TIF-UNIMI-2024-01

Higgs production at RHIC and the positivity
of the gluon helicity distribution

Daniel de Florian${}^{\,a}$, Stefano Forte${}^{\,b}$, Werner
Vogelsang${}^{\,c}$
a International Center for Advanced Studies (ICAS) and ICIFI,

ECyT-UNSAM, Campus Miguelete,

25 de Mayo y Francia, (1650) Buenos Aires, Argentina

b Tif Lab, Dipartimento di Fisica, Università di Milano and INFN,

Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy

c Institute for Theoretical Physics, Tübingen University, Auf der Morgenstelle
14,

72076 Tübingen, Germany

Abstract

We show that the negative polarized gluon distribution $\Delta g$ found in a
recent global next-to-leading order QCD analysis of the nucleon helicity
structure is incompatible with the fundamental requirement that physical
cross-sections must not be negative. Specifically, we show that the fact that
this polarized gluon strongly violates the positivity condition $|\Delta
g|\leq g$ in terms of the unpolarized gluon distribution $g$ leads to negative
cross-sections for Higgs boson production at RHIC as a physical process,
implying that this negative $\Delta g$ is unphysical.

Dedicated to the memory of Stefano Catani

Understanding the quark and gluon spin structure of the proton is a key focus
of modern nuclear and particle physics. An important component of this
endeavor is the precise determination of the proton helicity parton
distribution functions (PDFs). The gluon helicity PDF $\Delta g(x,\mu)$, in
particular, has received much attention in this context as its integral over
all momentum fractions $x$ measures the gluon spin contribution to the proton
spin and hence could hold the key to decomposing the proton spin into its
partonic contributions. A celebrated discovery was made in 2014, when it was
shown [1, 2] that data from the Relativistic Heavy Ion Collider (RHIC) [3]
provided evidence for a non-vanishing and positive $\Delta g$ in the region
$0.05\lesssim x\lesssim 0.2$. This finding was obtained on the basis of a
global next-to-leading order (NLO) QCD analysis of the world data on polarized
(semi-)inclusive deep-inelastic scattering and polarized $pp$ scattering. It
was confirmed in additional studies [4, 5] and later substantially
corroborated when further sets of RHIC data became available [3].

The observation of a (locally) positive $\Delta g$ was recently called into
question in Refs. [6, 7, 8]. Again in the context of an NLO analysis (in the
$\overline{\mathrm{MS}}$ scheme) the authors found PDFs featuring a negative
gluon helicity PDF, $\Delta g<0$. These PDFs are delivered as an ensemble of
replicas, quantifying the PDF uncertainty. Figure 1 shows the 78 replicas
corresponding to the PDFs of Ref. [7], available at [9] in LHAPDF [10] format.
We show results at factorization scales $\mu=\sqrt{10}$ GeV (left) and
$\mu=125$ GeV (right).

A striking feature of the PDFs with negative $\Delta g$ proposed in [6, 7, 8]
is that they strongly violate the positivity condition

$|\Delta g(x,\mu)|\,\leq g(x,\mu)$ (1)

at momentum fractions $x\gtrsim 0.25$. Indeed, as the authors state, the
negative $\Delta g$ PDFs are obtained only when the positivity condition is
relaxed in the analysis. The violation of the inequality (1) is evident from
Fig. 1 where we also show in both panels the replicas for the corresponding
unpolarized gluon distribution as obtained in the same analysis [7].

Figure 1:  PDF replicas of Ref. [7] for $-g(x,\mu)$ and $\Delta g(x,\mu)$ at
$\mu=\sqrt{10}$ GeV (left) and $\mu=125$ GeV (right).

Condition (1) arises of course from the fact that $g=g_{+}+g_{-}$ and $\Delta
g=g_{+}-g_{-}$ where $g_{+},g_{-}$ are the distributions for gluons with
positive or negative helicity inside a proton with positive helicity,
respectively. At leading order (LO) in perturbative QCD these can be regarded
as number densities and hence positive***Here and elsewhere we use the word
positive to mean non-negative., so Eq. (1) strictly holds. This is a
consequence of the fact that there exist physical processes for which at
leading order the physically measurable cross-section, which is a probability
and thus positive, is proportional to the PDFs. However, beyond LO the cross-
section is obtained by convoluting the PDF with a partonic cross-section
(coefficient function). The cross-section remains of course positive, but now
the coefficient function and the PDF depend on the factorization scheme and
hence are not necessarily separately positive, so strict positivity of
$g_{+},g_{-}$ does not need to hold any longer [15, 14, 11, 12, 13]. As a
result, solutions with $|\Delta g|\geq g$ are in principle formally possible.

Such violations of positivity however must have the size of the higher-order
corrections, because the possible violation of positivity of PDFs must be
compensated by the higher-order corrections to the coefficient functions so
that the physical cross-section remains positive. Indeed, In Ref. [15] next-
to-leading order (NLO) positivity bounds on polarized PDFs in the
$\overline{\rm MS}$ scheme were derived by requiring positivity of NLO cross-
sections. These NLO bounds were used to derive a bound on the polarized gluon
distribution in $x$ space [14], which was compared to the naive LO bound Eq.
(1) and found to differ from it at the percent level except at very small
$x\lesssim 10^{-3}$. In contrast to this, the violation of positivity of the
PDFs from Refs. [6, 7, 8] exhibited in Fig. 1 is much larger: in fact,
$|\Delta g|$ exceeds $g$ by a large factor. This suggests that these PDFs may
lead to unphysical predictions.

To see how this may happen, we recall how positivity bounds can be derived at
any perturbative order [15]. The derivation is based on the observation that
physically observable cross-sections – and theoretical predictions thereof –
are proportional to the number of observed events, and thus cannot be
negative. Using the spin-dependent cross-section for some reaction in
polarized $pp$ scattering as an example, we must have

$|A_{\mathrm{LL}}|\,\leq\,1\,,$ (2)

where

$A_{\mathrm{LL}}\,\equiv\,\frac{\Delta\sigma}{\sigma}\,\equiv\,\frac{\sigma_{++}-\sigma_{+-}}{\sigma_{++}+\sigma_{+-}}\,,$
(3)

with $\sigma_{++}$ ($\sigma_{+-}$) the cross-section when the two colliding
protons have the same (or opposite) helicities. (For simplicity, we are
considering a parity-conserving interaction). The condition (2) must apply to
any physical cross-section, regardless of whether it has been measured, or
even whether it is practically measured or measurable in an actual experiment.
Positivity bounds on any polarized PDF at, say NLO can then be derived by
imposing the condition Eq. (2) on a set of suitably chosen pairs of polarized
and unpolarized NLO cross-sections. For instance, the positivity bounds of
Ref. [15] were obtained by imposing the condition on polarized and unpolarized
deep-inelastic scattering, as well as for Higgs production in gluon–proton
scattering.

Because the bound Eq. (2) implies a bound on the polarized PDFs, it follows
that PDFs that violate this positivity bound lead to unphysical negative
cross-sections. Given the enormous violation of positivity for the gluon PDF
as seen in Fig. 1, one may immediately ask whether there could be a physical
observable for which an NLO prediction based on this gluon density violates
the condition Eq. (2). We will now show that this is indeed the case,
rendering the solutions of [6, 7, 8] with negative $\Delta g$ unphysical. A
suitable candidate for this purpose is an observable that is gluon-driven at
tree level, and probes the region $x\gtrsim 0.25$ of momentum fractions where
the gluon distribution in Fig. 1 violates positivity, such as the Higgs
production cross-section in $pp$ scattering. A dominant contribution to this
process is gluon-gluon fusion, $gg\rightarrow H$, through a top quark loop. To
lowest order in QCD no additional partonic channels involving incoming quarks
contribute to this process.

This process is of course of paramount importance at the LHC, and its total
cross-section has been studied in this context in great theoretical detail
(see Refs. [16, 17, 18, 19]). Here we will instead consider Higgs production
at RHIC energy, $\sqrt{S}=510$ GeV, using a range of Higgs masses $m_{H}$
between 100 GeV and 250 GeV. Although this is not relevant for canonical Higgs
phenomenology, it will allow us to access a perturbative cross-section in the
kinematic regime where $x\gtrsim 0.25$, since the lowest $x$-value probed in
the PDFs at leading order for a given Higgs mass is $m_{H}^{2}/S$, with
$\sqrt{S}$ the $pp$ center-of-mass energy.

The unpolarized and spin-dependent cross-sections for $pp\rightarrow HX$ may,
up to power corrections, be written in factorized form as

$\displaystyle\sigma^{pp\rightarrow H}$ $\displaystyle=$
$\displaystyle\sigma_{0}\sum_{i,j}\int_{\tau}^{1}dx_{1}\,\int_{\tau/x_{1}}^{1}dx_{2}\,f_{i}(x_{1},\mu)\,f_{j}(x_{2},\mu)\,\omega^{ij\rightarrow
H}\left(z=\frac{\tau}{x_{1}x_{2}},\alpha_{s}(\mu),\frac{\mu}{m_{H}}\right)\,,$
$\displaystyle\Delta\sigma^{pp\rightarrow H}$ $\displaystyle=$
$\displaystyle\sigma_{0}\sum_{i,j}\int_{\tau}^{1}dx_{1}\,\int_{\tau/x_{1}}^{1}dx_{2}\,\Delta
f_{i}(x_{1},\mu)\,\Delta f_{j}(x_{2},\mu)\,\Delta\omega^{ij\rightarrow
H}\left(z=\frac{\tau}{x_{1}x_{2}},\alpha_{s}(\mu),\frac{\mu}{m_{H}}\right),\;\;\;$
(4)

where $\tau=m_{H}^{2}/S$ and $\omega^{ij\rightarrow
H},\Delta\omega^{ij\rightarrow H}$ are normalized hard-scattering functions
that are computed in perturbation theory. They are defined as in Eq. (3) by
$\omega^{ij\rightarrow H}\equiv\frac{1}{2}(\omega^{ij\rightarrow
H}_{++}+\omega^{ij\rightarrow H}_{+-})$ and $\Delta\omega^{ij\rightarrow
H}\equiv\frac{1}{2}(\omega^{ij\rightarrow H}_{++}-\omega^{ij\rightarrow
H}_{+-})$, where $\omega^{ij\rightarrow H}_{\lambda_{i}\lambda_{j}}$ is the
cross section for incoming partons $i,j$ with helicities
$\lambda_{i},\lambda_{j}$. The normalization $\sigma_{0}$ is the same for
$\sigma^{pp\rightarrow H}$ and $\Delta\sigma^{pp\rightarrow H}$ and is given
by

$\sigma_{0}\,=\,\frac{\alpha_{s}^{2}|A|^{2}}{256\pi v^{2}}\,,$ (5)

with $\alpha_{s}$ the strong coupling and $v=246$ GeV the Higgs vacuum
expectation value. The factor $|A|^{2}$ results from the coupling of the two
gluons to the Higgs boson via a heavy-quark loop. Ignoring contributions from
charm and bottom quarks and keeping only the top quark of mass $m_{t}$, one
has, at lowest order (see [20, 21]):

$A\,=\,\tau_{q}\left(1+(1-\tau_{q})\arcsin^{2}(1/\sqrt{\tau_{q}})\right)\,,$
(6)

where $\tau_{q}\equiv 4m_{t}^{2}/m_{H}^{2}$. This expression was originally
obtained for the spin-averaged cross-section, but we have checked that it also
holds for the spin-dependent one. One may further assume that the top quark is
infinitely heavy. In the effective theory defined by this assumption, one has
$|A|^{2}=4/9$. In any case, the factor $|A|^{2}$ cancels in the spin
asymmetry. For simplicity, we have chosen the factorization and
renormalization scales to be the same in Eq. (S0.Ex1) and denoted them by
$\mu$. For our numerical results further below, we will set $\mu=m_{H}/2$, a
value that is known to lead to faster convergence of the perturbative
expansion of the Higgs cross-section [22]. However, none of our results
depends qualitatively on this choice.

