width which is $\,4\hbar^2/ma\lambda\,$.
This narrowness of the bands (compared with the large gaps) can be
understood as a consequence of the fact that at high energies, the
unit cells get decoupled since $\,\delta'\,$ approximates Neumann
boundary conditions.
The dependence of the discriminant of the necklace,
Eq.~(\ref{necklace discriminant}), on $\,k\,$ is, of course,
trigonometric. This implies that the band--gap structure is
periodic (or almost periodic) in $\,k\,$. In particular, both bands
and gaps tend to increase, linearly with $\,k\,$, at large energies,
as a consequence of the fact that $\,E=\hbar^2 k^2/2m\,$.
Furthermore, the gap to band ratio is almost periodic as well. This
behavior is qualitatively different from what one gets from periodic
potentials where the dependence of the discriminant on $\,k\,$ is not
trigonometric and where, on general grounds, one has $\,\Delta_{\rm
Period}(k) - 2\cos (ka)\to 0\;$ as $\,k\to\infty\,$. The special case
$\,L_1=L_2\,$ gives rise to an amusing situation where half of the gaps
(all the periodic ones) are closed.
Fig.2 shows the discriminant and the bands for the necklace of Fig.1 as
function of $\,k\,$.
%\special
The length ratio of the wire to scatterer is $\,5\,$. (The
discriminant is computed for the case where all four channels of the
scatterer have identical lengths). As one can see, even when the number
of channels is relatively small ($\,N_2=4\,$), and when the scatterers are
not really tiny ($\,L_1/L_2=5\,$), a feature of $\,\delta'\,$ emerges
in that the second gap is larger than the first, and the third gap is
larger than the second. The pattern reverses and then repeats
periodically. The figure shows one period. Taking $\,N_2\,$ larger
and $\,L_2\,$ smaller leads to gap increase for many more gaps.
One can actually make the gaps grow with energy on arbitrarily large
scales by taking increasingly complicated scatterers. For example, by
considering the limit of Eq.(\ref{reflection limit}), {\it i.e.},
$\,L_2\to 0\,$ and $\,L_2 N_2 = \beta\,$, the discriminant of the
necklace with $\,N_1=1,\;\, L_1=a\;$ obeys
\begin{eqnarray}
\Delta_{\rm Necklace} (k) &\!=\!& 2 \cos\big(k(a\!+\!L_2)\big)
+\left( N_2 + {1\over N_2}-2\right) \sin (kL_2) \sin(ka) \nonumber \\
\\ && \;\to\; 2\cos (ka) - \beta k \sin (ka)\,. \nonumber
\end{eqnarray}
The lower part of the spectrum coincides, therefore, with that of the
Kronig--Penney $\,\delta'\,$ model with $\,\lambda = \beta\,$. This
supports the point of view that the $\delta'$ model can be a useful
paradigm for certain geometric scatterers.
\vspace{3mm}
Let us now make a few comments regarding the proof of the Theorem,
which is an adaptation of a technique previously used by Simon and
Spencer \cite{SSp}. Replacing the $\,\delta'\,$ point scatterer at a
lattice point $\,n_0\,$ by a Neumann boundary condition, {\it i.e.}
setting $\,\lambda_{n_0}=\infty\,$, is a rank one perturbation (of
the resolvent) which decouples the right of $\,n_0\,$ from the left.
It is a general fact that the absolutely continuous spectrum is
stable under finite rank perturbations. The half lattice on the left
is essentially like a triangular well problem and so has discrete
spectrum. This means that the absolutely continuous spectrum is
determined from the half line to the right of $\,n_0\,$. We now
repeat sending $\,\lambda_{n_j}\,$ to infinity for a sequence of
points $\,n_j\,$, $\,j= j_0,j_0\!+1,\dots\;$, that march off to
infinity. Although for large $\,j\,$ the individual $\,\lambda
\delta_{n_j}'\,$ are close to a Neumann boundary condition, one can
actually not take any sequence of points. A judicious choice is
to take a partial sequence of the points $\,n_j \sim const +\gamma
(j+1/2)^2\,$, so that for large $\,j\,$, the Neumann decoupling takes
place at points that lie deep in the forbidden gaps of the periodic
problem, where the discriminant $\,\Delta (k_j)\sim {\cal O}(j)\,$ is
large. (The effective wave number and position are related by
conservation of energy: $\,n_j \sim \gamma (k_j a/\pi)^2\,$). Indeed,
one can show, via the asymptotics of Airy functions, that
$\,\big(H(a,\lambda,F)+iE_0\big)^{-1}\!- \big(H(a,\{\lambda_n\},F)
+iE_0\big)^{-1}\,$, where $\,E_0\,$ is a finite (real) constant,
$\,\lambda_n=\infty\,$ for an appropriate subsequence, and
$\,\lambda_n=\lambda\,$ otherwise, is a trace class operator. By the
Kuroda--Birman Theorem \cite{RS3} this gives the stability of the
absolutely continuous spectrum. Since $\,H(a,\{\lambda_n\},F)\,$
clearly has only pure point spectrum, the theorem follows.
