# Aharonov-Bohm effect in three-dimensional higher-order topological
insulators

Kun Luo  National Laboratory of Solid State Microstructures and school of
Physics, Nanjing University, Nanjing, 210093, China  Hao Geng  National
Laboratory of Solid State Microstructures and school of Physics, Nanjing
University, Nanjing, 210093, China  Collaborative Innovation Center of
Advanced Microstructures, Nanjing University, Nanjing 210093, China  Li Sheng
National Laboratory of Solid State Microstructures and school of Physics,
Nanjing University, Nanjing, 210093, China  Collaborative Innovation Center of
Advanced Microstructures, Nanjing University, Nanjing 210093, China  Wei Chen
Corresponding author<EMAIL_ADDRESS>National Laboratory of Solid State
Microstructures and school of Physics, Nanjing University, Nanjing, 210093,
China  Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China  D. Y. Xing  National Laboratory of Solid
State Microstructures and school of Physics, Nanjing University, Nanjing,
210093, China  Collaborative Innovation Center of Advanced Microstructures,
Nanjing University, Nanjing 210093, China

###### Abstract

The 1D hinge states are the hallmark of the 3D higher-order topological
insulators (HOTI), which may lead to interesting transport properties. Here,
we study the Aharonov-Bohm (AB) effect in the interferometer constructed by
the hinge states in the normal metal-HOTI junctions with a transverse magnetic
field. We show that the AB oscillation of the conductance can clearly manifest
the spatial configurations of such hinge states. The magnetic fluxes encircled
by various interfering loops are composed of two basic ones, so that the
oscillation of the conductance by varying the magnetic field contains
different frequency components universally related to each other.
Specifically, the four dominant frequencies $\omega_{x,y}$ and $\omega_{x\pm
y}$ satisfy the relations $\omega_{x\pm y}=\omega_{x}\pm\omega_{y}$, which
generally holds for different magnetic field, sample size, bias voltage and
weak disorder. Our results provide a unique and robust signature of the hinge
states and pave the way for exploring AB effect in the 3D HOTI.

##  I INTRODUCTION

Over the past two decades, topological phases of matter such as topological
insulator and superconductor have become an active research field of condensed
matter physics Qi and Zhang (2011); Hasan and Kane (2010). These materials are
characterized by the nontrivial band topology and the resultant gapless
(d-1)-dimensional edge states. Very recently, the concepts of higher-order
topological insulators (HOTI) and superconductors are theoretically proposed,
which are featured by the $(d-2)$-dimensional edge states Benalcazar  _et al._
(2017a, b); Song  _et al._ (2017); Langbehn  _et al._ (2017); Schindler  _et
al._ (2018a); Franca  _et al._ (2018); Park  _et al._ (2019); Khalaf (2018);
Vu  _et al._ (2020); Zhang  _et al._ (2020); Yan (2019a); Călugăru  _et al._
(2019); Yan (2019b); Yan  _et al._ (2018); Schindler  _et al._ (2018b).
Specifically, for the 3D HOTI there exist 1D gapless states along the hinges
of the sample, so-called hinge states, while the surface and the bulk states
are both insulating. Recent progresses have shown the evidences of the hinge
states in bismuth by the scanning-tunnelling spectroscopy and Josephson
interferometry Schindler  _et al._ (2018b), which pave the way for exploring
more intriguing properties of such topological states in the HOTI.

The 1D nature of the hinge states indicates that it is a good playground for
exploring various interference effects, such as Aharonov-Bohm (AB) and Fabry-
Pérot interferometers Liang  _et al._ (2001); van Wees  _et al._ (1989); Ji
_et al._ (2003); Ofek  _et al._ (2010); McClure  _et al._ (2009); Nakamura
_et al._ (2019). Actually, the chiral edge states of the quantum Hall phase
have become an important platform for the study of mesoscopic physics, in
which a variety of novel phenomena have been observed Ji  _et al._ (2003);
Henny  _et al._ (1999); Neder  _et al._ (2007); Weisz  _et al._ (2014) due to
its long coherence length and high adjustability. Compared with the chiral
edge states, the hinge states in the HOTI open additional possibilities for
the implementation of novel effects due to their 3D configurations, which
enrich the way of interfering in real space. Moreover, such effects cannot be
realized in any 2D systems, which in turn, can serve as the deterministic
evidence of the hinge states.

Figure 1: (a) The interferometer constructed by HOTI (green block) and normal
metal electrodes (orange blocks). The magnetic field $B$ is imposed in $x$-$y$
plane with a polar angle $\theta$. (b, c) Two elemental interfering loops with
encircled flux $\Phi_{x}$, $\Phi_{y}$. (d, e) Other two dominant interfering
loops with fluxes $\Phi_{x}+\Phi_{y}$, $|\Phi_{x}-\Phi_{y}|$.

The manifestation of the AB effect in an electron system is the periodic
oscillation of conductance as the closed trajectory of electrons encircles a
magnetic flux $\Phi$ Aharonov and Bohm (1959); Webb  _et al._ (1985); Holloway
_et al._ (2015); Lahiri  _et al._ (2018); Aleiner  _et al._ (2015);
Tserkovnyak and Halperin (2006). The dominant period of oscillation is equal
to the flux quantum $\Phi_{0}=h/e$, with a main frequency $2\pi/\Phi_{0}$. The
frequency of oscillation can be found by taking the fast Fourier transform
(FFT) of the conductance pattern Webb  _et al._ (1985); Holloway  _et al._
(2015); Lahiri  _et al._ (2018). Recently, AB effect has been used as an
effective way to detect edge states in various topological systems, such as
edge states of topological insulators Peng  _et al._ (2010); Bardarson  _et
al._ (2010); Zhang and Vishwanath (2010); Xypakis  _et al._ (2020), Majorana
fermions of topological superconductors Li  _et al._ (2012); Ueda and Yokoyama
(2014); Tripathi  _et al._ (2016); Bartolo  _et al._ (2020), surface states of
topological semimetal Wang  _et al._ (2016); Lin  _et al._ (2017), and non-
Abelian anyons of fractional quantum Hall systems Kim (2006); Halperin  _et
al._ (2011); Willett  _et al._ (2013); Nakamura  _et al._ (2019, 2020).

In this work, we investigate the AB effect in the interferometer composed of
the hinge states of the quadrangular HOTI by imposing an external magnetic
field. The insulating bulk and surface states indicate that the electron can
only propagate along the hinges of the sample, by which the enclosed magnetic
flux can lead to a coherent oscillation of the transmission probability.
Different from the edge states in any 2D systems, the 3D network of the hinge
states results in peculiar interfering trajectories, which relies not only on
the magnitude of the magnetic field but also on its orientation. The AB
interferometer is sketched in Fig. 1(a), where the HOTI is connected to two
leads made of the normal metal and a magnetic field
$\bm{B}=(B_{x},B_{y})=B(\cos\theta,\sin\theta)$ is applied in the $x$-$y$
plane with $\theta$ being the polar angle. The electrons injected from the
leads propagate along four chiral hinge states, which comprise a variety of
interfering loops; see Figs. 1(b)-1(e). The elemental interfering loops shown
in Figs. 1(b) and 1(c) are exactly the boundaries of the ($\pm$1,0,0) and
(0,$\pm$1,0) surfaces. The basic loops in Figs. 1(b) and 1(c) encircle a
magnetic flux of $\Phi_{x}=B\cos\theta S_{x}$ and $\Phi_{y}=B\sin\theta
S_{y}$, respectively, with $S_{x,y}$ the surface areas. Accordingly, the
frequency components $\omega_{x}=2\pi\cos\theta S_{x}/\Phi_{0}$ and
$\omega_{y}=2\pi\sin\theta S_{y}/\Phi_{0}$ naturally appear in the oscillating
pattern of the conductance as the magnetic field $B$ varies. Interestingly,
the magnetic flux in other interfering loops can all be interpreted by the two
elemental ones, among which two typical loops in Figs. 1(d) and 1(e) contain a
flux of $\Phi_{x\pm y}=\Phi_{x}\pm\Phi_{y}$, and the corresponding oscillating
frequency components satisfy $\omega_{x\pm y}=\omega_{x}\pm\omega_{y}$. It
turns out that the aforementioned four interfering loops and the corresponding
oscillating frequencies dominant the coherent oscillation of the conductance.
The relation $\omega_{x\pm y}=\omega_{x}\pm\omega_{y}$ generally holds
independent of various parameters such as the magnetic field, sample size and
the energy of electron, thus providing a universal and deterministic signature
of the hinge states and HOTI.

The rest of this paper is organized as follows. In Sec. II, we elucidate the
model of the HOTI adopted in our work. In Sec. III, we apply the scattering
matrix approach to analyze the coherent transport through the interferometer
and the AB oscillation of the conductance. Detailed numerical simulations on
the lattice model are conducted in Sec. IV, which verify the universality of
the physical results. Finally, a brief summary and outlook are given in Sec.
V.

##  II Model of HOTI

We adopt the model of 3D chiral HOTI introduced by Schindler et al. Schindler
_et al._ (2018a) as

$\begin{split}H_{\text{HOTI}}&=\left(M+t\sum_{i}{\cos
k_{i}}\right)\tau_{z}\sigma_{0}+\Delta_{1}\sum_{i}{\sin
k_{i}\tau_{x}\sigma_{i}}\\\ &\ \ \ +\Delta_{2}(\cos k_{x}-\cos
k_{y})\tau_{y}\sigma_{0},\end{split}$ (1)

where $\sigma_{i=x,y,z}$ and $\tau_{i}$ are the Pauli matrices acting on the
spin and orbital space, respectively. For $1<|M/t|<3$ and
$\Delta_{1},\Delta_{2}\neq 0$, the system lies in a chiral 3D HOTI phase. The
energy spectra are gapped in both the bulk and four surfaces parallel to the
$z$-axis. Importantly, the mass term is opposite in sign between adjacent
surfaces that results in the Jackiw-Rebbi-type bound states Jackiw and Rebbi
(1976) propagating only along the $\pm z$-direction, or the so-called
topological hinge states. Time-reversal symmetry is broken in Eq. (1), so that
the hinge states are unidirectional or chiral, without any backscattering
states within a given hinge. Notably, gapless Dirac cones protected by the
$\hat{C}_{4}^{z}\hat{T}$ symmetry persist on the surfaces perpendicular to the
$z$-axis Schindler  _et al._ (2018a). Therefore, it is beneficial to explore
pure signature of the hinge states through the transport in the $z$-direction.