As mentioned, to lowest order, $gg\rightarrow H$ is the only contributing
channel: because this is a $2\rightarrow 1$ reaction, it is characterized by
$\hat{s}=m_{H}^{2}$, corresponding to $z=1$, where $\hat{s}=x_{1}x_{2}S$ is
the partonic center-of-mass energy squared. Correspondingly, the partonic
hard-scattering function is given by a Dirac delta $\delta(1-z)$ at this
order. The NLO corrections to the partonic hard-scattering functions have been
computed in the $\overline{\mathrm{MS}}$ scheme in Refs. [20, 21] for the
spin-averaged and in [15] for the spin-dependent cross-section. For the
$gg$-channel we have, up to corrections of order $\alpha_{s}^{2}$:

$\displaystyle\omega^{gg\rightarrow H}\left(z,\alpha_{s},r\right)$
$\displaystyle=$
$\displaystyle\delta(1-z)\,+\,\frac{\alpha_{s}}{\pi}\left\\{\delta(1-z)\left(\frac{11}{2}+\pi^{2}\right)-\frac{11}{2}(1-z)^{3}\right.$
$\displaystyle+$
$\displaystyle\left.6\left(1-z+z^{2}\right)^{2}\left[2\left(\frac{\ln(1-z)}{1-z}\right)_{+}-\frac{\ln(z)}{1-z}-\frac{\ln(r^{2})}{(1-z)_{+}}\right]\right\\}\,,$
$\displaystyle\Delta\omega^{gg\rightarrow H}\left(z,\alpha_{s},r\right)$
$\displaystyle=$
$\displaystyle\delta(1-z)\,+\,\frac{\alpha_{s}}{\pi}\left\\{\delta(1-z)\left(\frac{11}{2}+\pi^{2}\right)+\frac{11}{2}(1-z)^{3}\right.$
(7) $\displaystyle+$
$\displaystyle\left.6z\left(2-3z+2z^{2}\right)\left[2\left(\frac{\ln(1-z)}{1-z}\right)_{+}-\frac{\ln(z)}{1-z}-\frac{\ln(r^{2})}{(1-z)_{+}}\right]\right\\}\,,$

where $r=\mu^{2}/m_{H}^{2}$, and where the $+$ distribution is defined in the
usual way. Starting at NLO, there are also two new partonic channels,
$qg\rightarrow Hq$ and $q\bar{q}\rightarrow Hg$. Their cross-sections are also
known at ${\cal O}(\alpha_{s})$ from [20, 21, 15]:

$\displaystyle\omega^{qg\rightarrow H}\left(z,\alpha_{s},r\right)$
$\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{\pi}\left\\{-\frac{1}{3}(1-z)(7-3z)+\frac{2}{3}\left(1+(1-z)^{2}\right)\left[\ln\frac{(1-z)^{2}}{zr^{2}}+1\right]\right\\}\,,$
$\displaystyle\Delta\omega^{qg\rightarrow H}\left(z,\alpha_{s},r\right)$
$\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{\pi}\left\\{(1-z)^{2}+\frac{2}{3}\left(1-(1-z)^{2}\right)\left[\ln\frac{(1-z)^{2}}{zr^{2}}+1\right]\right\\}\,,$
(8)

and

$\omega^{q\bar{q}\rightarrow
H}\left(z,\alpha_{s},r\right)\,=\,\frac{\alpha_{s}}{\pi}\,\frac{32}{27}\,(1-z)^{3}\,=\,-\Delta\omega^{q\bar{q}\rightarrow
H}\,.$ (9)

We now compute the spin asymmetry $A_{\mathrm{LL}}$ in Higgs production at
RHIC at $\sqrt{S}=510$ GeV, as a function of the Higgs mass. We adopt the PDF
set with the positivity-violating negative gluon distribution of Ref. [7],
used already for Fig. 1. We also use the unpolarized PDFs of [7] for the
denominator of the spin asymmetry: this allows us to effectively determine the
individual helicity cross-sections in Eq. (3) and thus check positivity on a
replica-by–replica basis. As a cross-check we have verified that for the LO
and NLO unpolarized cross-sections we recover the results given in Table 7 of
Ref. [22] when PDFs and parameters are chosen as in that paper. Figure 2
displays the asymmetry $A_{\mathrm{LL}}$ for the replicas shown in Fig. 1, on
a linear (left) and on a logarithmic (right) scale. A huge violation of the
physical positivity condition (2) is observed. Already for Higgs masses around
the physical value the asymmetry $A_{\mathrm{LL}}$ exceeds unity; at even
larger masses it easily reaches values of 10 or even 100. We are thus led to
the conclusion that the PDF set with the positivity-violating negative gluon
distribution cannot be regarded as physical as it leads to negative cross-
sections.

Evidently, the violation of physical positivity of $A_{\mathrm{LL}}$ seen in
Fig. 2 is a bulk property of all PDF replicas in a regime that is central for
the solutions with negative $\Delta g$ found in [6, 7, 8]. Hence, it cannot be
attributed to outliers, such as due to statistical fluctuations or large
uncertainties. It is important to note that the violation occurs in a
kinematic region corresponding to momentum fractions $x$ where the unpolarized
PDFs, and even the gluon PDF, are known rather accurately. For instance, at
$m_{h}=150$ GeV and with $\sqrt{S}=510$ GeV we have $x_{1}x_{2}\approx 0.09$
so at central rapidity $x_{1}=x_{2}\approx 0.3$. The uncertainty on the
unpolarized gluon, taking the conservative PDF4LHC21 combination, is about 5%
[23], so the violation of positivity cannot be reasonably attributed to
imperfect knowledge of the unpolarized gluon and its uncertainty. Indeed, we
have checked that replacing the unpolarized PDFs of [7] in the computation of
the asymmetry with the PDFs of the PDF4LHC21 set [23] and always taking the
largest of the 100 PDF4LHC21 replicas, which corresponds to a more than
three-$\sigma$ interval about the central gluon, we still get positivity
violation for $m_{H}\gtrsim 140$ GeV, exponentially increasing with Higgs mass
with a pattern analogous to the curves shown in Fig. 2.

For comparison, we also show in Fig. 2 (lower bands) the double-spin
asymmetries obtained for the sets of [7] with positive $\Delta g$, which show
a strikingly different behavior. In this case, about 1000 PDF replicas are
available and plotted. The vast majority of them satisfies positivity.
Violations of positivity are either in the tail of the distribution, or in
kinematic regimes dominated by very large values of $x$. In these regions, the
unpolarized gluon PDF is poorly known, and it is in fact extrapolated from
information at smaller $x$, so the positivity violation could be reabsorbed in
a change of unpolarized gluon PDF.

Figure 2:  Double-helicity asymmetry for Higgs production at RHIC as a
function of the Higgs mass, with a linear (left) or logarithmic (right) scale
on the vertical axis. The upper bands show $A_{{\mathrm{LL}}}$ as obtained for
the gluon distribution shown in Fig. 1, while the lower bands provide the
corresponding result for the sets of [7] with $\Delta g\geq 0$. In both plots,
the dashed lines show the physical limit given by $|A_{\mathrm{LL}}|=1$.

As mentioned, in a consistent quantum field theory negative cross-sections
cannot occur for any process, regardless of whether it is measurable in
practice or even in principle. However, in this case it is interesting to
observe that it is a physically measurable hadronic cross-section that is
predicted to be negative. Indeed, the gluon fusion process dominates the Higgs
production cross-section at all energies (see e.g. [25, 24]). We have in fact
checked explicitly, using MadGraph5_aMC@NLO [26] that the vector-boson fusion
process, which dominates Higgs productions in the quark channel
$qq^{\prime}\rightarrow qq^{\prime}H$, despite the presence of initial-state
valence quarks, gives a contribution to Higgs production in proton-proton
collisions that is more than an order of magnitude smaller than that by gluon
fusion also at RHIC energy. Hence the positivity violation seen in Fig. 2
leads to the prediction of a negative cross-section for inclusive Higgs boson
production in proton-proton collisions.

Along the same lines, one may wonder whether, given the large size of higher-
order corrections to Higgs production in gluon fusion, the violation of
physical positivity seen in Fig. 2 could be due to these NLO corrections, or
perhaps be alleviated by higher-order corrections. However, NLO corrections in
fact cancel to a very large degree in the spin asymmetry. Furthermore, the
channels with incoming quarks, $qg\rightarrow Hq$ and $q\bar{q}\rightarrow
Hg$, although nominally favored for very high $x$ thanks to the participation
of a valence quark PDF, remain sub-dominant and hence cannot re-instate
positivity. It would be straightforward to further improve the perturbative
framework by carrying out threshold resummation for the Higgs cross-section,
following the lines of [27]. This would, in fact, be required for an accurate
phenomenological study of Higgs production at RHIC. However, this, too, is
irrelevant for positivity since both the spin-averaged and the polarized
cross-section receive the same QCD corrections near partonic threshold
$\hat{s}\approx m_{H}^{2}$. In fact, the latter observation can be made more
general: the fact that the region of interest here is $x\gtrsim 0.25$ means
for a $pp$ collider that any relevant process is probed close to partonic
threshold. Given that at threshold the QCD corrections are dominated by soft
emission and that soft-gluon emission is spin-independent, for such a
kinematic regime the dominant QCD corrections will be very similar for general
polarized and unpolarized cross-sections, and even identical in some cases as
for color-singlet $2\rightarrow 1$ Higgs production. Therefore, spin
asymmetries are given by their LO expression to high accuracy. Thus a
significant violation of the LO positivity condition such as Eq. (1) in a
dominant partonic channel will automatically lead to the violation of
positivity in the physical hadronic cross-section and invalidate the
corresponding parton distributions.