\vspace{5mm}
\subsection*{Acknowledgments}
We benefited from discussions with F.~Gesztesy, S.~Jitomirskaya,
U.~Sivan and D.J.~Thouless. The research is supported by GIF and
DFG through SFB288, the Fund for the Promotion of Research at the
Technion and AS CR grant No.~14814. P.~E. expresses gratitude for
the hospitality extended to him at the Institute of Theoretical
Physics at the Technion and to E.~Schr\"odinger Institute in Vienna.
J.~A. acknowledges the hospitality of the IHES and Y.~L. the
hospitality of Barry Simon at Caltech.
\begin{thebibliography}{article}
\bibitem{point}
A.~Grossmann, R. H\o egh-Krohn, M.~Mebkhout, J. Math. Phys. {\bf
21}, 2376--2385 (1980); S.~Albeverio, F.~Gesztesy, R.~H\o egh-Krohn,
H.~Holden: {\it Solvable Models in Quantum Mechanics}, Springer,
Heidelberg (1988); Yu.N.~Demkov and V.N.~Ostrovskii: {\it Zero Range
Potentials and their Applications in Atomic Physics}, Plenum, New
York (1988).
\vspace{-1.8ex}
\bibitem{GMS}
V.~Grecchi, M.~Maioli and A.~Sacchetti, J. Phys. A~{\bf 26},
L379--L384 (1993); F.~Bentosela and V.~Grecchi,
Commun.~Math.~Phys.~{\bf 142}, 169--192 (1991).
\vspace{-1.8ex}
\bibitem{Ne}
For a recent review see, {\it e.g.}, G.~Nenciu, Rev.~Mod.~Phys.
{\bf 63}, 91--127 (1993), and references therein.
\vspace{-1.8ex}
\bibitem{CFKS}
H.I.~Cycon, R.G.~Froese, W.~Kirsch and B.~Simon: {\it Schr\"odinger
Operators}, Springer (1987).
\vspace{-1.8ex}
\bibitem{WK} M.I.~Weinstein and J.B.~Keller, SIAM J. Appl. Math. {\bf
45}, 941--958 (1987).
\vspace{-1.8ex}
\bibitem{Se}
P.~\v Seba, Rep. Math. Phys. {\bf 24}, 111--120 (1986); M.~Carreau,
J. Phys. A~{\bf 26}, 427--432 (1993); P.R.~Chernoff, R.~Hughes,
J.~Funct.~Anal. {\bf 111}, 92--117 (1993).
\vspace{-1.8ex}
\bibitem{MB}
E.E.~Mendez and G.~Bastard, Phys. Today, 34--42 (June 1993);
G.~Bastard, J.A.~Brum and R.~Ferreira, in {\it Solid State Physics}
{\bf 44}, 229 (H.Ehrenreich and D. Turnbull, ed.), Academic Press,
San Diego (1991), and references therein.
\vspace{-1.8ex}
\bibitem{Ao}
P.~Ao, Phys.~Rev.~B~{\bf 41}, 3998--4001 (1990).
\vspace{-1.8ex}
\bibitem{DSS}
F.~Delyon, B.~Simon and B.~Souillard, Phys.~Rev.~Lett.~{\bf 52},
2187--2189 (1984); Ann. Inst. Henri Poincar\'e {\bf 42}, 283--309
(1985).
\vspace{-1.8ex}
\bibitem{AEL}
J.~E.~Avron, P.~Exner and Y.~Last, in preparation.
\vspace{-1.8ex}
\bibitem{RS1}
M.~Reed, B.~Simon: {\it Methods of Modern Mathematical
Physics, I.~Functional Analysis}, Academic Press, New York (1980).
\vspace{-1.8ex}
\bibitem{Niu}
Q.~Niu, Phys.~Rev.~B~{\bf 40}, 3625 (1989).
\vspace{-1.8ex}
\bibitem{lr}
Left--reflection is related to right--reflection by conjugation. Here
we have chosen $\,r\,$ to stand for the left reflection.
\vspace{-1.8ex}
\bibitem{SSp}
B.~Simon, T.~Spencer, Commun.~Math.~Phys.~{\bf 125}, 113--125 (1989).
\vspace{-1.8ex}
\bibitem{RS3}
M.~Reed, B.~Simon: {\it Methods of Modern Mathematical
Physics, III.~Scattering Theory}, Academic Press, New York (1979).
\end{thebibliography}
\vspace{20mm}
\subsection*{Figure captions}
Fig.1. \quad A necklace made of geometric scatterers in the form of
``onions'', each made of four wires joint at two vertices. The
``onions'' are strung together by connecting wires.
\vspace{3mm}
\noindent
Fig.2 \quad The discriminant for the necklace of Fig.1 as function
of $\,k\,$. The length ratio of the connecting wires to the wires
making the ``onion'' is $\,5\,$. The two straight lines are at $\,\pm
2\, $ and the bands are drawn thick. One period (in $\,k\,$) is
shown and the pattern then repeats periodically.
\end{document}