##  III Scattering matrix analysis

In this section, we study the coherent transport of electrons through the
interferometer sketched in Fig. 1(a) based on the low-energy effective model
of the hinge states using the scattering matrix approach. The scattering
matrix of the whole interferometer can be obtained by combining those at two
normal metal-HOTI interfaces and the matrix of phase accumulation during
propagation in the hinge states. The matrix at the lower interface [cf. Fig.
1(a)] can be parameterized as

$S_{l}=\left(\begin{array}[]{cccc}r_{1}&r_{3}&t_{1}^{\prime}&t_{3}^{\prime}\\\
r_{2}&r_{4}&t_{2}^{\prime}&t_{4}^{\prime}\\\
t_{1}&t_{3}&r_{1}^{\prime}&r_{3}^{\prime}\\\
t_{2}&t_{4}&r_{2}^{\prime}&r_{4}^{\prime}\\\ \end{array}\right),$ (2)

which relates the incoming ($a_{l}$) and outgoing ($b_{l}$) waves in the
normal metal and the HOTI via $b_{l}=S_{l}a_{l}$. The matrix is assumed to be
$4\times 4$ such that two incoming/outgoing waves are taken into account on
both sides. For the HOTI, the number of channels corresponds to that of pairs
of the hinge states. The unitary condition $S_{l}S_{l}^{\dagger}=\text{I}$ is
ensured by the law of current conservation. Here, $t_{1,\cdots,4}$ are the
transmission amplitudes from the normal metal to the chiral hinge states of
the HOTI and $r_{1,\cdots,4}$ are the corresponding reflection amplitudes. The
scattering amplitudes corresponding to the incident waves from the hinge
states of HOTI are defined by
$t^{\prime}_{1,\cdots,4},r^{\prime}_{1,\cdots,4}$ in a similar way. The
scattering matrix for the upper interface can be defined as

$S_{u}=\left(\begin{array}[]{cccc}r_{1}^{u}&r_{3}^{u}&t_{1}^{u\prime}&t_{3}^{u\prime}\\\
r_{2}^{u}&r_{4}^{u}&t_{2}^{u\prime}&t_{4}^{u\prime}\\\
t_{1}^{u}&t_{3}^{u}&r_{1}^{u\prime}&r_{3}^{u\prime}\\\
t_{2}^{u}&t_{4}^{u}&r_{2}^{u\prime}&r_{4}^{u\prime}\\\ \end{array}\right).$
(3)

The phase modulation of the wave function due to the magnetic field can be
described by the matrix as

$S_{m}=\left(\begin{array}[]{cccc}0&0&e^{i\tilde{\phi_{1}}}&0\\\
0&0&0&e^{-i\tilde{\phi_{2}}}\\\ e^{-i\phi_{2}}&0&0&0\\\ 0&e^{i\phi_{1}}&0&0\\\
\end{array}\right)$ (4)

where the phases $\phi_{1},\phi_{2},\tilde{\phi_{1}},\tilde{\phi_{2}}$ are
related by the magnetic fluxes through
$\phi_{1}+\tilde{\phi_{1}}=\phi_{2}+\tilde{\phi_{2}}=\phi_{x}=2\pi\Phi_{x}/\Phi_{0},\phi_{1}-\tilde{\phi_{2}}=\phi_{2}-\tilde{\phi_{1}}=\phi_{y}=2\pi\Phi_{y}/\Phi_{0},\phi_{1}+\phi_{2}=\phi_{x+y}=2\pi\Phi_{x+y}/\Phi_{0}$
and $\tilde{\phi_{1}}+\tilde{\phi_{2}}=\phi_{x-y}=2\pi\Phi_{x-y}/\Phi_{0}$
with $\phi_{x,y}$ and $\phi_{x\pm y}$ being gauge invariant.

By combining three matrices $S_{l},S_{m},S_{u}$ in a standard way we obtain
the total scattering matrix for the whole system. Here, we focus on the
periods of the AB oscillation and an overall phase shift of the pattern is
unimportant. Therefore, we can choose $S_{l},S_{u}$ to be real for simplicity
which will not change the main results. For an electron incident from the
lower terminal, its transmission probability $T$ to upper terminal is obtained
after some algebra as

$\begin{split}T&=F^{-1}\Big{[}C+C_{X}\cos\phi_{x}+C_{Y}\cos\phi_{y}+C_{XY}\cos\phi_{x+y}+C_{XY}^{\prime}\cos\phi_{x-y}\Big{]},\\\
F&=M_{C}+M_{XY}\cos\phi_{x+y}+M_{XY}^{\prime}\cos\phi_{x-y}+M_{2X}\cos(2\phi_{x})+M_{2Y}\cos(2\phi_{y})-M_{X}\cos\phi_{x}-M_{Y}\cos\phi_{y},\end{split}$
(5)

where the explicit forms of the relevant parameters are given in Appendix A.
The numerator of the transmission in Eq. (5) shows that there are four
dominant periodic terms contributed by four interference loops in Figs.
1(b)-1(e) which correspond to four frequencies related by
$\omega_{x,y},\omega_{x\pm y}=\omega_{x}\pm\omega_{y}$. Note that such
relations are stabilized by the spatial configurations of the hinge states,
thus offer a clear and robust signature for its detection, which relies little
on the sample details and the energy. Although magnetic field in different
directions will change the values of frequencies it does not affect the
general relations between them.

Next, we provide numerical verification of such an observation using specific
scattering amplitudes. The AB oscillation of the conductance as a function of
$B$ and its FFT spectrum are shown in Figs. 2(a) and 2(b), respectively. The
polar angle of the magnetic field is set to $\theta=\pi/6$ and the unit of
frequency is chosen as $1/B_{0}$ with $B_{0}=\Phi_{0}/(2\pi S)$ and
$S=S_{x}=S_{y}$ the surface area. One can find multiple periods in the
conductance spectrum in Fig. 2(a). The FFT spectrum in Fig. 2(b) shows that
there are four dominant frequencies with
$\omega_{x}=0.86/B_{0},\omega_{y}=0.5/B_{0},\omega_{x-y}=0.36/B_{0},\omega_{x+y}=1.36/B_{0}$,
which conforms the aforementioned relation $\omega_{x\pm
y}=\omega_{x}\pm\omega_{y}$. Higher frequencies such as
$\omega_{2x},\omega_{2y}$ should also appear as that in the conventional 2D AB
effect. In stark contrast, the frequencies $\omega_{x\pm y}$ can only exist in
the 3D HOTI, which thus provides a unique evidence of the hinge states.

Figure 2: (a) The conductance pattern calculated by the scattering matrices.
(b) The FFT spectrum of the oscillation pattern. The relevant scattering
coefficients are set to
$t_{1}=t_{3}=t_{4}=r_{1}^{\prime}=r_{1}^{u}=r_{2}^{u}=\sqrt{0.4},t_{2}=r_{3}^{\prime}=\sqrt{0.3},r_{1}=r_{2}^{\prime}=\sqrt{0.2},r_{2}=r_{3}=r_{4}=t_{1}^{u}=t_{2}^{u}=t_{3}^{u}=\sqrt{0.1},r_{4}^{\prime}=t_{4}^{u}=-\sqrt{0.1},r_{3}^{u}=r_{4}^{u}=-\sqrt{0.4}.$

##  IV Lattice model simulation

Based on the scattering matrix analysis, we see that there are four dominant
frequencies satisfying universal relations $\omega_{x\pm
y}=\omega_{x}\pm\omega_{y}$. In this section, we perform numerical simulation
to give rigorous results. We write the model in Eq. (1) on a cubic lattice as
Levitan and Pereg-Barnea (2020)

$\displaystyle H^{\text{Lattice}}_{\text{HOTI}}$
$\displaystyle=\sum_{i}{c_{i}^{\dagger}M\sigma_{0}\tau_{z}c_{i}}$ (6)
$\displaystyle+\Bigg{\\{}\sum_{i}{{c_{i+x}^{\dagger}\bigg{[}\frac{e^{i\varphi_{x}}}{2}(\Delta_{2}\sigma_{0}\tau_{y}+t\sigma_{0}\tau_{z}+i\Delta_{1}\sigma_{x}\tau_{x})\bigg{]}c_{i}}}$
$\displaystyle-\sum_{i}{c_{i+y}^{\dagger}(\Delta_{2}\sigma_{0}\tau_{y}+t\sigma_{0}\tau_{z}+i\Delta_{1}\sigma_{y}\tau_{x})c_{i}}$
$\displaystyle+\sum_{i}{c_{i+z}^{\dagger}\frac{e^{i\varphi_{z}}}{2}(t\sigma_{0}\tau_{z}+i\Delta_{1}\sigma_{z}\tau_{x})c_{i}}+h.c.\Bigg{\\}},$

where
$c_{i}=(c_{a,\uparrow,i},c_{b,\uparrow,i},c_{a,\downarrow,i},c_{b,\downarrow,i})$
are the annihilate operators at lattice site $i$ with two spin
($\uparrow,\downarrow$) and two orbit ($a,b$) components. The Peierls phase
$\varphi_{x,z}=\frac{e}{\hbar}\int_{r_{i}}^{r_{j}}\bm{A}(\bm{r})\cdot
d\bm{r}$, where $\bm{A}(\bm{r})=(B_{y}z,0,B_{x}y)$ is the vector potential
under Landau gauge. The lattice model of the normal metal electrodes is

$\displaystyle H_{\text{NM}}$
$\displaystyle=\sum_{i}{(-6t+U)c_{i}^{\dagger}c_{i}}$ (7)
$\displaystyle+\sum_{i}{t(c_{i+x}^{\dagger}c_{i}+c_{i+y}^{\dagger}c_{i}+c_{i+z}^{\dagger}c_{i})}+h.c..$

Figure 3: (a) Conductance oscillation for different incident energy and angle
$\theta$ by the lattice simulation. An offset $2e^{2}/h$ is imposed to
adjacent curves for clarity. (b, c) Corresponding FFT spectra of the
conductance patterns. Four dominant frequencies are marked by the dashed
lines. The model parameters are set to
$t=-1,M=2.3,\Delta_{1}=0.8,\Delta_{2}=0.5,U=2$. The lattice of the HOTI is set
to a $30a\times 30a\times 30a$ cube, where $a=1$ is the lattice constant.