In conclusion, by considering Higgs boson production at RHIC we have shown
that previously proposed scenarios for the proton’s polarized parton
distribution functions with a large negative gluon polarization lead to
unphysical negative cross-sections. Reassuringly, such scenarios appear to be
strongly disfavored by RHIC data for direct-photon [28] and dijet [3]
production not included in the analyses of Refs. [6, 7, 8]. Amusingly, the
Higgs production process that we have considered is in fact not hypothetical
at all: based on our results, we estimate that about half a dozen Higgs bosons
should have been produced at RHIC during its lifetime with $510$ GeV running.

## Acknowledgments

This work originated from discussions at the Center for Frontiers in Nuclear
Science (CFNS) workshop “Precision QCD predictions for ep Physics at the EIC
(II)”, Stony Brook, September 18-22, 2023. We are grateful to M. Zaro for
helpful remarks on the VBF process, and to I. Borsa, J. Karpie, R. Sassot, N.
Sato and M. Stratmann for discussions and to N. Sato for comments on a first
draft. The work of D.de F. has been partially supported by Conicet, ANPCyT and
the Alexander von Humboldt Foundation. W.V. is supported by Deutsche
Forschungsgemeinschaft (DFG) through the Research Unit FOR 2926 (project
409651613).

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# On the Schrödinger Operator with a Periodic PT-symmetric Matrix Potential

O. A. Veliev
Dogus University, Istanbul, Turkey.
e-mail<EMAIL_ADDRESS>
###### Abstract

In this article we obtain asymptotic formulas for the Bloch eigenvalues of the
operator $L$ generated by a system of Schrödinger equations with periodic PT-
symmetric complex-valued coefficients. Then using these formulas we classify
the spectrum $\sigma(L)$ of $L$ and find a condition on the coefficients for
which $\sigma(L)$ contains all half line $[H,\infty)$ for some $H.$

Key Words: Non-self-adjoint differential operator, PT-symmetric coefficients,
Periodic matrix potential.

AMS Mathematics Subject Classification: 34L05, 34L20.

##  1 Introduction and Preliminary Facts

Let $L(Q)$ be the differential operator generated in the space
$L_{2}^{m}(-\infty,\infty)$ of the vector functions by the differential
expression

$-y^{{}^{\prime\prime}}+Qy,$ (1)

where $Q=\left(q_{i,j}\right)$ is a $m\times m$ matrix with the PT-symmetric
$\pi$-periodic locally square integrable entries $q_{i,j}.$ In other words,

$\overline{q_{i,j}\left(-x\right)}=q_{i,j}\left(x\right),\text{
}q_{i,j}\left(x+\pi\right)=q_{i,j}\left(x\right),\text{ }q_{i,j}\in
L_{2}\left[0,\pi\right].$ (2)

It is well-known that [1, 5, 7] the spectrum $\sigma(L(Q))$ of the operator
$L(Q)$ is the union of the spectra $\sigma(L_{t}(Q))$ of the operators
$L_{t}(Q)$ for $t\in(-1,1]$ generated in $L_{2}^{m}[0,\pi]$ by the
differential expression (1) and the quasiperiodic conditions

$y^{{}^{\prime}}\left(\pi\right)=e^{i\pi
t}y^{{}^{\prime}}\left(0\right),\text{ }y\left(\pi\right)=e^{i\pi
t}y\left(0\right).$ (3)

Note that $L_{2}^{m}(a,b)$ is the set of the vector functions
$f=\left(f_{1},f_{2},...,f_{m}\right)$ with $f_{k}\in L_{2}(a,b)$ for
$k=1,2,...,m.$ The norm $\left\|\cdot\right\|$ and inner product
$(\cdot,\cdot)$ in $L_{2}^{m}(a,b)$ are defined by

$\left\|f\right\|=\left(\int\nolimits_{(a,b)}\left|f\left(x\right)\right|^{2}dx\right)^{\frac{1}{2}},\text{
}(f,g)=\int\nolimits_{(a,b)}\left\langle
f\left(x\right),g\left(x\right)\right\rangle dx,$

where $\left|\cdot\right|$ and $\left\langle\cdot,\cdot\right\rangle$ are the
norm and inner product in $\mathbb{C}^{m}.$ For $t\in(-1,1]$ the spectra
$\sigma(L_{t}(Q))$ of the operators $L_{t}(Q)$ consist of the eigenvalues
called the Bloch eigenvalues of $L(Q)$. Any eigenfunction $\Psi_{\lambda(t)}$
corresponding to the Bloch eigenvalue $\lambda(t)$ is called the Bloch
function. We say that $\Psi_{\lambda(t)}$ is a normalized Bloch Function if
its $L_{2}^{m}(0,\pi)$ norm is $1$.

Let us introduce some preliminary results and describe briefly the scheme of
the paper. The eigenvalues of the operator $L_{t}(Q)$ are the roots of the
characteristic determinant

$\Delta(\lambda,t)=\det(Y_{j}^{(\nu-1)}(\pi,\lambda)-e^{i\pi
t}Y_{j}^{(\nu-1)}(0,\lambda))_{j,\nu=1}^{2}=$ (4) $e^{i2m\pi
t}+f_{1}(\lambda)e^{i(2m-1)\pi t}+f_{2}(\lambda)e^{i(2m-2)\pi
t}+...+f_{2m-1}(\lambda)e^{i\pi t}+1$

which is a polynomial of $e^{i\pi t}$ with entire coefficients
$f_{1}(\lambda),f_{2}(\lambda),...$, where $Y_{1}(x,\lambda)$ and
$Y_{2}(x,\lambda)$ are the solutions of the matrix equation

$-Y^{{}^{\prime\prime}}(x)+Q\left(x\right)Y(x)=\lambda Y(x)$

satisfying $Y_{1}(0,\lambda)=O_{m}$, $Y_{1}^{{}^{\prime}}(0,\lambda)=I_{m}$
and $Y_{2}(0,\lambda)=I_{m}$, $Y_{2}^{{}^{\prime}}(0,\lambda)=O_{m}$ (see [6]
Chapter 3). Here $O_{m}$ and $I_{m}$ are $m\times m$ zero and identity
matrices respectively. It is clear that

$\varphi_{k,1,t}=\left(\begin{array}[c]{c}\frac{e^{i\left(2k+t\right)x}}{\sqrt{\pi}}\\\
0\\\ \vdots\\\ 0\end{array}\right),$
$\varphi_{k,2,t}=\left(\begin{array}[c]{c}0\\\
\frac{e^{i\left(2k+t\right)x}}{\sqrt{\pi}}\\\ \vdots\\\
0\end{array}\right),...,\varphi_{k,m,t}=\left(\begin{array}[c]{c}0\\\
\vdots\\\ 0\\\ \frac{e^{i\left(2k+t\right)x}}{\sqrt{\pi}}\end{array}\right)$
are the normalized eigenfunctions of the operator $L_{t}(O_{m})$ corresponding
to the eigenvalue $\left(2k+t\right)^{2}$. If $t\neq 0,1,$ then the
multiplicity of the eigenvalue $\left(2k+t\right)^{2}$ is $m$ and the
corresponding eigenspace is
$E_{k}(t)=span\left\\{\varphi_{k,1,t},\varphi_{k,2,t},...,\varphi_{k,m,t}\right\\}$.
In the cases $t=0$ and $t=1$ the multiplicity of the nonzero eigenvalues
$\left(2k\right)^{2}$ and $\left(2k+1\right)^{2}$ is $2m$ and the
corresponding eigenspaces are

$E_{k}(0)=span\left\\{\varphi_{n,j,0}:n=k,-k;\text{ }j=1,2,...m\right\\}$ and

$E_{k}(\pi)=span\left\\{\varphi_{n,j,1}:n=k,-(k+1);\text{
}j=1,2,...m\right\\}$ respectively.

The brief scheme of the paper is the following. First we note that (see
Theorem 1) if $\lambda$ is an eigenvalue of multiplicity $p$ of the operator
$L_{t}(Q),$ then $\overline{\lambda}$ is also an eigenvalue of the same
multiplicity of $L_{t}(Q).$ This is the characteristic property of the
Schrödinger operator with PT-symmetric potential. Then, in Theorem 2 we prove
that there exists a constant $c$ such that the eigenvalues $\lambda(t)$ of the
operator $L_{t}(Q)$ lie on the $c$ neighborhoods of the eigenvalues
$(2n+t)^{2}$ of $L_{t}(O_{m})$ for $n\in\mathbb{Z}$. For this we use the
formula

$\left(\lambda(t)-\left(2n+t\right)^{2}\right)\left(\Psi_{\lambda(t)},\varphi_{n,s,t}\right)=\left(Q\Psi_{\lambda(t)},\varphi_{n,s,t}\right),$
(5)

which can be obtained from the equality
$L_{t}\left(Q\right)\Psi_{\lambda(t)}=\lambda(t)\Psi_{\lambda(t)}$ by
multiplying both sides by $\varphi_{n,s,t}(x)$ and using
$L_{t}\left(O_{m}\right)\varphi_{n,s,t}(x)=\left(2n+t\right)^{2}\varphi_{n,s,t}(x).$
In the other words, first we consider the operator $L_{t}(O_{m})$ for an
unperturbed operator and the operator of multiplication by $Q$ for a
perturbation.

Then to obtain a sharp asymptotic formulas we consider the operator $L_{t}(Q)$
as perturbation of $L_{t}(A),$ where

$A=\int\nolimits_{(0,\pi)}Q\left(x\right)dx,$ (6)

by $Q-A,$ that is, we take the operator $L_{t}(A)$ for an unperturbed operator
and the operator of multiplication by $Q-A$ for a perturbation. Therefore
first we analyze the eigenvalues and eigenfunctions of $L_{t}(A)$.

