# Preserving the Sun from the Cold
by a Perfectly Reflecting Dyson Sphere

Don N. Page
Department of Physics
4-183 CCIS
University of Alberta
Edmonton, Alberta T6G 2E1
Canada Internet address<EMAIL_ADDRESS>
(2022 July 21)

###### Abstract

Some entities, such as humans, survive longest if their environment is neither
too hot nor too cold, and the sun is no exception. It is rather obvious that
if the sun were enclosed inside a thermally conducting sphere surrounded by a
heat bath kept much hotter than the present central temperature of the sun,
its nuclear burning would occur faster, so that the sun would last for a
shorter time. It is less obvious that if the sun were surrounded by a
perfectly reflecting sphere to prevent its radiation from escaping to cold
empty space, it could actually last longer. Here I shall show that this is the
case for such a sphere at least somewhat larger than the present solar radius.
This naïvely paradoxical result is a consequence of the negative specific heat
of many gravitating systems, so as the energy emitted by the sun is reflected
back to increase the thermal energy, the sun expands and its central
temperature goes down rather than up and reduces the nuclear burning rate, so
that the sun can last much longer than five billion years, for a lifetime
growing roughly exponentially with the cube root of the radius of the
perfectly reflecting sphere.

##  1 Introduction

A ‘Dyson sphere’ [1] is a hypothetical construction by an advanced
civilization around a star to collect far more of the stellar radiation than
that intercepted by the civilization’s planet (an idea attributed by Dyson [2,
3] to Olaf Stapledon [4]; another influence may have been J. D. Bernal [5]).
It has generally been thought of as a way for the civilization to utilize more
of the stellar radiation, without considering the effect on the evolution of
the star (which would generally be negligible in the more nearly realistic
case in which the Dyson sphere intercepts only a small fraction of the stellar
radiation).

Much of the research on Dyson spheres has been on their effect on the
appearance of the system from far away, in order to provide a guide for
potentially observational consequences for us. Very little has been written on
the effects on the central star. The most recent papers that I am aware of on
this are those of Huston and Wright [6, 7], which consider the effects on
central stars from Dyson spheres that reflect just part of the stellar
radiation back to the star. Because of the negative gravitothermal heat
capacity of stars, this influx of energy from the reflected radiation causes
the star to expand and cool, reducing its luminosity, but generally only by
less than a factor of two for the stars and partial reflections considered in
those papers.

However, here I shall abandon all pretext of practicality and consider the
thought experiment of what would happen if one did surround the sun by a
perfectly reflecting spherical shell that did not let out any of the solar
radiation. Naïvely one might expect preventing the radiation from escaping
would cause the sun to heat up and increase the rate of nuclear burning,
reducing its lifetime. Nevertheless, the negative specific heat of
gravitational systems can cause the sun’s central temperature actually to cool
as more solar energy is reflected back in, as well as swelling up the sun to
reduce the central density, so that for any perfectly reflecting shell
sufficiently larger than the present solar radius, the sun can burn slower and
last much longer, all the way up to the timescale of baryon decay for shell
radii comparable to 1 au.

##  2 Solar Parameters

The present sun is sufficiently condensed that its total gravitational energy,
particle thermal energy, and radiation energy (this total being denoted by
$E$, not including the rest mass energy of the particles) is too far negative
for the sun to attain an isothermal configuration inside any perfectly
reflecting sphere larger than the present solar radius $R_{\odot}$ before
nuclear burning increases this energy $E$ sufficiently. However, the time for
the enclosed sun to increase $E$ sufficiently by nuclear burning so that an
isothermal configuration can be achieved is comparable to the Ritter-Kelvin-
Helmholtz timescale [8] of about 16 million years (see below for the
evaluation), which is much shorter than even the present calculated future
lifetime of the sun (about 8 billion years, 5 during the Main Sequence [9]),
so I shall ignore this relatively short time and just estimate the time for
the sun to burn its remaining hydrogen to helium when it is an enclosed
isothermal plasma. By then the temperature will be so hot that the remaining
nuclear burning to iron will take much less time than the time of hydrogen
burning, so I shall also ignore the time to burn helium to iron.