The AB interferometer constructed by the 3D HOTI (green block) and the normal
metal leads (two orange blocks) is shown in Fig. 1(a). The blue arrowed lines
denote the chiral hinge states. The cross section of the HOTI in the $x$-$y$
plane is set as $30a\times 30a$ and that for the normal metal leads is
$5a\times 5a$ with $a$ being the lattice constant. The magnetic field $B$
exists only in the HOTI region and is parallel to the $x$-$y$ plane.

Consider an electron impinging from the normal metal towards the HOTI with its
energy lying in the bulk gap ($\simeq$0.7$|t|$) and surface gap
($\simeq$0.32$|t|$) of the HOTI, so that only the hinge channels are available
for propagation. Backscattering can occur at the interfaces, giving rise to
various interference loops. The two terminal conductance $G$ is calculated
using KWANT package Groth  _et al._ (2014) and the AB conductance oscillation
for different incident energy ($ie$) and polar angle $\theta$ of magnetic
field are shown in Fig. 3(a) (curves are offset by $2e^{2}/h$ for clarity). To
get the dominant frequencies, we perform FFT calculation whose spectra are
shown in Figs. 3(b) and 3(c).

The numerical results are consistent with those by the scattering matrix
analysis in Figs. 2(b). For different incident energies, the oscillation
patterns in Fig. 3(a) look in stark difference. However, the dominant
frequencies remain almost the same; see Fig. 3(b). To check the general
relations between oscillation frequencies, we first locate two notable peaks
$\omega_{x}$ and $\omega_{y}$ by the dashed lines and the other two peaks
$\omega_{x\pm y}=\omega_{x}\pm\omega_{y}$ are marked accordingly in Fig. 3(b).
One can see that the dominant peaks match the dashed lines very well apart
from a small deviation from $\omega_{x+y}$ for $ie=0.1$ which is attributed to
the limit of numerical calculations. For different polar angle $\theta$,
similar results can be seen in Fig. 3(c). Although the location of peaks
change for different $\theta$, the general relations between them persist.

Figure 4: (a) $\omega_{x+y}$ and (b) $\omega_{x-y}$ as a function of elemental
frequencies $\omega_{x}$ and $\omega_{y}$. Black and orange dots denote
numerical results for different length of the HOTI and polar angle $\theta$ of
the magnetic field. The reference planes satisfy $\omega_{x\pm
y}=\omega_{x}\pm\omega_{y}$. Insets: side view of the plots which reveal the
deviation of the dots from the planes. The parameters are the same as those in
Fig. 3.

In Fig. 4, we present more general results by varying both the polar angle
$\theta$ and the thickness of the HOTI in the $z$-direction. Each pair of
parameters generate one point in both Figs. 4(a) and 4(b), with its
coordinates extracted in the same way as done in Fig. 3(b). The reference
planes therein correspond to the frequency rule $\omega_{x\pm
y}=\omega_{x}\pm\omega_{y}$. One can see that the numerical results labeled by
the black and orange dots are well located around the reference planes, which
indicates the universality of the frequency rule. Note that there are a few
dots of negative frequencies in Fig. 4(b) for $\omega_{x}<\omega_{y}$. In
experiments, one should rather measure $|\omega_{x}-\omega_{y}|$ instead.

Figure 5: (a) Conductance patterns with different disorder strength $\alpha$.
(b) FFT spectra of the conductance. The incident energy is $ie=0.1$ and the
polar angle of the field is $\theta=\pi/6$. Other parameters are the same as
those in Fig. 3.

Disorder generally exists in real samples and it can be expected that
topological chiral hinge states and thus the AB effect should be robust, the
same as that in the quantum Hall edge states. We show numerical results for
the disorder distributed in the whole HOTI region in Fig. 5 with different
strength $\alpha$. For weak disorder strength $\alpha<0.7|t|$ (the gap of the
bulk states), the oscillation pattern and frequency rules $\omega_{x\pm
y}=\omega_{x}\pm\omega_{y}$ retain. For strong disorder $\alpha>0.7|t|$, the
oscillation pattern quenches stemming from disorder induced coupling between
the surface/bulk states and the hinge states. Therefore, as long as the
disorder in the sample of HOTI is not too strong, the AB effect can be
hopefully observed. Similar conclusion also holds for the surface roughness.
Although the AB effect is quite robust against the disorder effect, the
observation should be carried out within the phase coherence length of the
system. The dephasing effect always reduces the visibility of the coherent
oscillation until it vanishes Golizadeh-Mojarad and Datta (2007); Lahiri  _et
al._ (2018). One more remark is that the interference here is all of the AB
type without Al’tshuler-Aronov-Spivak (AAS) type contribution Al’Tshuler  _et
al._ (1981); Sharvin and Sharvin (1981); Al’tshuler  _et al._ (1982). The
model in Eq. (1) breaks time-reversal symmetry so that the AAS effect is
absent.

##  V Summary and Outlook

In summary, we have investigated the AB effect in the chiral hinge states of
the 3D HOTI. Due to the spatial configurations of the hinge states, new types
of interfering loops appear compared with the 2D AB interference. Importantly,
we predict a universal relationship $\omega_{x\pm y}=\omega_{x}\pm\omega_{y}$
between the dominant oscillating frequencies, which offers a unique signature
of the hinge states as well as the HOTI. Our study can be generalized
straightforwardly to AB effect in the 3D HOTI with helical hinge states.

Note added. Recently, we became aware of a related work Li  _et al._ (2021),
which focuses on different aspects.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
under Grant No. 12074172 (W.C.), No. 11674160 and No. 11974168 (L.S.), the
startup grant at Nanjing University (W.C.), the State Key Program for Basic
Researches of China under Grants No. 2017YFA0303203 (D.Y.X.) and the Excellent
Programme at Nanjing University.

##  Appendix A specific forms of coefficient for analysis calculation

In Eq. (5) of the main text, the coefficients are expressed as

$\begin{split}C&=(t_{1}^{2}+t_{3}^{2})(W_{0}^{2}+W_{1}^{2}+W_{2}^{2}+X_{0}^{2}+X_{1}^{2}+X_{2}^{2})+(t_{2}^{2}+t_{4}^{2})(Y_{0}^{2}+Y_{1}^{2}+Y_{2}^{2}+Z_{0}^{2}+Z_{1}^{2}+Z_{2}^{2}),\\\
C_{X}&=2(W_{0}W_{1}+X_{0}X_{1})(t_{1}^{2}+t_{3}^{2})+2(Y_{0}Y_{1}+Z_{0}Z_{1})(t_{2}^{2}+t_{4}^{2})+2(W_{0}Y_{2}+W_{2}Y_{0}+X_{0}Z_{2}+X_{2}Z_{0})(t_{1}t_{2}+t_{3}t_{4}),\\\
C_{Y}&=2(W_{0}W_{2}+X_{0}X_{2})(t_{1}^{2}+t_{3}^{2})+2(Y_{0}Y_{2}+Z_{0}Z_{2})(t_{2}^{2}+t_{4}^{2})+2(W_{0}Y_{1}+W_{1}Y_{0}+X_{0}Z_{1}+X_{1}Z_{0})(t_{1}t_{2}+t_{3}t_{4}),\\\
C_{XY}&=2(W_{0}Y_{0}+W_{0}Y_{0})(t_{1}t_{2}+t_{3}t_{4}),\\\
C_{XY}^{\prime}&=2(W_{1}W_{2}+X_{1}X_{2})(t_{1}^{2}+t_{3}^{2})+2(Y1Y_{2}+Z_{1}Z_{2})(t_{2}^{2}+t_{4}^{2})+2(W_{1}Y_{1}+W_{2}Y_{2}+X_{1}Z_{1}+X_{2}Z_{2})(t_{1}t_{2}+t_{3}t_{4}),\\\
M_{XY}&=2M_{1}M_{4}+2M_{2}M_{3},\ \ \ \ \
M_{XY}^{\prime}=2M_{1}M_{3}+2M_{2}M_{4},\ \ \ \ \ M_{2X}=2M_{1}M_{2},\ \ \ \ \
M_{2Y}=2M_{3}M_{4},\\\ M_{X}&=2M_{0}M_{1}+2M_{0}M_{2},\ \ \ \ \
M_{Y}=2M_{0}M_{3}+2M_{0}M_{4},\ \ \ \ \
M_{C}=M_{0}^{2}+M_{1}^{2}+M_{2}^{2}+M_{3}^{2}+M_{4}^{2},\end{split}$ (8)

which contain the parameters defined by the elements of the scattering
matrices as