Using (2) and the substitution $t=-x$ one can get the equality

$\overline{\int_{0}^{\pi}q_{i,j}\left(x\right)dx}=\int_{0}^{\pi}q_{i,j}\left(-x\right)dx=-\int_{0}^{-\pi}q_{i,j}\left(t\right)dt=\int_{0}^{\pi}q_{i,j}\left(t\right)dt$

which means that

$\int_{0}^{\pi}q_{i,j}\left(x\right)dx\in\mathbb{R}$

for all $i$ and $j$. Hence the entries of the matrix $A$ are the real numbers.
Therefore, the eigenvalues of the matrix $A$ consist of the real eigenvalues
and the pairs of the conjugate complex numbers. The distinct eigenvalues of
$A$ are denoted by $\mu_{1},\mu_{2},...,\mu_{p}.$ Without loss of generality
we denote the real distinct eigenvalues by $\mu_{1}<\mu_{2}<...<\mu_{s}$ and
the nonreal distinct eigenvalues by$\ \mu_{s+1},\mu_{s+2},...,\mu_{p}.$ It
readily implies that the spectrum of $L(A)$ consists of the real half line
$[\mu_{1},\infty)$ and nonreal half lines

$[\mu_{j}+a:\text{ }a\in[0,\infty)$ (7)

for $j=s+1,s+2,...,p.$

In Theorem 3 we find a sharp and uniform, with respect to the quasimomenta
$t\in(-1,1],$ asymptotic formula for the Bloch eigenvalues of $L(Q)$ in term
of the eigenvalues of the matrix $A$ for any matrix potential $Q$ with locally
square integrable entries. This asymptotic formula implies that the spectrum
of $L(Q)$ is asymptotically close to the spectrum of the operator $L(A).$
Therefore, if the matrix $A$ has no real eigenvalues then the spectrum of the
operator $L(Q)$ with general potential $Q$ approaches the nonreal half lines
(7), which implies that the real component $\sigma(L(Q))\cap\mathbb{R}$ of the
spectrum $\sigma(L(Q))$ is contained in a finite interval $[a,b]$ (see Theorem
4). Then we prove that, if the entries of $Q$ are the PT-symmetric functions,
and the matrix $A$ has a real eigenvalue with odd multiplicity, then
$\sigma(L(Q))$ contains the main part (in sense of (41)) of $[0,\infty)$ (see
Theorem 6). For this we consider the multiplicity of the Bloch eigenvalues
(see Theorem 5). Note that if $m$ is an odd number, then the matrix $A$ has a
real eigenvalue with odd multiplicity, because the total sum of the
multiplicities of the eigenvalues $\mu_{1},\mu_{2},...,\mu_{p}$ is $m$ and the
nonreal eigenvalues are the pairs of the conjugate complex numbers with the
same multiplicity. Therefore Theorem 6 implies that if $m$ is an odd number
and (2) holds, then the spectrum of $L(Q)$ contains the main part of
$[0,\infty)$ (see Corollary 1). In Theorem 7 we find a condition on the real
eigenvalues of $A$ for which the spectrum of $L(Q)$ contains all half line
$[H,\infty)$ for some $H$.

Finally, note that in the papers (see [4, 9, 11]) we investigated the non-
self-adjoint Schrödinger operator with a periodic matrix potential. However,
as far as I know, this paper is the first paper about the Schrödinger operator
with a PT-symmetric periodic matrix potential. In this paper, we investigate
the influence of PT-symmetry on spectrum of the operator $L(Q)$.

##  2 On the Bloch eigenvalues and Spectrum

First let us note that in the case $m=1,$ it is well-known that if $\lambda\in
L_{t}(Q),$ then $\overline{\lambda}\in L_{t}(Q)$ (see [3, 8, 12]). Taking into
account that this obvious fact was not formulated for arbitrary $m$, we prove
the following more general statement.

###### Theorem 1

If $\lambda$ is an eigenvalue of multiplicity $v$ of the operator $L_{t}(Q),$
then $\overline{\lambda}$ is also an eigenvalue of the same multiplicity of
$L_{t}(Q).$

Proof. First let us note that $\lambda\in\sigma\left(L_{t}(Q)\right)$ if and
only if there exists a solution $\Psi(\cdot,\lambda)$ of the equation

$-y^{{}^{\prime\prime}}(x)+Q\left(x\right)y(x)=\lambda y(x)$ (8)

satisfying the equality

$\Psi(x+\pi,\lambda)=e^{it}\Psi(x,\lambda)$ (9)

for all $x\in(-\infty,\infty).$ Indeed if
$\lambda\in\sigma\left(L_{t}(Q)\right),$ then there exists a solution
$\Phi(x,\lambda)$ of (8) satisfying (3). Therefore the solution $y(x,\lambda)$
of (8) defined by

$y(x,\lambda)=\Psi(x+\pi,\lambda)-e^{it}\Psi(x,\lambda)$

satisfies the initial conditions $y(0,\lambda)=y^{\prime}(0,\lambda)=0.$ Then
by the uniqueness theorem $y(x,\lambda)=0$ for all $x\in(-\infty,\infty),$
that is, (9) holds.

Now suppose that there exists a solution $\Psi(x,\lambda)$ of (8) satisfying
(9). Then it is clear that $\Psi(x,\lambda)$ satisfies (3). It means that
$\lambda\in\sigma\left(L_{t}(Q)\right).$

Now we are ready to prove the theorem. If $\lambda$ is an eigenvalue of
$L_{t}(Q),$ then there exists a solution $\Psi(x,\lambda)$ of (8) satisfying
(9). Then using (2) one can readily see that the function $\Phi(x)$ defined by
$\Phi(x,\lambda)=\overline{\Psi(-x,\lambda)}$ satisfies the equation

$-y^{{}^{\prime\prime}}(x)+Q(x)y(x)=\overline{\lambda}y(x)$

and the equality

$\Phi(x+\pi)=\overline{\Psi(-x-\pi,\lambda)}=\overline{e^{-it}\Psi(-x,\lambda)}=e^{it}\Phi(x).$

It means that $\overline{\lambda}$ is also an eigenvalue of $L_{t}(Q)$ and
$\Phi$ is the corresponding eigenfunction.

Instead of the equation (8) using the equations

$-\Psi_{j}^{{}^{\prime\prime}}(x,\lambda)+Q\left(x\right)\Psi_{j}(x,\lambda)-\lambda\Psi_{j}(x,\lambda)=\Psi_{j-1}(x,\lambda)$
(10)

for the associated functions
$\Psi_{1}(x,\lambda),\Psi_{2}(x,\lambda),...,\Psi_{s}(x,\lambda),$ where
$j=1,2,...,s$ and $\Psi_{0}(x,\lambda)=\Psi(x,\lambda)$ and repeating the
above argument we conclude that if $y(x,\lambda)$ is an associated function
corresponding to the eigenfunction $\Psi(x,\lambda),$ then
$\overline{y(-x,\lambda)}$ is an associated eigenfunction corresponding to the
eigenfunction $\Phi(x,\lambda).$ Therefore the multiplicities of the
eigenvalues $\lambda$ and $\overline{\lambda}$ are the same. The theorem is
proved.

Now we estimate the Bloch eigenvalues. First we prove the following simple
estimation for the Bloch eigenvalues.

###### Theorem 2

$(a)$ There exists a constant $M$ such that

$\sup_{\lambda\in\sigma(L(Q)),\text{
}x\in[0,\pi]}\left|\Psi_{\lambda}(x)\right|<M,$ (11)

where $\Psi_{\lambda}$ is a normalized Bloch function corresponding to the
eigenvalue $\lambda.$

$(b)$ All eigenvalues of the operator $L_{t}(Q)$ lie on the union of the disks

$D\left((2n+t)^{2},c\right):=\\{\lambda\in\mathbb{C}:\left|\lambda-(2n+t)^{2}\right|<c\\}$
(12)

for $n\in\mathbb{Z},$ where $t\in(-1,1]$, $c=MB$ and

$B=\left(\int\limits_{0}^{\pi}\left(\sum_{i,j=1}^{m}\left|q_{i,j}\left(x\right)\right|^{2}\right)dx\right)^{1/2}.$
(13)

Proof. $(a)$ Let $\Psi_{\lambda(t)}$ be a normalized eigenfunction (Bloch
function) corresponding to the eigenvalue $\lambda(t)$ of the operator
$L_{t}(Q).$ There exists $n$ such that
$(2n-1)^{2}\leq\operatorname{Re}\lambda(t)<(2n+1)^{2}.$ Then it is clear that
there exists a constant $C$ such that

$\sum_{k\in\left(\mathbb{Z}\backslash
B(n)\right)}\frac{1}{\left|\lambda(t)-(2k+t)^{2}\right|}<C$ (14)

for all $t\in(-1,1],$ where $B(n)=\left\\{\pm\left(n-1\right),\pm
n,\pm\left(n+1\right)\right\\}.$ In the decomposition

$\Psi_{\lambda(t)}=\sum_{\begin{subarray}{c}k\in B(n),\\\
s=1,2,...,m\end{subarray}}\left(\Psi_{\lambda(t)},\varphi_{k,s,t}\right)\varphi_{k,s,t}+\sum_{\begin{subarray}{c}k\in\left(\mathbb{Z}\backslash
B(n)\right),\\\
s=1,2,...,m\end{subarray}}\left(\Psi_{\lambda(t)},\varphi_{k,s,t}\right)\varphi_{k,s,t}$

of $\Psi_{\lambda(t)}$ by the orthonormal basis

$\left\\{\varphi_{k,s,t}:k\in\mathbb{Z}\text{ },\text{ }s=1,2,...,m\right\\}$
(15)

replacing $\left(\Psi_{\lambda(t)},\varphi_{k,s,t}\right)$ for
$k\in\left(\mathbb{Z}\backslash B(n)\right)$ by

$\frac{\left(\Psi_{\lambda(t)},Q^{\ast}\varphi_{k,s,t}\right)}{\lambda(t)-(2k+t)^{2}}$

(see (5)) and then using (14), (13) and the obvious inequalities

$\left|\left(\Psi_{\lambda(t)},Q^{\ast}\varphi_{k,s,t}\right)\right|\leq\left\|Q^{\ast}\varphi_{k,s,t}\right\|\leq\frac{1}{\sqrt{\pi}}B$

we obtain the proof of (11).