Once the sun has emitted enough energy that it becomes an isothermal ball of
plasma, it will initially have negative specific heat, so that as $E$
increases, the temperature $T$ will decrease, down to a minimum determined by
the mass $M_{\odot}$ of the sun, the radius $R$ of the perfectly reflecting
sphere, and the mean mass $m$ of the particles in the sun. Beyond this point,
as $E$ continues to increase, the temperature will go back up. The regime
around the time of minimum $T$ will be a bottleneck for the nuclear burning,
where the energy generation rate is lowest and most of the lifetime will be
spent. (Since the nuclear burning rate for fixed composition depends not only
strongly on the temperature but also proportionally to the average square of
the density over the configuration, which continually decreases as the energy
$E$ increases and the self-gravitating plasma becomes less centrally
condensed, the minimum nuclear burning rate occurs somewhat after the
temperature reaches its minimum.)

To get the mean mass $m=\rho/n$ of the massive particles in the sun (e.g., not
including the photons, but only the particles with nonrelativistic thermal
energies), where $\rho$ is the mass density and $n$ is the total particle
number density, for simplicity I shall neglect the mass of the electrons,
assume a hydrogen nucleus mass fraction $X=0.70$ (taken all to be protons), a
helium nucleus mass fraction $Y=0.28$ (taken all to be alpha particles), and
mass fraction $Z=0.02$ of ‘metals’ (nuclei heavier than helium; here I shall
take this small fraction to be oxygen-16) [9]. Approximating the mass of each
baryon as the proton mass $m_{p}$ and neglecting the mass of the electrons,
and setting the total number density to be $n=n_{X}+n_{Y}+n_{Z}+n_{e}$ with
free proton number density $n_{X}$, alpha-particle number density $n_{Y}$,
oxygen-16 nucleus number density $n_{Z}$, and electron number density
$n_{e}\approx n_{X}+2n_{Y}+8n_{Z}$ for neutrality (the approximation coming
from assuming that the metal nuclei all are those of oxygen-16, each with 8
protons), one gets $\rho_{X}\approx m_{p}n_{X}$, $\rho_{Y}\approx
4m_{p}n_{Y}$, $\rho_{Z}\approx 16m_{p}n_{Z}$, $\rho_{e}\approx 0$, and hence
$\rho=\rho_{X}+\rho_{Y}+\rho_{Z}+\rho_{e}\approx m_{p}(n_{X}+4n_{Y}+16n_{Z})$
and $n\approx 2n_{X}+3n_{Y}+9n_{Z}$. With $X=\rho_{X}/\rho\approx 0.70$,
$Y=\rho_{Y}/\rho\approx 0.28$, and $Z=\rho_{Z}/\rho\approx 0.02$, this leads
to

$m=\frac{\rho}{n}\approx\frac{m_{p}}{2X+(3/4)Y+(9/16)Z}\approx\frac{m_{p}}{1.62125}\approx
1.03\times 10^{-27}\,\mathrm{kg}.$ (1)

Now if along with this estimate for the mean particle mass $m$, we also use
the solar parameters [10] $M_{\odot}=1.988\,41(4)\times 10^{30}$ kg,
$GM_{\odot}=1.327\,124\,400\,18(9)\times 10^{20}$ m3 s-2, nominal solar
equatorial radius $R_{\odot}=6.957\times 10^{8}$ m, nominal solar luminosity
$L_{\odot}=3.828\times 10^{26}$ W, and the Boltzmann constant $k\equiv
1.380\,649\times 10^{-23}$ J/K, we can calculate various other parameters for
the sun, such as the total number of massive particles in the sun,