$\begin{split}W_{0}&=t_{1}^{u},W_{1}=t_{3}^{u}r_{2}^{\prime}r_{1}^{u}-t_{1}^{u}r_{2}^{\prime}r_{3}^{u},W_{2}=t_{3}^{u}r_{4}^{\prime}r_{2}^{u}-t_{1}^{u}r_{4}^{\prime}r_{4}^{u},\\\
X_{0}&=t_{2}^{u},X_{1}=t_{4}^{u}r_{2}^{\prime}r_{1}^{u}-t_{2}^{u}r_{2}^{\prime}r_{3}^{u},X_{2}=t_{4}^{u}r_{4}^{\prime}r_{2}^{u}-t_{2}^{u}r_{4}^{\prime}r_{4}^{u},\\\
Y_{0}&=t_{3}^{u},Y_{1}=t_{1}^{u}r_{3}^{\prime}r_{4}^{u}-t_{3}^{u}r_{3}^{\prime}r_{2}^{u},Y_{2}=t_{1}^{u}r_{1}^{\prime}r_{3}^{u}-t_{3}^{u}r_{1}^{\prime}r_{1}^{u},\\\
Z_{0}&=t_{4}^{u},Z_{1}=t_{2}^{u}r_{3}^{\prime}r_{4}^{u}-t_{4}^{u}r_{3}^{\prime}r_{2}^{u},Z_{2}=t_{2}^{u}r_{1}^{\prime}r_{3}^{u}-t_{4}^{u}r_{1}^{\prime}r_{1}^{u},\\\
M_{0}&=1+r_{1}^{\prime}r_{1}^{u}r_{4}^{\prime}r_{4}^{u}+r_{3}^{\prime}r_{2}^{u}r_{2}^{\prime}r_{3}^{u}-r_{1}^{\prime}r_{3}^{u}r_{4}^{\prime}r_{2}^{u}-r_{3}^{\prime}r_{4}^{u}r_{2}^{\prime}r_{1}^{u},\\\
M_{1}&=r_{2}^{\prime}r_{3}^{u},M_{2}=r_{3}^{\prime}r_{2}^{u},M_{3}=r_{4}^{\prime}r_{4}^{u},M_{4}=r_{1}^{\prime}r_{1}^{u}.\end{split}$
(9)

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# Half Line Titchmarsh-Weyl $m$ Functions of Vector-Valued Discrete
Schrödinger Operators

Keshav Raj Acharya  Department of Mathematics, Embry-Riddle Aeronautical
University, 1 Aerospace Blvd., Daytona Beach, FL 32114, U.S.A.
<EMAIL_ADDRESS>and  Matt McBride  Department of Mathematics and Statistics,
Mississippi State University, 175 President’s Cir., Mississippi State, MS
39762, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We show that the half-line $m$ functions associated with the vector-valued
Schrödinger operators are the elements in the Siegel upper half space. We
introduce a metric on the space of $m$ functions associated to the vector-
valued discrete Schródinger operators. Then we show that the action of
transfer matrices on these $m$ functions is distance decreasing.

##  1\. Introduction

The spectral theory of one dimensional Schrödinger operator has been well
studied. The extension of the theory to higher dimensions has yet to be
developed and is of interest of many researchers. There is some literature
which deals with the Schrödinger equations in higher dimensions with matrix-
valued potentials, for example see [3, 4, 6, 5, 7]. Likewise, as the theory of
Titchmarsh-Weyl $m$ functions provide elegant methods in describing the
spectrum of the associated operators, it is natural to study these functions
for vector-valued discrete Schrödinger operators. Some basic theory has been
developed in [1]. In order to study these $m$ functions we need to study
matrix-valued Herglotz functions. In this regard, there has been some studies
about the matrix-valued Herglotz functions, for example see [8]. Our goal in
this paper is to discuss some basic properties of half-line Titchmarsh-Weyl
$m$ functions associated to the vector-valued discrete Schrödinger operators.

We begin by considering an equation of the form,

$y(n+1)+y(n-1)+B(n)y(n)=zy(n)\,,\,z\in\mathbb{C}$ (1.1)

where for fixed $n$,
$y(n)=(y_{1}(n),y_{2}(n),\ldots\,,y_{d}(n))^{t}\in\mathbb{C}^{d}$ and
$B(n)\in\mathbb{R}^{d\times d}$ is a symmetric matrix. Here $\\{y(n)\\}$ and
$\\{B(n)\\}$ are respectively vector-valued and matrix-valued sequences. One
can consider $B(n)$ as a Hermitian matrix but for the sake of simplicity we
only consider $B(n)$ as a real symmetric matrix. In this paper, we give
insight into half-line $m$ functions and discuss properties these functions
enjoy. Then we describe the action of transfer matrices associated to equation
(1.1) on these functions. We then introduce the metric on the space of $m$
functions an extended analogous of hyperbolic metric in complex upper half
plane. Finally we observe the distance properties. The paper is organized as
follows: We present the preliminaries in Section 2. In Section 3, we define
the half-line $m$ functions following the definition in [1], present very
important properties. Then we define a metric on the space of $m$ functions
and show that action of transfer matrices on $m$ functions is distance
decreasing.

##  2\. Preliminaries

Similar to the one dimensional space, we consider the equation (1.1) in a
Hilbert space. Let $\ell^{2}(\mathbb{N},\mathbb{C}^{d})$ be the Hilbert space
of square summable vector-valued sequences. That is

$\ell^{2}(\mathbb{N},\mathbb{C}^{d})=\left\\{y:\sum_{n=1}^{\infty}y(n)^{*}y(n)<\infty\right\\}\,.$

Equation (1.1) induces a vector-valued discrete Schrödinger operator $J$ on
$\ell^{2}(\mathbb{N},\mathbb{C}^{d})$ as

$(Jy)(n)=\left\\{\begin{aligned} &y(n+1)+y(n-1)+B(n)y(n)&&\textrm{if }n>1\\\
&y(1)+B(1)y(2)&&\textrm{if }n=1\,.\end{aligned}\right.$

Notice from this we see that $J$ is a bounded self-adjoint operator.

Define the difference expression
$\tau:\ell^{2}(\mathbb{N},\mathbb{C}^{d})\to\ell^{2}(\mathbb{N},\mathbb{C}^{d})$
by

$\tau(u(n))=u(n+1)+u(n-1)+B(n)u(n)$ (2.1)

which looks like $J$ but acts on any vector-valued sequence. In fact, for any
$z\in\mathbb{C}$, a solution $u(n)$ of $(\tau-z)u=0$ is uniquely determined by
the values $u(n_{0})$ and $u(n_{0}+1)$, therefore the solution space is a $2d$
dimensional vector space of sequences. It is shown in [1] that there are
precisely $d$ linearly independent solutions to equation (1.1) that are in
$\ell^{2}(\mathbb{N},\mathbb{C}^{d})$.

One first observation is to connect equation (1.1) with a second order
differential expression, similar to the one in one dimensional space, see
[12]. Indeed, define the following first order difference operator
$\Delta:\ell^{2}(\mathbb{N},\mathbb{C}^{d})\to\ell^{2}(\mathbb{N},\mathbb{C}^{d})$
by

$(\Delta u)(n)=u(n+1)-u(n).$

An immediate calculation yields the adjoint to $\Delta$, namely
$\Delta^{*}:\ell^{2}(\mathbb{N},\mathbb{C}^{d})\to\ell^{2}(\mathbb{N},\mathbb{C}^{d})$
via

$(\Delta^{*}u)(n)=u(n-1)-u(n)\,.$

Using the definition of $\tau$, consider the following calculations:

$\displaystyle-(\Delta^{*}\Delta u)(n)$
$\displaystyle=-\Delta^{*}(u(n+1)-u(n))=-u(n)+u(n+1)+u(n-1)-u(n)$
$\displaystyle=(\tau u)(n)-(B(n)+2I)u(n)\,,$

and

$\displaystyle-(\Delta\Delta^{*}u)(n)$
$\displaystyle=-\Delta(u(n-1)-u(n))=u(n+1)+u(n-1)-2u(n)$ $\displaystyle=(\tau
u)(n)-(B(n)+2I)u(n)\,.$

Notice that this is similar to the vector-valued Sturm-Liouville differential
expression

$-\frac{d^{2}}{dx^{2}}(y)+V(x)y$

where $y$ is in some properly chosen domain and $V$ is a potential function.

For any two vector valued sequences $\\{x(n)\\},\\{v(n)\\}$, recall that the
Wronskian is defined by

$W_{n}(x(n),y(n))=x(n+1)^{t}y(n)-x(n)^{t}y(n+1)\,.$ (2.2)

It can be easily shown that if $u(n)$ and $v(n)$ are any two solutions to
equation (1.1), then $W_{n}(u(n),v(n))$ is independent of $n$. This definition
can be easily generalized to $d\times d$ matrix-valued sequences $\\{X(n)\\}$
and $\\{Y(n)\\}$ as follows:

$W_{n}(X(n),Y(n))=X(n+1)^{t}Y(n)-X(n)^{t}Y(n+1)\,.$ (2.3)

In this case $W_{n}(X(n),Y(n))$ is a $d\times d$ matrix-valued function. It
was shown in [1] that if $X(n)$ and $Y(n)$ are matrix valued solutions to
equation (1.1), then is $W_{n}(X(n),Y(n))$ independent of $n$. Using Abel
summation we arrive at Green’s formula:

$\sum_{j=n}^{m}\left(X(j)^{*}(\tau Y)(j)-(\tau
Y)(j)^{*}Y(j)\right)=W_{n-1}(\overline{X}(n-1),Y(n-1))-W_{m}(\overline{X}(m),Y(m))\,.$
(2.4)

Please see [1] for a detailed computation of formula (2.4).

##  3\. Half line Weyl $m$ functions

A $d\times 2d$ matrix-valued sequence $Y(n)$ is said to be a fundamental
matrix solution to equation (1.1) if it satisfies equation (1.1) in matrix
form and columns of $Y(n)$ form a set of linearly independent solutions to the
vector form to equation (1.1).