$(b)$ Suppose that there exists an eigenvalue $\lambda(t)$ of $L_{t}(Q)$ lying
out of $D(\left(2n+t)^{2},c\right)$ for all $n\in\mathbb{Z}$. Then the
inequality

$\left|\lambda(t)-(2n+t)^{2}\right|\geq c$

holds for all $n\in\mathbb{Z}$. Using the Parseval’s equality for the
orthonormal basis (15) and then (5) we get

$1=\sum_{\begin{subarray}{c}n\in\mathbb{Z},\text{ }\\\
s=1,2,...,m\end{subarray}}\left|\left(\Psi_{\lambda(t)},\varphi_{n,s,t}\right)\right|^{2}\leq\sum_{\begin{subarray}{c}n\in\mathbb{Z},\text{
}\\\
s=1,2,...,m\end{subarray}}\frac{\left|\left(Q\Psi_{\lambda(t)},\varphi_{n,s,t}\right)\right|^{2}}{c^{2}}=\frac{\left\|Q\Psi_{\lambda(t)}\right\|^{2}}{c^{2}},$
(16)

where

$\left\|Q\Psi_{\lambda(t)}\right\|^{2}=\int\limits_{0}^{\pi}\left|Q(x)\Psi_{\lambda(t)}(x)\right|^{2}dx.$

On the other hand, it follows from (11) that

$\left|Q(x)\Psi_{\lambda(t)}(x)\right|^{2}<\left(\sum_{i,j=1}^{m}\left|q_{i,j}\left(x\right)\right|^{2}\right)M^{2}.$

Therefore by (13) we have

$\left\|Q\Psi_{\lambda(t)}\right\|^{2}<c^{2}.$ (17)

Using (17) in (16) we get a contradiction $1<1.$ The theorem is proved.

Now to obtain a sharp and uniform with respect to $t\in(-1,1]$ asymptotic
formula we consider the operator $L_{t}(Q)$ as perturbation of $L_{t}(A)$ and
analyze the spectrum of the operators $L_{t}(A),$ where $A$ is defined in (6).
For this we introduce the following notations. Suppose the matrix $A$ has $p$
distinct eigenvalues $\mu_{1},\mu_{2},...,\mu_{p}$ with multiplicities
$m_{1},m_{2},...,m_{p}$ respectively, where $m_{1}+m_{2}+...+m_{p}=m$. Let
$u_{j,1},$ $u_{j,2},...u_{j,s_{j}}$ be the linearly independent eigenvectors
corresponding to the eigenvalue $\mu_{j}.$ Denote by $u_{j,s,1},$
$u_{j,s,2},...u_{j,s,r_{j,s}-1}$ the associated vectors corresponding to the
eigenvector $u_{j,s},$ such that

$\left(A-\mu_{j}I\right)u_{j,s,k}=u_{j,s,k-1}$

for $k=1,2,...,r_{j,s}-1,$ where $u_{j,s,0}=u_{j,s}.$ Note that $r_{j,s}$ is
called the multiplicity of the eigenfunction $u_{j,s}$ and
$r_{j,1}+r_{j,2}+...+r_{j,s_{j}}=m_{j}.$ The number $r_{j}$ defined by
$r_{j}=\max_{s}r_{j,s}$ is a maximum multiplicity of the eigenfunctions
corresponding to the eigenvalue $\mu_{j}.$ It is not hard to see that the
eigenvalues, eigenfunctions and associated functions of $L_{t}(A)$ are

$\mu_{k,j}(t)=\left(2k+t\right)^{2}+\mu_{j},\text{
}\Phi_{k,j,s}(x)=u_{j,s}e^{i\left(2k+t\right)x},\text{
}\Phi_{k,j,s,r}(x)=u_{j,s,r}e^{i\left(2k+t\right)x}$ (18)

respectively, since they satisfy the equation obtained from (10) by replacing
$Q$ to $A.$ Similarly, the eigenvalues, eigenfunctions, and associated
functions of $L_{t}^{\ast}(A)$ are $\overline{\mu_{k,j}},$

$\Phi_{k,j,s}^{\ast}(x)=u_{j,s}^{\ast}e^{i\left(2k+t\right)x}\text{
}\And\Phi_{k,j,s,r}^{\ast}(x)=u_{j,s,r}^{\ast}e^{i\left(2k+t\right)x},$ (19)

where $u_{j,s}^{\ast}=$ $u_{j,s,0}^{\ast}$ and $u_{j,s,r}^{\ast}$ are the
eigenvector and associated vector of $A^{\ast}$ corresponding to
$\overline{\mu_{j}}.$ To obtain the sharp asymptotic formulas we use the
formula

$(\lambda-\mu_{n,j})^{r+1}(\Psi,\Phi_{n,j,s,r}^{\ast})=\sum_{q=0}^{r}(\lambda-\mu_{n,j})^{q}((Q-A)\Psi,\Phi_{n,j,s,q}^{\ast})$
(20)

of [9] (see (15) of [9]), where $\Psi$ is a normalized eigenfunction of
$L_{t}(Q)$ corresponding to the eigenvalue $\lambda$ and estimate the
multiplicands $(\Psi,\Phi_{n,j,s,r}^{\ast})$ and
$((Q-A)\Psi,\Phi_{n,j,s,q}^{\ast})$ in the left-hand side and the right-hand
side of (20). For this we use the following proposition which can be proved by
direct calculations. Note that (20) was obtained from the equality
$L_{t}\left(Q\right)\Psi=\lambda\Psi$ by multiplying both sides by
$\Phi_{n,j,s,r}^{\ast}.$

###### Proposition 1

Let $A(k,t)$ be $\left\\{k\right\\},$ $\left\\{\pm k\right\\},$
$\left\\{k,-k-1\right\\}$ and $\left\\{k,-k+1\right\\}$ respectively if
$t\in\left((-2/3,-1/3)\cup(1/3,2/3)\right),$ $t\in(-1/3,1/3),$ $t\in(2/3,1)$
and $t\in(-1,-2/3).$ If $\left|k\right|>1,$ then

$\left|(2n+t)^{2}-(2k+t)^{2}\right|\geq\frac{4}{3}\left(2\left|k\right|-1\right)$

for all $n\in\left(\mathbb{Z}\backslash A(k,t)\right).$

In the following estimations we use the positive constants denoted by $c_{k}$
for $k=1,2,...$, independent on $t,$ whose exact values are inessential.

###### Lemma 1

Let $\Psi$ be a normalized eigenfunction of $L_{t}(Q)$ corresponding to the
eigenvalue $\lambda$ lying in the disk $D(\left(2k+t)^{2},c\right),$ where

$\left|k\right|>1,\frac{4}{3}\left(2\left|k\right|-1\right)>3c$ (21)

and $c$ is defined in (12). Then there exists $n\in A(k,t)$ such that

$\left|(\Psi,\Phi_{n,j,s,r}^{\ast})\right|\geq c_{1}$ (22)

for some $j,s,r$ and

$\left|((Q-A)\Psi,\Phi_{n,j,s,q}^{\ast})\right|\leq
c_{2}\left(\frac{1}{|k|}+q_{k}\right),$ (23)

for all $j,s,q$, where

$q_{k}=\max\\{\mid q_{i,j,s}\mid:i,j=1,2,...m;\text{ }s\in\left\\{\pm
2k,\pm(2k+1),\pm(2k-1)\right\\},$

$q_{i,j,s}=\frac{1}{\sqrt{\pi}}\int\limits_{0}^{\pi}q_{i,j}\left(x\right)e^{-2isx}dx,$

and $q_{i,j}(x)$ is the entry of the matrix $Q(x)$.

Proof. By Proposition 1 we have

$|\lambda-\left(2n+t\right)^{2}|\geq\frac{4}{3}\left(2\left|k\right|-1\right)-c>2c$
(24)

for all $t\in(-1,1]$ and $n\in\left(\mathbb{Z}\backslash A(k,t)\right)$ if
$\lambda\in D(\left(2k+t)^{2},c\right)$ and $k$ satisfies conditions (21).
Therefore using (5) Bessel inequality and (17) we obtain

$\sum\limits_{\begin{subarray}{c}n\in\left(\mathbb{Z}\backslash
A(k,t)\right)\\\
s=1,2,...,m\end{subarray}}\left|\left(\Psi,\varphi_{n,s,t}\right)\right|^{2}=\sum\limits_{\begin{subarray}{c}n\in\left(\mathbb{Z}\backslash
A(k,t)\right)\\\ s=1,2,...,m\end{subarray}}\frac{\left|\left(\Psi
Q,\varphi_{n,s,t}\right)\right|}{\left|\lambda-\left(2\pi
n+t\right)^{2}\right|}\leq\frac{\left\|\Psi
Q\right\|^{2}}{4c^{2}}\leq\frac{1}{4}.$

It with the Parsaval’s equality implies that there exist $n\in A(k,t)$ and
$i\in\left\\{1,2,...,m\right\\}$ such that

$\left|\left(\Psi,\varphi_{n,i,t}\right)\right|>c_{3}.$

Since the system of the root vectors of the matrix $A^{\ast}$ is a basis of
$\mathbb{C}^{m}$ and (19) holds, the last inequality implies that (22) holds
for some $j,s,r$.

Now to prove (23) we estimate the term $((Q-A)\Psi,\varphi_{n,i,t})$ and take
(19) into account. Using the decomposition

$\Psi=\sum\limits_{v\in\mathbb{Z},\text{
}s=1,2,...,m}\left(\Psi,\varphi_{v,s,t}\right)\varphi_{v,s,t}$

of $\Psi$ by the orthonormal basis (15) we obtain.

$\left((Q-A)\Psi,\varphi_{n,i,t}\right)=\sum\limits_{\begin{subarray}{c}v\in\left(\mathbb{Z}\backslash\left\\{n\right\\}\right),\\\
\text{
}s=1,2,...,m\end{subarray}}q_{i,s,n-v}\left(\Psi,\varphi_{v,s,t}\right).$ (25)

The right-hand side of (25) is the sum of

$S_{1}=:\sum_{v\in A(k,t)\backslash\left\\{n\right\\};\text{
}s=1,2,...,m}q_{i,s,n-v}\left(\Psi,\varphi_{v,s,t}\right)$

and

$S_{2}=:\sum\limits_{v\in\left(\mathbb{Z}\backslash A(k,t)\right);\text{
}s=1,2,...,m}q_{i,s,n-v}\left(\Psi,\varphi_{v,s,t}\right).$

Here $\Psi$ and $\varphi_{v,s,t}$ are the normalized eigenfunctions and the
set $A(k,t)\backslash\left\\{n\right\\}$ consist of at most one number
Moreover, it follows from the definition of $A(k,t)$ that

$(n-v)\in\left\\{\pm 2k,\pm(2k+1),\pm(2k-1)\right\\}$

for all $n\in A(k,t)$ and
$v\in\left(A(k,t)\backslash\left\\{n\right\\}\right).$ Therefore, using the
definition of $q_{k}$ we obtain

$\left|S_{1}\right|\leq mq_{k}.$ (26)

Now let us estimate $S_{2}.$ By (5) and (24) we have

$\left|S_{2}\right|\leq
c_{4}\frac{1}{|k|}\sum\limits_{\begin{subarray}{c}v\in\left(\mathbb{Z}\backslash
A(k,t)\right),\\\
i=1,2,...,m\end{subarray}}\left|q_{s,i,n-v}\right|\left|\left(Q\Psi,\varphi_{v,s,t}\right)\right|.$

Now using Schwards inequality of the space $l_{2}$ and taking (17) into
account we obtain

$\left|S_{2}\right|\leq c_{5}\frac{1}{|k|}.$ (27)

Thus (23) follows from (26), (27) and (19).