Write $Y=(U,V)$ where $U$ and $V$ are $d\times d$ matrix-valued solutions to
equation (1.1). One of the goals of this paper is to discuss the half line $m$
function. Thus, we consider the equation given by (1.1) on
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. Let $\mathbb{N}_{-}=\\{0,1,\ldots
n\\}$ and $\mathbb{N}_{+}=\\{n+1,N+2,\ldots\\}$. Suppose $U$ and $V$ satisfy
the following initial conditions at $n$:

$\displaystyle U(n)=-I,$ $\displaystyle V(n)=0$ $\displaystyle U(n+1)=0,$
$\displaystyle V(n+1)=I\,.$

Define the Weyl $m$ functions on $\mathbb{N}_{-}$ and $\mathbb{N}_{+}$ by

$F_{\pm}(n)=U(n)\pm M_{\pm}(n)V(n)$ (3.1)

where we require that $F_{-}(0)=0$ and
$F_{+}\in\ell^{2}(\mathbb{N},\mathbb{C}^{d\times d})$. Notice that the columns
of $F_{\pm}(n)$ form a linearly independent set of solutions to equation (1.1)
and therefore for each $n$, $F_{\pm}(n)$ are invertible. For
$z\in\mathbb{C}^{+}$, these half-line $m$ functions are uniquely determined
and are given by:

$M_{+}(n,z)=-F_{+}(n+1,z)F_{+}(n,z)^{-1},\ \
M_{-}(n,z)=F_{-}(n+1,z)F_{-}(n,z)^{-1}\,.$ (3.2)

The theory of these $m$ functions has been very useful tool in the Spectral
theory of Schrödinger operators. There are numerous literatures dealing with
these $m$ functions for one dimensional Schrödinger operators, see [12] as an
example, for Jacobi operators. There are also some articles which discuss
about the Weyl $m$ functions for matrix-valued Schrödinger operators in
continuous case, see [3, 4, 5, 7, 8]. Our goal is to discuss some basic
properties of the $m$ functions defined by (3.2).

First we show that $M_{\pm}(n)$ can be expressed in terms of a resolvent
operator. As such we introduce the following notation. Let $J_{\pm}$ be the
restriction of $J$ to the space $\ell^{2}(\mathbb{N}_{\pm},\mathbb{C}^{d})$
and denote the Dirac delta type vector-valued sequences as follows:

$\displaystyle\bbdelta_{1}^{1}=\left\\{\begin{bmatrix}1\\\ 0\\\ \vdots\\\
0\end{bmatrix},\begin{bmatrix}0\\\ 0\\\ \vdots\\\
0\end{bmatrix},\dots\right\\},\bbdelta_{2}^{1}=\left\\{\begin{bmatrix}0\\\
1\\\ \vdots\\\ 0\end{bmatrix},\begin{bmatrix}0\\\ 0\\\ \vdots\\\
0\end{bmatrix},\dots\right\\},\ldots,$
$\displaystyle\bbdelta_{d}^{1}=\left\\{\begin{bmatrix}0\\\ 0\\\ \vdots\\\
1\end{bmatrix},\begin{bmatrix}0\\\ 0\\\ \vdots\\\
0\end{bmatrix},\dots\right\\},\ldots,\bbdelta_{d}^{n}=\left\\{\begin{bmatrix}0\\\
0\\\ \vdots\\\ 0\end{bmatrix},\dots\begin{bmatrix}0\\\ 0\\\ \vdots\\\
1\end{bmatrix},\dots\right\\}$

Similarly for matrix-valued sequence denote:

$\mathbb{\Delta}_{1}=\left\\{I,0\ldots\right\\},\ldots,\mathbb{\Delta}_{n}=\left\\{0,\ldots,I,\ldots\right\\}$

where $I$ and $0$ are the $d\times d$ identity and null matrix respectively.
We have the following proposition:

######  Proposition 3.1.

Let $z\in\mathbb{C}^{+}$.

$M_{+}(n,z)=\langle\mathbb{\Delta}_{n+1},(J_{+}-zI)^{-1}\mathbb{\Delta}_{n+1}\rangle=\Big{(}\langle\bbdelta_{i}^{n+1},(J_{+}-zI)^{-1}\bbdelta_{j}^{n+1}\rangle\Big{)}$
(3.3)

###### Proof.

Let $g_{j}(k,z)=(J_{+}-zI)^{-1}\bbdelta_{j}^{n+1}(k)$. Notice that
$g_{j}(k,z)$, for $j=1,\ldots,d$ and for $k>n+1$ are solutions to equation
(1.1) and belong to $\ell^{2}(N_{+},\mathbb{C}^{d})$. Define $G(k,z)$ by

$G(k,z)=\Big{(}g_{1}(k,z),g_{2}(k,z),\dots,g_{d}(k,z)\Big{)}\,.$

From this definition it follows that $G(k,z)$ is a matrix valued solution to
equation (1.1) for $k>n+1$ and $G(k,z)\in\ell^{2}(N_{+},\mathbb{C}^{d\times
d})$. Thus there exists an invertible constant matrix $C\in\mathbb{C}^{d\times
d}$ such that

$G(k,z)=F(k,z)C\,.$ (3.4)

Since $F(n,z)=-I$ and $\ G(n,z)=-C$, it follows that

$M_{+}(n,z)=-F_{+}(n+1,z)F(n,z)^{-1}=-G(n+1,z)G(n,z)^{-1}=G(n+1,z)C^{-1}\,.$

To find the matrix $C$, we need only to compare the values at $n+2$ place.
Notice that

$(J_{+}-zI)G(n+1,z)=(\bbdelta_{1}^{1},\bbdelta_{2}^{1},\dots\bbdelta_{d}^{1})$

and hence $(J_{+}-zI)G(n+1,z)=I$. Therefore,

$G(n+2,z)=(z-B(n+1))G(n+1,z)+I\,.$ (3.5)

Moreover, since $F(n,z)$ is a solution to equation (1.1) we have

$F(n+2,z)=(z-B(n+1))F(n+1,z)-F(n,z)\,.$

From equation (3.4) it follows that

$G(n+2,z)=(z-B(n+1))F(n+1,z)C-F(n,z)C\,.$ (3.6)

Finally, comparing equations (3.5) and (3.6) we deduce that $-F(n,z)C=I$.
Since $F(n,z)=-I$, it follows that $C=I$. Therefore, we have that
$M_{+}(n,z)=G(n+1,z)$.

Thus the matrix $M_{+}(n,z)$ by comparing the values at $k=n+1$,

$G(n+1,z)=F(n+1,z)C$

Now we find the entries of the matrix solution $G(n+1,z)$. Indeed, for each
$j=1,\ldots,d$, the components $g_{ij(n+1,z)}$ for

$g_{j}(n+1,z)=\begin{bmatrix}g_{1j}(n+1,z)\\\ g_{2j}(n+1,z)\\\ \vdots\\\
g_{dj}(n+1,z)\end{bmatrix}$

are given by

$g_{ij(n+1,z)}=\langle\bbdelta_{i}^{n+1},(J_{+}-zI)^{-1}\bbdelta_{j}^{n+1}\rangle$

for $i=1,\ldots,d$. It follows that

$M_{+}(n,z)=\langle\mathbb{\Delta}_{n+1},(J_{+}-zI)^{-1}\mathbb{\Delta}_{n+1}\rangle=\Big{(}\langle\bbdelta_{i}^{n+1},(J_{+}-zI)^{-1}\bbdelta_{j}^{n+1}\rangle\Big{)}\,.$

This completes the proof. ∎

Eqn. (3.3) suggests that $\overline{M_{+}(n,z)}=M_{+}(n,\bar{z}).$
We now show that $M_{\pm}(n,z)$ are matrix-valued Herglotz functions.

######  Lemma 3.2.

For any $z\in\mathbb{C}^{+}$ and any $n$, $M_{\pm}(n,z)$ are symmetric and
satisfy the relation

$M_{\pm}(n,z)+M_{\pm}(n\mp 1,z)^{-1}\mp\left(zI-B(n)\right)=0\,.$ (3.7)

###### Proof.

We first apply the Wronskian to show that $M_{\pm}(n)$ are symmetric. Indeed,
for $M_{-}(n,z)$, consider the following calculation:

$\displaystyle
M_{-}(n,z)^{t}-M_{-}(n,z)=\Big{(}F_{-}(n+1,z)F_{-}(n,z)^{-1}\Big{)}^{t}-F_{-}(n+1,z)F_{-}(n,z)^{-1}$
$\displaystyle=\Big{(}F_{-}(n,z)^{-1}\Big{)}^{t}F_{-}(n+1,z)^{t}-F_{-}(n+1,z)F_{-}(n,z)^{-1}$
$\displaystyle=\Big{(}F_{-}(n,z)^{-1}\Big{)}^{t}\Big{(}F_{-}(n+1,z)^{t}F_{-}(n,z)-F_{-}(n,z)^{t}F_{-}(n+1,z)\Big{)}F_{-}(n,z)^{-1}$
$\displaystyle=\Big{(}F_{-}(n,z)^{-1}\Big{)}^{t}W_{n}(F_{-}(n,z),F_{-}(n,z))F_{-}(n,z)^{-1}\,.$

Since $F_{-}(n,z)$ is a solution to equation (1.1),
$W_{n}(F_{-}(n,z),F_{-}(n,z))$ is independent of $n$. It therefore follows
that $W_{n}(F_{-}(n,z),F_{-}(n,z))=W_{0}(F_{-}(0,z),F_{-}(0,z))=0$. This in
turn implies that $M_{-}(n,z)$ is symmetric. A similar calculation shows that
$M_{+}(n,z)$ is also symmetric. All that remains is to show that
$M_{\pm}(n,z)$ satisfies equation (3.7).

Indeed, from equation (1.1), we have:

$F_{+}(n+1,z)+F_{+}(n-1,z)+\big{(}B(n)-zI\big{)}F_{+}(n,z)=0$

and hence

$F_{+}(n+1,z)F_{+}(n,z)^{-1}+F_{+}(n-1,z)F_{+}(n,z)^{-1}+\big{(}B(n)-zI\big{)}=0\,.$

This implies that

$M_{+}(n,z)+M_{+}(n-1,z)^{-1}+zI-B(n)=0\,.$

An analogous calculation shows that for $M_{-}(n,z)$ we have

$M_{-}(n,z)+M_{-}(n-1,z)^{-1}+B(n)-zI=0\,.$

This completes the proof. ∎

Recall the imaginary part of $M_{\pm}(n,z)$ is given by:

$\operatorname{Im}M_{\pm}(n,z)=\frac{1}{2i}\left(M_{\pm}(n,z)-M_{\pm}(n,z)^{*}\right)\,.$

We have the following proposition:

######  Proposition 3.3.

For any $z\in\mathbb{C}^{+}$, the matrices $\operatorname{Im}M_{\pm}(n,z)$ are
positive.

###### Proof.

First we show that $\operatorname{Im}M_{-}(n,z)$ is a positive matrix. Using
Green’s formula, equation (2.4) with $X(n)$ and $Y(n)$ are replaced by
$F_{-}(n,z)$ we get

$\displaystyle\sum_{j=1}^{n}\left(F_{-}(j,z)^{*}(\tau F_{-})(j,z)-(\tau
F_{-})(j,z)^{*}F_{-}(j,z)\right)=$ (3.8)
$\displaystyle\qquad=W_{0}(\overline{F_{-}(0,z)},F_{-}(0,z))-W_{n}(\overline{F_{-}(n,z)},F_{-}(n,z))\,.$

Since $F_{-}(0,z)=0$ and $(\tau F_{-})(j,z)=zF_{-}(j,z)$, equation (3.8)
simplifies to

$(z-\overline{z})\sum_{j=1}^{n}F_{-}(j,z)^{*}F_{-}(j,z)=-W_{n}(\overline{F_{-}(n,z)},F_{-}(n,z))\,.$
(3.9)

Since $F_{-}(n,z)$ can be decomposed into the fundamental solutions $U(n,z)$
and $V(n,z)$, that is $F_{-}(n,z)=U(n,z)-M_{-}(n,z)V(n,z)$, the linearity of
Wronskian implies that

$\displaystyle(z-\bar{z})\sum_{j=1}^{n}F_{-}(j,z)^{*}F_{-}(j,z)=$ (3.10)
$\displaystyle=-W_{n}(\overline{U(n,z)-M_{-}(n,z)V(n,z)},U(n,z)-M_{-}(n,z)V(n,z))$
$\displaystyle=W_{n}(U(n,\overline{z}),U(n,z))+W_{n}(U(n,\overline{z}),M_{-}(n,z)V(n,z))$
$\displaystyle\qquad+W_{n}(M_{-}(n,\overline{z})V(n,\overline{z}),U(n,z))-W_{n}(M_{-}(n,\overline{z})V(n,\overline{z}),M_{-}(n,z)V(n,z))\,.$

Using definition of the Wronskian and the boundary conditions at $n$ and $n+1$
we obtain that

$\displaystyle W_{n}(U(n,\overline{z}),U(n,z))=0$ $\displaystyle
W_{n}(M_{-}(n,\overline{z})V(n,\overline{z}),M_{-}(n,z)V(n,z))=0\,.$

It follows that

$\displaystyle W_{n}(U(n,\overline{z}),M_{-}(n,z)V(n,z))=M_{-}(n,z)$
$\displaystyle
W_{n}(M_{-}(n,\overline{z})V(n,\overline{z}),U(n,z))=-M_{-}(n,\overline{z})\,.$

Hence equation (3.10) becomes

$(z-\overline{z})\sum_{j=1}^{n}F_{-}(j,z)^{*}F_{-}(j,z)=M_{-}(n,z)-M_{-}(n,\overline{z})\,.$

Since $z\in\mathbb{C}^{+}$, we have $(z-\overline{z})/2i>0$ and thus it
follows that $\operatorname{Im}M_{-}(n,z)$ is a positive matrix.

Similarly, to show $\operatorname{Im}M_{+}(n,z)$ is a positive matrix, we
replace $X(n)$ and $Y(n)$ both with $F_{+}(n,z)$ in equation (2.4). Since
$F_{+}(n,z)$ is defined on $\mathbb{N}_{+}$, for $N\geq n+1$ we have

$\displaystyle\sum_{j=n+1}^{N}\left(F_{+}(j,z)^{*}(\tau F_{+})(j,z)-(\tau
F_{+})(j,z)^{*}F_{+}(j,z)\right)$ (3.11)
$\displaystyle\quad=W_{n}(\overline{F_{+}(n,z)},F_{+}(n,z))-W_{N}(\overline{F_{+}(N,z)},F_{+}(N,z))\,.$

Since $F_{+}(z)\in\ell^{2}(\mathbb{N},\mathbb{C}^{d\times d})$ for every
$z\in\mathbb{C}^{+}$, taking the limit as $N\rightarrow\infty$ yields zero on
the left-hand side of equation (3.11). Moreover since the Wronskian of
solutions to equation (1.1) is independent of $N$ we get

$\lim_{N\to\infty}W_{N}(\overline{F_{+}(N,z)},F_{+}(N,z))=0\,.$

Thus using a similar argument as above it follows that

$(z-\overline{z})\sum_{j=1}^{n}F_{+}(j,z)^{*}F_{+}(j,z)=M_{+}(n,z)-M_{+}(n,\overline{z})\,.$

Therefore $\operatorname{Im}M_{+}(n,z)$ is a positive matrix completing the
proof. ∎

The fractional linear transformation have been used in analyzing the theory of
$m$ functions. More specifically, the transfer matrices associated to the
Schrödinger equations are fractional linear transformation. For the matrix-
valued Schrödhinger operators, the transfer matrices are considered as matrix-
valued fractional linear transformation. For any $T\in\mathbb{C}^{2d\times
2d}$ of the form:

$T=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$

where are $A$, $B$, $C$, and $D$ are $d\times d$ matrices, a matrix-valued
fractional linear transformation is a map $T:\mathbb{C}^{d\times
d}\rightarrow\mathbb{C}^{d\times d}$ defined by

$T(Z)=(AZ+B)(CZ+D)^{-1}\,,$ (3.12)

for all $Z\in\mathbb{C}^{d\times d}$ for which $T$ is well defined. We observe
that the transfer matrices associated to the equation (1.1) are in fact
matrix-valued fractional linear transformation of the form (3.12).

If $F(n)$ is a matrix solution to equation (1.1) we have

$\begin{bmatrix}F(n+1)\\\ \mp F(n)\end{bmatrix}=\begin{pmatrix}zI-B(n)&\pm
I\\\ \mp I&0\end{pmatrix}\begin{bmatrix}F(n)\\\ \mp F(n-1)\end{bmatrix}\,,$
(3.13)

where $B(n)$ is a $d\times d$ symmetric matrix. The matrices

$T_{\pm}(n)=\begin{pmatrix}zI-B(n)&\pm I\\\ \mp I&0\end{pmatrix}$ (3.14)

given in equation (3.13) are called transfer matrices which describe the
evolution of the vectors

$\begin{bmatrix}F(n+1)\\\ \mp F(n)\end{bmatrix}$

under iteration of $T_{\pm}(n)$. These transfer matrices $T_{\pm}(n)$ can be
considered as complex matrix-valued linear transformations (3.12) acting on
the space of $m$ functions by using (3.14):

$T_{\pm}(n)M_{\pm}(n)=(zI-B(n))(M_{\pm}(n)\pm I)(\mp IM_{\pm}(n))^{-1}\,.$

The transfer matrices also relate the $m$ functions at $n$ and $n-1$ as shown
in the following lemma.

######  Lemma 3.4.

For any $z\in\mathbb{C}^{+}$, $M_{\pm}(n)=T_{\pm}(n)M_{\pm}(n-1)$.

###### Proof.

Since $F_{\pm}(n)$ are solutions to equation (1.1) we have,

$F_{\pm}(n+1,z)+F_{\pm}(n-1,z)+B(n)F_{\pm}(n,z)=zF_{\pm}(n,z)\,.$

Consider the following calculation:

$\displaystyle\begin{pmatrix}z-B(n)&I\\\ -I&0\end{pmatrix}M_{+}(n-1)$
$\displaystyle=\begin{pmatrix}z-B(n)&I\\\
-I&0\end{pmatrix}\begin{bmatrix}F_{+}(n,z)\\\ -F_{+}(n-1,z)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}(z-B(n))F_{+}(n,z)-F_{+}(n-1,z)\\\
-F_{+}(n,z)\end{bmatrix}=\begin{bmatrix}F_{+}(n+1,z)\\\
-F_{+}(n,z)\end{bmatrix}$
$\displaystyle=-F_{+}(n+1,z)F_{+}(n,z)^{-1}=M_{+}(n)\,$

where the last line follows from the usual interpretation of matrix-valued
linear transformations. This shows the lemma is true for $M_{+}(n)$. A similar
calculation shows

$M_{-}(n)=T_{-}(n)M_{-}(n-1)\,.$

This completes the proof. ∎

The matrices $T_{\pm}(n)$ satisfy the following symplectic identity:

######  Lemma 3.5.

For all $n\in\mathbb{N}$ and any symmetric $d\times d$ matrix $B$,
$T_{\pm}(n)$ satisfy the following identities:

  1. (1)

$T_{\pm}(n)^{t}\begin{pmatrix}0&I\\\
-I&0\end{pmatrix}T_{\pm}(n)=\begin{pmatrix}0&I\\\ -I&0\end{pmatrix}$.

  2. (2)

$\begin{pmatrix}I&0\\\ 0&-I\end{pmatrix}T_{+}(n)\begin{pmatrix}I&0\\\
0&-I\end{pmatrix}=T_{-}(n)$.