Now we are ready to consider the Bloch eigenvalues in detail. The following
theorem follows from (20) and Lemma 1.

###### Theorem 3

If (21) holds, then the eigenvalues of the operator $L_{t}(Q)$ lying in
$D(\left(2k+t)^{2},c\right)$ are contained in $\varepsilon_{k}$ neighborhood
$D\left(\mu_{n,j},\varepsilon_{k}\right)$ of the eigenvalues $\mu_{n,j}(t)$ of
$L_{t}(A)$ for $j=1,2,...,m$ and $n\in A(k,t),$ where $\varepsilon_{k}\leq
c_{6}(\mid\frac{1}{k}\mid+q_{k})^{1/r_{j}}.$

Proof. Dividing (20) by $(\Psi,\Phi_{n,j,s,r}^{\ast}),$ and using (22) and
(23) we obtain

$(\lambda-\mu_{n,i}(t))^{r+1}=\sum_{q=0}^{r}(\lambda-\mu_{n,i}(t))^{q}O\left(\frac{1}{|k|}+q_{k}\right),$

where $r+1\leq r_{i}$, from which we obtain the proof of the theorem.

Now using these results we classify the spectrum of $L(Q)$. Theorems 2 and 3
imply that the spectrum of $L(Q)$ is asymptotically close to the spectrum of
the operator $L(A),$ since $\varepsilon_{k}\rightarrow 0$ as
$k\rightarrow\infty.$ It is clear that if the Fourier coefficients $q_{i,j,k}$
of the entries $q_{i,j}$ of $Q$ is $O(1/k),$ for example if the entries are
the continuous functions, then $\varepsilon_{k}=O(1/k).$ The following theorem
immediately follows from Theorems 2 and 3.

###### Theorem 4

If the matrix $A$ has no real eigenvalues, then the real component of the
spectrum of $L(Q)$ is contained in a finite interval $[a,b].$

Proof. If the eigenvalues $\mu_{1},\mu_{2},...,\mu_{p}$ of the matrix $A$ are
nonreal numbers then there exists $c_{7}$ such that

$\varepsilon_{k}<\min_{j=1,2,...p}\left|\operatorname{Im}\mu_{j}\right|$ (28)

and (21) holds for $\left|k\right|>c_{7}.$ Inequality (28) implies that the
disks $D\left(\mu_{n,j}(t),\varepsilon_{k}\right)$ for $\left|k\right|>c_{7},$
$n\in\mathbb{Z}$ and $j=1,2,...,p$ have no intersection points with the real
lines. Therefore using Theorems 2 and 3 we obtain that the real component of
the spectrum of $L(Q)$ is contained in the bounded set
${\textstyle\bigcup\limits_{\left|k\right|\leq
c_{7}}}D(\left(2k+t)^{2},c\right).$ It implies the proof of the theorem.

Now we consider the cases when the matrix $A$ has the real eigenvalues and
investigate the real component of the spectrum of $L(q)$. Recall that (see the
end of the introduction) the real and nonreal distinct eigenvalues are denoted
respectively by $\mu_{1}<\mu_{2}<...<\mu_{s}$ and$\
\mu_{s+1},\mu_{s+2},...,\mu_{p}$. To investigate the real spectrum of $L(Q)$,
we use Theorems 2 and 3 and find the conditions on $k$ and $t$ such that the
boundary of the closed disk
$\overline{D\left(\mu_{k,j}(t),\varepsilon_{k}\right)}$ for $j\leq s$ belong
to the resolvent set of the operator $L_{t}(Q).$ Since
$\mu_{k,i}(t)=\mu_{-k,i}(-t)$ (see (18)), it is enough to study the disks
$\overline{D\left(\mu_{k,i}(t),\varepsilon_{k}\right)}$ for $k\geq 0$ and
$t\in(-1,1].$ We consider the closed disks
$\overline{D\left(\mu_{n,j}(t),\varepsilon_{k}\right)}$ for $j\leq s$, $n\in
A(k,t)$ and $k\geq N_{1},$ where $N_{1}$ is a positive integer such that if
$k>N_{1},$ then

$\displaystyle\text{ }\delta_{k}$ $\displaystyle<c,$ (29)
$\displaystyle\frac{4}{3}\left(2k-1\right)$
$\displaystyle>3c+\mid\mu_{j}\mid,$ (30) $\displaystyle\delta_{k}$
$\displaystyle<\min_{j=s+1,s+2,...p}\left|\operatorname{Im}\mu_{j}\right|,\text{
}\delta_{k}<\mid\mu_{j}-\mu_{i}\mid$ (31)

for $j\leq s$, $i\leq s$ and $i\neq j$, where
$\delta_{k}=2\max\left\\{\varepsilon_{k},\varepsilon_{-k},\varepsilon_{-k-1},\varepsilon_{-k+1}\right\\}$

###### Lemma 2

Suppose that $k>N_{1}$, $1\leq j\leq s$ and $t\in(-1,1].$ Then

$(a)$ The closed disk $\overline{D\left(\mu_{k,j}(t),\varepsilon_{k}\right)}$
has no common points with the disks $\overline{D(\left(2n+t)^{2},c\right)}$
and $\overline{D\left(\mu_{k,i}(t),\varepsilon_{k}\right)}\ $for $n\notin
A(k,t)$ and $i\neq j$.

$(b)$ The disk $\overline{D\left(\mu_{k,j}(t),\varepsilon_{k}\right)}$ has no
common points with the disks
$\overline{D\left(\mu_{n,i}(t),\varepsilon_{n}\right)}$ for $n\in A(k,t)$ and
$s+1\leq i\leq p$ for all $t\in(-1,1].$ The disk
$\overline{D\left(\mu_{k,j}(t),\varepsilon_{k}\right)}$ has no common points
with the disks $\overline{D\left(\mu_{n,i}(t),\varepsilon_{n}\right)}$ for
$n\in A(k,t)$ and $1\leq i\leq s$ if

$t\in\left((-1,1]\backslash U(j,k,\delta_{k})\right),$ (32)

where $U(j,k,\delta_{k})=U(j,k,i,-k,\delta_{k})\cup
U(j,k,i,-k-1,\delta_{k})\cup U(j,k,i,-k+1,\delta_{k}),$

$U(j,k,i,-k,\delta_{k})=\left(\frac{\mu_{i}-\mu_{j}-\delta_{k}}{8k},\frac{\mu_{i}-\mu_{j}+\delta_{k}}{8k}\right),$
$U(j,k,i,-k-1,\delta_{k})=\left(1+\frac{\mu_{i}-\mu_{j}-\delta_{k}}{4(2k+1)},1+\frac{\mu_{i}-\mu_{j}+\delta_{k}}{4(2k+1)}\right)$

and

$U(j,k,i,-k+1,\delta_{k})=\left(-1+\frac{\mu_{i}-\mu_{j}-\delta_{k}}{4(2k+1)},-1+\frac{\mu_{i}-\mu_{j}+\delta_{k}}{4(2k+1)}\right).$

Proof. $(a)$ Using Proposition 1 and (30) one can easily verify that the
distance between the centres of the disks
$\overline{D\left(\mu_{k,j}(t),\varepsilon_{k}\right)}$ and
$\overline{D(\left(2n+t)^{2},c\right)}$ for $n\notin A(k,t)$ is greater than
$3c.$ On the other hand, by (29) the total sum of the radii of these disks is
less than $2c.$ Therefore these disks have no common points.

It follows from (31) that the distance $\mid\mu_{j}-\mu_{i}\mid$ between the
centres of the disks $\overline{D\left(\mu_{k,j}(t),\varepsilon_{k}\right)}$
$\overline{D\left(\mu_{k,i}(t),\varepsilon_{k}\right)}$ is greater than the
total sum of their radii. That is why they also have no common point.

$(b)$ The proof of the first statement follows from the first inequality of
(31). Now we prove the second statement. By the definition of $A(k,t)$ it is
enough to prove that the disks
$\overline{D\left(\mu_{k,j}(t),\varepsilon_{k}\right)}$ and
$\overline{D\left(\mu_{n,i}(t),\varepsilon_{n}\right)}$ have no common points
for $n=-k,$ $n=-k-1$ and $n=-k+1$ respectively if $t\in\left((-1,1]\backslash
U(j,k,i,-k,\delta_{k})\right)$, $t\in\left((-1,1]\backslash
U(j,k,i,-k-1,\delta_{k})\right)$ and $t\in\left((-1,1]\backslash
U(j,k,i,-k+1,\delta_{k})\right),$ where $i=1,2,...,s.$ This can be easily
verified by the direct calculations of the differences
$\mu_{k,j}(t)-\mu_{-k,i}(t)$, $\mu_{k,j}(t)-\mu_{-k-1,i}(t)$ and
$\mu_{k,j}(t)-\mu_{-k+1,i}(t)$ and using the definitions of the sets

$U(j,k,i,-k,\delta_{k}),$ $U(j,k,i,-k-1,\delta_{k})$ and
$U(j,k,i,-k+1,\delta_{k}).$ The lemma is proved.

Now using this lemma we prove the following theorem which plays a crucial role
in the investigation of the real component of the spectrum of $L(Q).$

###### Theorem 5

Suppose that $k>N_{1}$, $j\leq s$ and (32) hold.

$(a)$ If $\mu_{j}$ is an eigenvalue of $A$ of multiplicity $v$ then the number
of the eigenvalues (counting multiplicity) of $L_{t}\left(Q\right)$ lying in
$D\left(\mu_{k,j}(t),\varepsilon_{k}\right)$ is $v$. Moreover, if $\lambda$ is
an eigenvalue of multiplicity $v$ of the operator $L_{t}(Q)$ lying in
$D\left(\mu_{k,j},\varepsilon_{k}\right),$ then $\overline{\lambda}$ is also
an eigenvalue of the same multiplicity of $L_{t}(Q)$ lying in
$D\left(\overline{\mu_{k,j}},\varepsilon_{k}\right).$

$(b)$ If $\mu_{j}$ is a simple eigenvalue of $A$, then the eigenvalue of
$L_{t}\left(Q\right)$ lying in $U\left(\mu_{k,j},\varepsilon_{k}\right)$ is a
simple and real eigenvalue.