###### Proof.

These identities follow from matrix multiplication. Indeed we have

$\displaystyle T_{\pm}(n)^{t}\begin{pmatrix}0&I\\\
-I&0\end{pmatrix}T_{\pm}(n)$ $\displaystyle=\begin{pmatrix}zI-B(n)&\pm I\\\
\mp I&0\end{pmatrix}^{t}\begin{pmatrix}0&I\\\
-I&0\end{pmatrix}\begin{pmatrix}zI-B(n)&\pm I\\\ \mp I&0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}zI-B(n)&\mp I\\\ \pm
I&0\end{pmatrix}\begin{pmatrix}\mp I&0\\\ -zI+B(n)&\mp I\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&I\\\ -I&0\end{pmatrix}\,.$

This shows the first identity. Similarly we have

$\displaystyle\begin{pmatrix}I&0\\\
0&-I\end{pmatrix}T_{+}(n)\begin{pmatrix}I&0\\\ 0&-I\end{pmatrix}$
$\displaystyle=\begin{pmatrix}I&0\\\
0&-I\end{pmatrix}\begin{pmatrix}zI-B(n)&I\\\
-I&0\end{pmatrix}\begin{pmatrix}I&0\\\ 0&-I\end{pmatrix}$
$\displaystyle=\begin{pmatrix}zI-B(n)&-I\\\ I&0\end{pmatrix}\,,$

showing the second identity and completing the proof. ∎

The matrix in the first identity is a special symplectic matrix and often
appears. We give it the following label:

$J=\begin{pmatrix}0&I\\\ -I&0\end{pmatrix}\,.$

It follows from Lemma 3.5 that $T_{\pm}(n)\in SL(2d,\mathbb{C})$ which is the
group of $2d\times 2d$ complex symplectic matrices. Therefore, these are
matrix-valued functions mapping complex upper half plane $\mathbb{C}^{+}$ to
$SL(2d,\mathbb{C}).$ From Lemma 3.2 and Proposition 3.3 that for any $n$ and
any $z\in\mathbb{C}^{+},$ $M_{\pm}(n,z)$ are symmetric and have positive
definite imaginary part. Thus these are matrices in $\mathcal{S}_{d}$, the
Siegel upper half plane:

$\mathcal{S}_{d}=\\{Z\in\mathbb{C}^{d\times
d}:Z=X+iY,\,X^{t}=X,\,Y^{t}=Y,\,Y>0\\}\,.$

If $T\in SL(2d,\mathbb{R})$, the mapping $Z\mapsto T(Z)$ in equation (3.12) is
a well defined map and is a group action on $\mathcal{S}_{d}$, see [2] for
details. The transfer matrices $T_{\pm}(n)$, however, are complex symplectic
matrices acting on the space of $m$ functions. [2] provided a necessary
condition for a complex symplectic matrix $T\in SL(2d,\mathbb{C})$ so that the
map $Z\mapsto T(Z)$ is well defined. Here we show that the map $Z\mapsto
T_{\pm}(n,z)(Z)$ on $\mathcal{S}_{d}$ is well defined. More precisely we have
the following lemma.

######  Lemma 3.6.

For any $z\in\mathbb{C}^{+}$, if $B(n)$ is any real symmetric matrix, then
$T_{\pm}(n,z)(Z)\in\mathcal{S}_{d}$ for all $Z\in\mathcal{S}_{d}$.

###### Proof.

For any $z\in\mathbb{C}^{+}$, the second identity in Lemma 3.5, implies
$T_{+}(n,z)$ and $T_{-}(n,z)$ are conjugate to each other. Therefore, it is
enough to show that $T_{-}(n,z)(Z)\in\mathcal{S}_{d}$. Write the transfer
matrix $T_{-}(n,z)$ as the following product of matrices:

$T_{-}(n,z)=\begin{pmatrix}I&-B(n)\\\ 0&I\end{pmatrix}\begin{pmatrix}I&zI\\\
0&I\end{pmatrix}\begin{pmatrix}0&-I\\\ I&0\end{pmatrix}:=T_{B}T_{z}T_{J}\,.$

Since $J$ is a symplectic matrix, so is $-J$. Moreover the matrix $T_{B}\in
SL(2d,\mathbb{R})$. So $T_{J}(Z)$ and $T_{B}(W)$ are in $\mathcal{S}_{d}$ for
any $Z$ and $W$ in $\mathcal{S}_{d}$. Claim: $T_{z}(Z)\in\mathcal{S}_{d}$ for
any $Z\in\mathcal{S}_{d}$. But this follows easily since $T_{z}(Z)=Z+zI$ which
is clearly in $\mathcal{S}_{d}$. ∎

Next we observe the distance properties of Weyl $m$ functions. In one
dimensional space, hyperbolic metrics are commonly used to analyze the value
distribution for solutions of Schrödinger equations.

Define the following map on $\mathcal{S}_{d}$,
$d_{\infty}:\mathcal{S}_{d}\times\mathcal{S}_{d}\to\mathbb{R}$ via

$d_{\infty}(Z_{1},Z_{2})=\inf_{Z(t)}\int_{0}^{1}F_{Z(t)}(\dot{Z}(t))\,dt\,,\,\,Z_{1},Z_{2}\in\mathcal{S}_{d}$
(3.15)

where

$F_{Z}(W)=\|Y^{-1/2}WY^{-1/2}\|\,,$ (3.16)

and where the infimum is taken over all differentiable paths $Z(t)$ joining
$Z_{1}$ to $Z_{2}$. Here $Y$ is a positive definite matrix and possesses the
square root and $Y^{-1/2}=(Y^{1/2})^{-1}=(Y^{-1})^{1/2}$. The norm in equation
(3.16) is the operator norm of matrices acting on $\mathbb{C}^{d}$.

A calculation shows that $d_{\infty}$ is a metric on $\mathcal{S}_{d}$ and
hence $(\mathcal{S}_{d},d_{\infty})$ is a metric space.

######  Lemma 3.7.

For any $d\times d$ positive definite matrix with real entries and any
positive $\lambda$:

$A^{1/2}(A+\lambda I)^{-1/2}=(A(A+\lambda I)^{-1})^{1/2}=((I+\lambda
A^{-1})^{-1})^{1/2}\,.$

Moreover, if $I-\lambda A$ is positive, then

$\|A^{1/2}(A+\lambda I)^{-1/2}\|^{2}<\frac{1}{1+\lambda^{2}}\,.$

###### Proof.

The first equation follows from the continuous functional calculus with the
square root function on $[0,\infty)$ and the fact that $\lambda>0$. The norm
inequality now follows from the first equation and the usual norm estimate on
a Neumann series of operators. ∎

For any $T\in SL(2d,\mathbb{R})$, when considering the map from (3.12) as a
group action, [2] showed that this action, in fact, is distance preserving
with respect to the metric $d_{\infty}$. That is,

######  Theorem 3.8.

For any $T\in SL(2d,\mathbb{R})$ and any $W_{1},W_{2}\in\mathcal{S}_{d}$ we
have

$d_{\infty}(T(W_{1}),T(W_{2}))=d_{\infty}(W_{1},W_{2})\,.$

In general if $T\in SL(2d,\mathbb{C})$, Theorem 3.8 need not be true. However,
the map given in equation (3.12) with transfer matrices $T_{\pm}(n)$ is
distance decreasing. We have the following theorem.

######  Theorem 3.9.

Let $z\in\mathbb{C}^{+}$ and suppose $W_{j}=T_{-}(0,z)Z_{j}$ for $Z_{1}$ and
$Z_{2}$ in $\mathcal{S}_{d}$. Then

$d_{\infty}(T_{-}(n,z)W_{1},T_{-}(n,z)W_{2})\leq\frac{1}{1+y^{2}}\,d_{\infty}(W_{1},W_{2})\,,$
(3.17)

for all $n\in\mathbb{N}$.

###### Proof.

As in the proof of Lemma 3.6 Write $T_{-}(n,z)=T_{B}T_{z}T_{J}$. Since $T_{B}$
and $T_{J}$ are in $SL(2d,\mathbb{R})$, and the fact that $T_{z}$ has range in
$\mathcal{S}_{d}$, by Theorem 3.8, we get

$\displaystyle d_{\infty}(T_{-}(n,z)(W_{1}),T_{-}(n,z)(W_{2}))$
$\displaystyle=d_{\infty}(T_{B}T_{z}T_{J}(W_{1}),T_{B}T_{z}T_{J}(W_{2}))$
(3.18) $\displaystyle=d_{\infty}(T_{z}T_{J}(W_{1}),T_{z}T_{J}(W_{2}))\,.$

Moreover since $T_{J}\in SL(2d,\mathbb{R})$ we have

$d_{\infty}(T_{J}(W_{1}),T_{J}(W_{2}))=d_{\infty}(W_{1},W_{2})\,.$

It therefore follows that $T_{-}(n,z)(W)\in\mathcal{S}_{d}$.

Let $U_{j}=T_{J}(W_{j})=JW_{j}$, for $j=1,2$. Then $T_{z}(U_{j})=U_{j}+zI$.
Let $U(t)$ be a length minimizing path between $U_{1}$ and $U_{2}$. Then
$U(t)+zI$ is a path between $U_{1}+zI$ and $U_{2}+zI$. Moreover

$\frac{d}{dt}(U(t)+zI)=\dot{U(t)}\,.$

Suppose $V(t)=\operatorname{Im}U(t)$ then
$\operatorname{Im}\big{(}U(t)+zI\big{)}=V(t)+yI.$ We have the following
calculation:

$\displaystyle F_{(U(t)+zI)}(\dot{U(t)})=$
$\displaystyle=\|(V(t)+yI)^{-1/2}\dot{U(t)}(V(t)+yI^{-1/2}\|$
$\displaystyle=\|(V(t)+yI)^{-1/2}V(t)^{1/2}V(t)^{-1/2}\dot{U(t)}V(t)^{-1/2}V(t)^{1/2}(V(t)+yI)^{-1/2}\|$
$\displaystyle\leq\|(V(t)+yI)^{-1/2}V(t)^{1/2}\|\cdot\|V(t)^{-1/2}\dot{U(t)}V(t)^{-1/2}\|\cdot\|V(t)^{1/2}(V(t)+yI)^{-1/2}\|\,.$

Since $V(t)^{1/2}$ and $(V(t)+yI)^{-1/2}$ are symmetric, it follows that

$\|(V(t)+yI)^{-1/2}V(t)^{1/2}\|=\|V(t)^{1/2}(V(t)+yI)^{-1/2}\|\,.$

Therefore, from the symmetry and the above inequality, we have

$F_{(U(t)+zI)}(\dot{U(t)})\leq\|(V(t)+yI)^{-1/2}V(t)^{1/2}\|^{2}\|V(t)^{-1/2}\dot{U(t)}V(t)^{-1/2}\|\,.$
(3.19)

Next, since $U_{j}=JW_{j}$, for $j=1,2$, $U(t)$ must be of the form $JW(t)$
where $W(t)$ is a path between $W_{1}$ and $W_{2}$. However, since
$W_{j}=T_{-}(0,z)Z_{j}$, $W$ is of the form $W=T_{-}(0,z)Z$ where
$Z(t)=X(t)+iY(t)$ is a path between $Z_{1}$ and $Z_{2}$. By viewing the
multiplication as the group action, we have the following

$\displaystyle W(t)$ $\displaystyle=T_{-}(0,z)Z(t)=\begin{pmatrix}zI-B(0)&I\\\
-I&0\end{pmatrix}Z(t)=zI-B(0)-Z(t)^{-1}$
$\displaystyle=zI-B(0)-\overline{Z(t)}|Z(t)|^{-1}\,,$

where $|Z|=X^{2}+Y^{2}$. Since $Y$ is positive and
$\operatorname{Im}W(t)=yI+Y(X^{2}+Y^{2})^{-1}$, it follows that
$\operatorname{Im}W(t)-yI$ is also positive. Thus by applying the continuous
functional calculus to $\operatorname{Im}W(t)-yI$ with the function $f(x)=1/x$
on $(0,\infty)$, we get

$(\operatorname{Im}W(t))^{-1}-\frac{1}{y}I<0\,.$ (3.20)

Since $U(t)=JW(t)=-W(t)^{-1}$, by applying the continuous functional calculus
again we have

$U(t)=-W(t)^{-1}=-\overline{W(t)}|W(t)|^{-1}=-\overline{W(t)}\left((\operatorname{Re}W(t))^{2}+(\operatorname{Im}W(t))^{2}\right)^{-1}\,.$

Therefore,

$\operatorname{Im}U(t)=\operatorname{Im}W(t)\left((\operatorname{Re}W(t))^{2}+(\operatorname{Im}W(t))^{2}\right)^{-1}\,.$

Since

$(\operatorname{Im}W(t))^{2}\leq(\operatorname{Re}W(t))^{2}+(\operatorname{Im}W(t))^{2}\,,$

using the functional calculus again with the function $f(x)=1/x$ on
$(0,\infty)$ and using inequality (3.20) we obtain,

$\operatorname{Im}W(t)\left((\operatorname{Re}W(t))^{2}+(\operatorname{Im}W(t))^{2}\right)^{-1}\leq\operatorname{Im}W(t)^{-1}<\frac{1}{y}I\,.$

This implies that $I-y\operatorname{Im}U(t)$ is positive and hence
$I-y\operatorname{Im}V(t)$ is positive. By Lemma 3.7 we get

$\|(V+yI)^{-1/2}V^{1/2}\|<\frac{1}{1+y^{2}}\,.$

By using the above equation (3.19) becomes

$\displaystyle F_{(U(t)+zI)}(\dot{U(t)})$
$\displaystyle\leq\|(V(t)+yI)^{-1/2}V(t)^{1/2}\|^{2}\|V(t)^{-1/2}\dot{U(t)}V(t)^{-1/2}\|$
$\displaystyle\leq\frac{1}{1+y^{2}}F_{(U(t))}(\dot{U(t)})\,.$

Integrating the above inequality yields:

$\int_{0}^{1}F_{(U(t)+z)}(\dot{U(t)})\,dt\leq\frac{1}{1+y^{2}}\int_{0}^{1}F_{(U(t))}(\dot{U(t)})\,dt\,.$

Then taking the infimum over all such paths $U(t)$ we obtain the following:

$d_{\infty}(T_{z}U_{1},T_{z}U_{2})\leq\frac{1}{1+y^{2}}\,d_{\infty}(U_{1},U_{2})\,.$

Again since $J\in SL(2d,\mathbb{R})$, we have

$d_{\infty}(U_{1},U_{2})=d_{\infty}(T_{J}W_{1},T_{J}W_{2})=d_{\infty}(W_{1},W_{2})\,.$

Using the previous two equations in equation (3.18) we obtain:

$d_{\infty}(T_{-}(n,z)W_{1},T_{-}(n,z)W_{2})\leq\frac{1}{1+y^{2}}\,d_{\infty}(W_{1},W_{2})$

which completes the proof. ∎

Let $P_{-}(n,z)=T_{-}(n,z)\cdots T_{-}(1,z)$. Then, since $T_{-}(j,z)$ are
symplectic matrices for $j=1,\ldots,n$ and $SL(2d,\mathbb{C})$ is a group, it
follows that $P_{-}(n,z)\in SL(2d,\mathbb{C})$. We have the following
corollary.

######  Corollary 3.10.

For any $z\in\mathbb{C}^{+}$,

$d_{\infty}(M_{-}(n,z),P_{-}(n,z)M_{+}(0,z))\leq\frac{1}{(1+y^{2})^{n}}\,d_{\infty}(M_{-}(0,z),M_{+}(0,z))\,.$

###### Proof.

For any $z\in\mathbb{C}^{+}$ and $n\in\mathbb{N}$ by Lemma 3.4 we have

$M_{-}(n,z)=P_{-}(n,z)M_{-}(0,z)\,.$

Hence the result follows from Theorem 3.9. ∎

We define another $m$ function on the left half line $(-\infty,n)$. Assume
that $F_{-}(n,z)$ is a square summable matrix-valued solution to equation
(1.1) defined by equation (3.1) on $(-\infty,n)$. Define
$\widetilde{M}_{-}(n,z)$ by the following:

$\widetilde{M}_{-}(n,z)=-F_{-}(n-1,z)F_{-}(n,z)^{-1}\,.$

Similar to the proofs in Lemma 3.2 and Proposition 3.3, we see that
$\widetilde{M}_{-}(n,z)$ is symmetric and
$\operatorname{Im}\widetilde{M}_{-}(n,z)$ is a positive operator. In addition,
$\widetilde{M}_{-}(n,z)$ and $M_{-}(n,z)$ are related by the following
equation:

$\widetilde{M}_{-}(n,z)=M_{-}(n,z)-zI+B(n)\,.$

For fixed $n\in\mathbb{N}$, the mappings $z\mapsto M_{\pm}(n,z)$ are matrix
valued Herglotz functions. Set $M_{+}(z)=M_{+}(0,z)$. Since $M_{+}$ is a
matrix-valued Herglotz function, by Riesz-Herglotz representation, there
exists a matrix-valued measure $\mu$ on the bounded Borel subset of
$\mathbb{R}$ so that

$M_{+}(z)=C+Dz+\int_{\mathbb{R}}\left(\frac{1}{\lambda-z}-\frac{\lambda}{1+\lambda^{2}}\right)\,d\mu_{+}\,,$

for some constants $C$ and $D$. In addition, $M_{+}(z)$ has finite normal
limits, that is

$M_{+}(\lambda\pm i0)=\lim_{\varepsilon\downarrow 0}M_{+}(\lambda\pm
i\varepsilon)$

for almost all $\lambda\in\mathbb{R}$, see [3, 8]. More on matrix-valued
Herglotz functions are explained in the same papers. By the Lebesgue
Decomposition Theorem, the matrix-valued Borel measure $\mu+$ can be
decomposed as:

$\mu_{+}=\mu_{+,ac}+\mu_{+,sc}+\mu_{+,pp}\,,$

where $\mu_{+,ac}$ is the measure that is absolutely continuous with respect
to $\mu_{+}$, $\mu_{+,sc}$ is the singular continuous part of $\mu_{+}$ and
$\mu_{+,pp}$ is the pure point part of $\mu_{+}$. Let $\Sigma$, $\Sigma_{ac}$,
$\Sigma_{sc}$, and $\Sigma_{pp}$ be the topological support of $\mu_{+}$,
$\mu_{+,ac}$, $\mu_{+,sc}$, and $\mu_{+,pp}$ respectively.

The following theorem from [8] showed the connection between the matrix-valued
Herglotz functions and the essential support of corresponding spectral
measures. We state it without proof.

######  Theorem 3.11.

$\displaystyle\Sigma_{ac}$
$\displaystyle=\bigcup_{r=1}^{d}\left\\{\lambda\in\mathbb{R}:\lim_{\varepsilon\downarrow
0}M_{+}(\lambda+i\varepsilon)\textrm{ exists,
}\operatorname{rank}\operatorname{Im}(M_{+}(\lambda+i0))=r\right\\}$
$\displaystyle\Sigma_{sc}$
$\displaystyle=\bigcup_{r=1}^{d}\left\\{\lambda\in\mathbb{R}:\lim_{\varepsilon\downarrow
0}\operatorname{Im}(\operatorname{tr}M_{+}(\lambda+i\varepsilon))=\infty,\,\lim_{\varepsilon\downarrow
0}\varepsilon\operatorname{tr}M_{+}(\lambda+i\varepsilon)=0\right\\}$
$\displaystyle\Sigma_{pp}$
$\displaystyle=\bigcup_{r=1}^{d}\left\\{\lambda\in\mathbb{R}:\operatorname{rank}\lim_{\varepsilon\downarrow
0}\varepsilon M_{+}(\lambda+i\varepsilon)=r\right\\}\,.$

Describing the absolutely continuous spectrum is one of the main goals in
studying vector-valued discrete Schrödinger operators. Remling successfully
did this for one dimensional Schrödinger operators, [11], and in his renowned
paper [10], he describes it for Jacobi matrices. The techniques utilized there
involve the concept of reflectionless. An operator is said to be
reflectionless on a Borel set $B\subseteq\mathbb{R}$ if the associated
Herglotz functions satisfy $m_{+}(x)=-\overline{m_{-}(x)}$ for almost all
$x\in B$ in the Lebesgue sense. This definition naturally extends to our
matrix-valued Herglotz fuctions $M_{\pm}$. The above theorem is also utilized
in describing the absolutely continuous spectrum as yields precise limit
conditions on the Herglotz functions. Describing the absolutely continuous
spectrum is an extremely difficult question and this is one of our main goals
in future work.

Acknowledgment: The first author was partially supported by the Office of
Sponsored Research at Embry-Riddle Aeronautical University, Florida.

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