Proof. $(a)$ First we prove that the boundary of
$D\left(\mu_{k,j},\varepsilon_{k}\right)$ lies in the resolvent set of
$L_{t}\left(Q\right).$ By Theorem 2 the eigenvalues of $L_{t}\left(Q\right)$
lie in the disks $D(\left(2n+t)^{2},c\right)$ for $n\in\mathbb{Z}.$ By Lemma
2$(a)$ the eigenvalues lying $D(\left(2n+t)^{2},c\right)$ for
$n\in\left(\mathbb{Z}\backslash A(k,t)\right)$ do not lie in the boundary of
$D\left(\mu_{k,j},\varepsilon_{k}\right).$ It remains to prove the following
statement. The eigenvalues lying $D(\left(2n+t)^{2},c\right)$ for $n\in
A(k,t)$ do not lie in the boundary of
$D\left(\mu_{k,j},\varepsilon_{k}\right).$ To prove this statement we consider
the following three cases separately:
$t\in\left((-2/3,-1/3)\cup(1/3,2/3)\right),$ $t\in(-1/3,1/3)$, $t\in(2/3,1]$
and $t\in(-1,-2/3).$

If $t\in\left((-2/3,-1/3)\cup(1/3,2/3)\right)$ then
$A(k,t)=\left\\{k\right\\}$ (see definition of $A(k,t)$ in Proposition1).
Therefore the proof of the statement follows from Lemma 2$(a).$

If $t\in(-1/3,1/3),$ then $A(k,t)=\left\\{\pm k\right\\}.$ The case $n=k$
follows from Lemma 2$(a).$ Let us consider the case $n=-k.$ By Theorem 3 the
eigenvalues of $L_{t}\left(Q\right)$ lying in the disks
$D(\left(-2k+t)^{2},c\right)$ are contained in
$D\left(\mu_{n,j},\varepsilon_{-k}\right)$ for $j=1,2,...,p$ and $n\in
A(-k,t).$ Since $A(-k,t)=\left\\{\pm k\right\\}$ we need to consider the disks
$D\left(\mu_{k,j},\varepsilon_{-k}\right)$ and
$D\left(\mu_{k,j},\varepsilon_{-k}\right).$ The proof of the statement for
$D\left(\mu_{k,j},\varepsilon_{-k}\right)$ and
$D\left(\mu_{-k,j},\varepsilon_{-k}\right)$ follows from Lemma 2$(a)$ and
Lemma 2$(b)$ respectively.

If $t\in(2/3,1]$ and $t\in(-1,-2/3)$ then $A(k,t)=\left\\{k,-k-1\right\\}$ and
$A(k,t)=\left\\{k,-k+1\right\\}$ respectively. Moreover
$A(-k-1,t)=\left\\{-k-1,k\right\\}$ and $A(-k+1,t)=\left\\{-k+1,k\right\\}.$
Therefore repeating the proof of the case $t\in(-1/3,1/3)$ we get the proof of
this statement for the cases $t\in(2/3,1]$ and $t\in(-1,-2/3).$

Thus the circle
$\left\\{\lambda\in\mathbb{C}:\left|\lambda-\mu_{k,j}\right|=\varepsilon_{k}\right\\}$
lies in the resolvent set of $L_{t}\left(Q\right).$ Consider the following
family of operators

$L_{\varepsilon}=L_{t}(A)+\varepsilon(Q-A),\text{ }0\leq\varepsilon\leq 1.$

Repeating the proof of the case $\varepsilon=1,$ one can easily verify that
the circle
$\left\\{\lambda\in\mathbb{C}:\left|\lambda-\mu_{k,j}\right|=\varepsilon_{k}\right\\}$
lies in the resolvent set of $L_{\varepsilon}$ for $\varepsilon\in[0,1].$
Therefore, taking into account that the family $L_{\varepsilon}$ is
halomorphic (in the sense of [2]) with respect to $\varepsilon,$ we obtain
that the number of the eigenvalues of $L_{\varepsilon}$ lying inside of
$\left\\{\lambda\in\mathbb{C}:\left|\lambda-\mu_{k,j}\right|=\varepsilon_{k}\right\\}$
are the same for all $\varepsilon\in[0,1]$. Since the operator
$L_{0}=L_{t}(A)$ has $v$ eigenvalues (counting the multiplicity) inside
$\left\\{\lambda\in\mathbb{C}:\left|\lambda-\mu_{k,j}\right|=\varepsilon_{k}\right\\}$
the operator $L_{t}(Q)$ has also $v$ eigenvalues inside of this circle. Since
$\mu_{j}$ for $j\leq s$ is a real number, if$\ \lambda\in
D\left(\mu_{k,j},\varepsilon_{k}\right)$ then $\overline{\lambda}$ $\in
D\left(\mu_{k,j},\varepsilon_{k}\right)$ too. Therefore, the last statement of
$(a)$ follows from Theorem 1.

$(b)$ It follows from $(a)$ that if $\mu_{j}$ is a simple eigenvalue of $A$,
then the operator $L_{t}(Q)$ has a unique eigenvalue (counting multiplicity)
in the disk $D(\mu_{k,1}(t),\varepsilon_{k}).$ It means that
$\lambda_{k,j}(t)$ is a simple eigenvalue. Moreover, if $\mu_{j}$ is a real
number and $\lambda_{k,j}(t)$ is a nonreal eigenvalue of $L_{t}\left(Q\right)$
lying in $D\left(\mu_{k,j}(t),\varepsilon_{k}\right)$ then
$\overline{\lambda_{k,j}(t)}$ is also an eigenvalue of $L_{t}\left(Q\right)$
lying in $D\left(\mu_{k,j}(t),\varepsilon_{k}\right),$ which contradicts to
the above uniqueness. Note also that if $\mu_{j}$ is a simple eigenvalue of
$A,$ then $r_{j}=1$ and hence $\varepsilon_{k}\leq
c_{6}(\mid\frac{1}{k}\mid+q_{k})$ (see Theorem 3). The theorem is proved.

Now using Theorem 5 we investigate the real component of the spectrum of
$L(Q).$ This investigation is based on the following idea. First we consider
the set

$\left\\{\mu_{k,j}(t):t\in\left((-1,1]\backslash
U(j,k,\delta_{k})\right)\right\\},$ (33)

where $U(j,k,\delta_{k})$ is defined in (32). By the definition of
$\mu_{k,j}(t)$ the set

$\left\\{\mu_{k,j}(t):t\in(-1,1]\right\\}$ is the interval
$(\mu_{j}+(2k-1)^{2},\mu_{j}+(2k+1)^{2}].$ The union
$\left\\{\mu_{k,j}(t):k>N_{1},t\in(-1,1]\right\\}$ of these interval for
$k>N_{1}$ is a half line $(\mu_{j}+(2N_{1}+1)^{2},\infty).$ Since the set
$U(j,k,\delta_{k})$ is the union of the small intervals for large $k,$ the
union of the sets (33) for $k>N_{1}$ contains the main part of the half line
$(\mu_{j}+(2N_{1}+1)^{2},\infty)$. Note that $U(j,k,\delta_{k})$ consist of
the intervals defined in Lemma 2$(b).$ Since $\delta_{k}\rightarrow 0$ as
$k\rightarrow\infty,$ there exists $N(j)>N_{1}$ such that if $k\geq N(j)$,
then these intervals are pairwise disjoint. Thus we have

$(-1,1]\backslash
U(j,k,\delta_{k})=[a_{1},b_{1}]\cup[a_{2},b_{2}]\cup...\cup[a_{l},b_{l}],$
(34)

for $k\geq N(j),$ where $a_{1}<b_{1}<a_{2}<b_{2}<...$ and the sum of the
length of these intervals is asymptotically close to the length of $(-1,1].$
Namely

$2-{\textstyle\sum\limits_{i=1,2,...l}}\left(b_{i}-a_{i}\right)<c_{9}\frac{\delta_{k}}{k}.$
(35)

Note that if $a_{1}=-1,$ then in (34) the interval $[a_{1},b_{1}]$ should be
replaced by $(a_{1},b_{1}].$ This replacement does not change the
investigation. Therefore, without loss of generality, we assume that (34)
holds. Since $\mu_{k,j}$ is an increasing function on $(-1,1],$ the image
$\mu_{k,j}([a_{i},b_{i}])$ of the interval $[a_{i},b_{i}]$ is the interval
$[\mu_{k,j}(a_{i}),\mu_{k,j}(b_{i})].$ Therefore

$\mu_{k,j}\left((-1,1]\backslash
U(j,k,\delta_{k})\right)={\textstyle\bigcup\nolimits_{i=1,2,...,l}}[\mu_{k,j}(a_{i}),\mu_{k,j}(b_{i})]$
(36)

for $k\geq N(j).$ Using (35) and definition of the function $\mu_{k,j}$ we see
that the length of the intervals in (36) is asymptotically close to the length
of $(\mu_{j}+(2k-1)^{2},\mu_{j}+(2k+1)^{2}]$ in the sense that

$8k-{\textstyle\sum_{i=1,2,...l}}\left(\mu_{k,j}\left(b_{i})-\mu_{k,j}(a_{i}\right)\right)<c_{10}\delta_{k}\rightarrow
0$ (37)

as $k\rightarrow\infty.$ Thus the set ${\textstyle\bigcup\limits_{k\geq
N(j)}}\mu_{k,j}\left((-1,1]\backslash U(j,k,\delta_{k})\right)$ contains the
main part of the half line $(\mu_{j}+(2N(j)-1)^{2},\infty).$

Now, using Theorem 5 and (37) we prove that the spectrum of $L(Q)$ contains
the main part of the half line $[0,\infty)$. First let us explain it in the
simplest case, when $\mu_{j}$ is a simple eigenvalue of $A.$ Then by Theorem
5$(b)$ the eigenvalue $\lambda_{k,j}(t)$ of $L_{t}\left(Q\right)$ lying in
$U\left(\mu_{k,j},\varepsilon_{k}\right)$ is a simple and real eigenvalue.
Moreover, the simplicity of the eigenvalue $\lambda_{k,j}(t)$ implies that it
continuously depend on $t.$ As a result the set

$\left\\{\lambda_{k,j}(t):t\in\left((-1,1]\backslash
U(j,k,\delta_{k})\right)\right\\}$ (38)

contains the main part of the set (33). Therefore the union of the sets (38)
for $k\geq N(j)$ contain the main part of the half line
$(\mu_{j}+(2N(j)-1)^{2},\infty)$.

Now let us discuss it in more general case when $\mu_{j}$ is a real eigenvalue
of odd multiplicity $v$. Then by Theorem 5$(a)$ the operator $L_{t}(Q)$ has
$v$ eigenvalues in the neighborhood of $\mu_{k,j}(t).$ All these eigenvalues
can not be nonreal, since they are pairwise conjugate number and $v$ is an odd
number. Thus the operator $L_{t}(Q)$ has a real eigenvalue. In the following
theorem we prove that these real eigenvalues fill the main part of the half
line $[0,\infty).$

###### Theorem 6

Suppose that the matrix $A$ has a real eigenvalue $\mu_{j}$ of odd
multiplicity.

$(a)$ Then the real component of the spectrum of $L(Q)$ contains the
subintervals

$\Gamma_{k,j,i}=[\mu_{k,j}(a_{i})+\delta_{k},\mu_{k,j}(b_{i})-\delta_{k}]$
(39)

for $i=1,2,...l$ of the interval
$\Gamma_{k,j}=(\mu_{j}+(2k-1)^{2},\mu_{j}+(2k+1)^{2}]$, where $k\geq N(j).$
These subintervals are pairwise disjoint and satisfy the inequality

$\mu(\Gamma_{k,j})-\sum_{i=1,2,...l}\mu(\Gamma_{k,j,i})<c_{11}\delta_{k},$
(40)

where $\mu(E)$ denotes the measure of the set $E$ and
$\delta_{k}=O(\mid\frac{1}{k}\mid+q_{k})^{1/r_{j}}$

$(b)$ The spectrum of $L(Q)$ contains the main part of $[0,\infty)$ in the
sense that

$\lim_{n\rightarrow\infty}\frac{\mu([0,n]\backslash\sigma(L(Q)))}{\mu(\sigma(L(Q))\cap[0,n])}=0.$
(41)

Proof. Let $\mu_{j}$ be a real eigenvalue of multiplicity $v$ of the matrix
$A,$ where $v$ is an odd number. Then by Theorem 5 the disk
$D\left(\mu_{k,j}(t),\delta_{k}\right)$ for $t\in[a_{i},b_{i}]$ and
$i=1,2,...l$ contains $v$ eigenvalues (counting the multiplicity) of
$L_{t}(Q),$ where $[a_{i},b_{i}]$ is defined in (34). Let us denote they by
$\lambda_{k,1}(t),\lambda_{k,2}(t),...,\lambda_{k,v}(t).$ Consider the
unordered $v$-tuple
$\left\\{\lambda_{k,1}(t),\lambda_{k,2}(t),...,\lambda_{k,v}(t)\right\\}.$ It
follows from (4) that this unordered $v$-tuple depend continuously (in sense
of Theorem 5.2 of [2]) on the parameter $t\in[a_{i},b_{i}].$ Then by Theorem
5.2 of [2] there exist $p$ single-valued continuous functions
$\lambda_{1}(t),\lambda_{2}(t),...,\lambda_{v}(t)$ the value of which
constitute the $v$-tuple
$\left\\{\lambda_{k,1}(t),\lambda_{k,2}(t),...,\lambda_{k,v}(t)\right\\}$ for
$t\in[a_{i},b_{i}].$

Now we prove that

$[\mu_{k,j}(a_{i})+\delta_{k},\mu_{k,j}(b_{i})-\delta_{k}]\subset\left(\cup_{s=1}^{v}\gamma_{s}\right)\subset\sigma(L(Q)),$
(42)

where $\gamma_{s}$ is the curve
$\left\\{\lambda_{s}(t):t\in[a_{i},b_{i}]\right\\}.$ Assume the converse. Then
there exists

$a\in[\mu_{k,j}(a_{i})+\delta_{k},\mu_{k,j}(b_{i})-\delta_{k}]\backslash\left(\cup_{s=1}^{v}\gamma_{s}\right).$

It means that the continuous curves $\gamma_{s}$ extend from
$\lambda_{s}(a_{i})\in U\left(\mu_{k,j}(a_{i}),\delta_{k}\right)$ to
$\lambda_{s}(b_{i})\in U\left(\mu_{k,j}(b_{i}),\delta_{k}\right)$ pas above or
below of the point $a,$ since $\operatorname{Re}\lambda_{s}(a_{i})\
<a<\operatorname{Re}\lambda_{s}(b_{i}).$ Moreover by Theorem 5$(a)$ if
$\gamma_{s}$ passes above of $a$ then there exist
$j\in\left\\{1,2,...,v\right\\}$ such that $j\neq s$ and $\gamma_{j}$ passes
below of$\ a.$ It implies that the number $v$ of the curves $\gamma_{s}$ is an
even number. It contradicts to the assumption that $v$ is an odd number. Thus
(42) is proved and the subintervals $\Gamma_{k,j,i}$ defined in (39) are the
subsets of the spectrum of $L(Q)$. Estimation (40) follows from (37). Note
that $N(j)$ can be chosen so that
$\mu_{k,j}\left(b_{i})-\mu_{k,j}(a_{i}\right)-2\delta_{k}>0$ for $k\geq N(j).$

The proof of (41) follows from $(a).$

###### Corollary 1

If $m$ is an odd number, then the real component of the spectrum of $L(Q)$
contains the main part of $[0,\infty)$ and (41) holds.

Now we find a condition on the eigenvalues of the matrix $A$ for which the the
real component of the spectrum of $L(Q)$ contains a half line $[H,\infty)$ for
some $H.$

###### Theorem 7

If the matrix $A$ has at least three real eigenvalues $\mu_{j_{1}},$
$\mu_{j_{2}}$ and $\mu_{j_{3}}$ of odd multiplicities such that

$\min_{i_{1},i_{2},i_{3}}\left(diam(\\{\mu_{j_{1}}+\mu_{i_{1}},\mu_{j_{2}}+\mu_{i_{2}},\mu_{j_{3}}+\mu_{i_{3}}\\})\right)=d\neq
0,$ (43)

where $i_{k}=1,2,...,s$ for $k=1,2,3$ and

$diam(E)=\sup_{x,y\in E}\mid x-y\mid,$

then there exists a number $H$ such that $[H,\infty)\in\sigma(L(Q))$.

Proof. By Theorem 6$(a)$ we have

$\sigma(L(Q))\supset\left(S(j_{1},N(j_{1}))\cup S(j_{2},N(j_{2}))\cup
S(j_{3},N(j_{3}))\right),$ (44)

where

$S(j,N)={\textstyle\bigcup\limits_{i=1,2,...l;\text{ }k\geq
N}}[\mu_{k,j}(a_{i})+\delta_{k},\mu_{k,j}(b_{i})-\delta_{k}]$ (45)

Using the definition of $U(j,k,\delta_{k})$ one can easily verify that there
exists $\alpha_{k}$ such that $\alpha_{k}\rightarrow 0$ as
$k\rightarrow\infty$ and

$\mu_{k,j}(U(j,k,\delta_{k}))\subset{\textstyle\bigcup\limits_{n=-1,0,1;\text{
}i=1,2,...,s}}C(k,j,i,\alpha_{k},n),$ (46)

where

$C(k,j,i,a,n)=\\{x\in\mathbb{R}:\mid
x-((2k+n))^{2}-\frac{\mu_{i}+\mu_{j}}{2}\mid<a\\}.$ (47)

Then we have

${\textstyle\bigcup\limits_{k\geq
N(j_{p})}}\\{\mu_{k,j_{p}}(t):t\in(-1,1]\backslash
U(j_{p},k,\alpha_{k})\\}\supset(h,\infty)\backslash{\textstyle\bigcup\limits_{\begin{subarray}{c}n=-1,0,1;\\\
k\geq N\\\ i=1,2,...,s\end{subarray}}}C(k,j_{p},i,\alpha_{k},n),$

where $h=(2N(j_{p})-1)^{2}+\mu_{j_{p}}$ and $p=1,2,3.$ Therefore using
(44)-(47) we obtain that there exist $H>h$ and $\beta_{k}$ such that
$\beta_{k}\rightarrow 0$ as $k\rightarrow\infty$ and

$\sigma(L(q))\supset(H,\infty)\backslash{\textstyle\bigcup\limits_{n=-1,0,1;\text{
}k\geq N;\text{ }i=1,2,...,s}}C(k,j_{p},i,\beta_{k},n)$ (48)

for all $p=1,2,3,$ where
$N\geq\max\left\\{N(j_{1}),N(j_{2}),N(j_{3})\right\\}.$ Now we prove that

$[H,\infty)\subset\sigma(L(q)).$ (49)

By (48) to prove (49) it is enough to show that the set

${\textstyle\bigcap\limits_{p=1,2,3}}\left({\textstyle\bigcup\limits_{n=-1,0,1;\text{
}k\geq N;\text{ }i=1,2,...,s}}C(k,j_{p},i,\beta_{k},n)\right)$ (50)

is empty. Since $\beta_{k}\rightarrow 0$ the number $N$ can be chosen so that
$4\beta_{k}<d$ for $k\geq N$. If the set (50) contains an element $x,$ then

$x\in{\textstyle\bigcup\limits_{n=-1,0,1;\text{ }k\geq N;\text{
}i=1,2,...,s}}C(k,j_{p},i,\beta_{k},n)$

for all $p=1,2,3.$ Using this and the definition of $C(k,j_{p},i,a,n)$ (see
(47)), we obtain that there exist $k\geq N;$ $n\in\left\\{-1,0,1\right\\}$ and
$i_{p}\in\left\\{1,2,...,s\right\\}$ such that

$\mid x-(\pi(2k+n))^{2}-\frac{\mu_{j_{p}}+\mu_{i_{p}}}{2}\mid<\beta_{k}$

for all $p=1,2,3$. This implies that

$\left|\left(\mu_{j_{q}}+\mu_{i_{q}}\right)-\left(\mu_{j_{p}}+\mu_{i_{p}}\right)\right|<4\beta_{k}<d$

for all $p,q=1,2,3$, where $p\neq q.$ It contradicts the condition (43).
Theorem is proved.

Note that in paper [10] we proved Theorem 7 for the self-adjoint operator
$L(Q)$ under additional condition that $\mu_{j_{1}},$ $\mu_{j_{2}}$ and
$\mu_{j_{3}}$ are the simple eigenvalues. In this paper we prove it for the
non-self-adjoint operator $L(Q)$ and without the simplicity condition.

##  3 Data Availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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