$N_{\odot}=\frac{M_{\odot}}{m}\approx 1.93\times 10^{57},$ (2)

the average density of the sun,

$\rho_{\odot}\equiv\frac{3M_{\odot}}{4\pi R_{\odot}^{3}}\approx 1\,410\
\mathrm{kg}/\mathrm{m}^{3},$ (3)

the solar luminosity per mass

$\epsilon_{\odot}\equiv\frac{L_{\odot}}{M_{\odot}}\approx 1.925\times 10^{-4}\
\mathrm{W}/\mathrm{kg}=1.925\times 10^{-4}\ \mathrm{m}^{2}/\mathrm{s}^{3}$ (4)

(which is about 1/2500 of an estimate of my own basal metabolic rate per mass
and about 1/1000 of $c^{2}$ multiplied by the present Hubble expansion rate,
so at its present luminosity the sun would emit about 0.1% of its mass energy
in a Hubble time), a characteristic energy associated with the solar
parameters,

$E_{\odot}\equiv\frac{GM_{\odot}^{2}}{2R_{\odot}}=1.897\times 10^{41}\
\mathrm{J},$ (5)

a characteristic solar temperature,

$T_{\odot}\equiv\frac{E_{\odot}}{kN_{\odot}}\equiv\frac{GM_{\odot}m}{2kR_{\odot}}\approx
7.131\times 10^{6}\ \mathrm{K},$ (6)

and a characteristic solar time,

$t_{\odot}\equiv\frac{E_{\odot}}{L_{\odot}}\equiv\frac{GM_{\odot}^{2}}{2R_{\odot}L_{\odot}}\approx
1.570\times 10^{7}\ \mathrm{years}.$ (7)

This time is essentially the Ritter-Kelvin-Helmholtz timescale [8], the time
for the sun to emit an amount of energy equal to the kinetic energy of its
particles, which would also be an estimate for the future lifetime of the sun
if no more nuclear reactions occurred. The fact that the sun has lasted about
291 times as long, $4.568\times 10^{9}$ years [9, 11], is due to the nuclear
burning of hydrogen into helium, which provides the heat and pressure to hold
the sun up for billions of years against gravitational collapse. If the sun
were placed inside a perfectly reflecting sphere, it would take a time of the
order of $t_{\odot}$ for the energy $E$ inside (not counting the decreasing
rest mass energy of the particles as protons are converted to alpha particles,
releasing the energy that gives the increase in $E$), which starts off
negative because of the dominance of the negative gravitational potential
energy, to become sufficiently close to zero (at a negative value depending on
the radius of the reflecting shell, and on the mass inside, which I shall
always approximate as $M_{\odot}$) that the self-gravitating plasma can become
isothermal. Because $t_{\odot}$ is so much smaller than the nuclear burning
times, I shall ignore the time needed for the sun to evolve to an isothermal
ball of plasma inside the reflecting sphere, though the details of that
evolution could make another interesting research project, perhaps modeled on
the papers of Macy Huston and Jason Wright cited earlier [6, 7].

##  3 Properties of Self-Gravitating Spherically-Symmetric Isothermal Perfect
Fluid Balls

Now let us consider the properties of an self-gravitating isothermal ball of
perfect fluid, such as the plasma of photons, electrons, protons, alpha
particles, and oxygen nuclei that I shall take as an approximation for the
solar composition (with oxygen standing in for all the nuclei heavier than
alpha particles). The ratio of the Schwarzschild radius of the sun to its
present radius is

$\frac{R_{S}}{R_{\odot}}=\frac{2GM_{\odot}}{c^{2}R_{\odot}}=\frac{2\,953.250\,076\,50(20)\
\mathrm{m}}{695\,700\,000\ \mathrm{m}}\approx 4.245\times 10^{-6},$ (8)

which is very small, so we can use the equations of nonrelativistic Newtonian
gravitation, since I shall restrict to reflecting shell radii $R\geq
R_{\odot}$. The energy $E_{r}=(4\pi/3)R^{3}aT^{4}$ of thermal electromagnetic
radiation, with radiation energy density constant $a=\pi^{2}k^{4}/[15(\hbar
c)^{3}]\approx 7.566\times 10^{-16}$ J/(m3K4), can make a significant
contribution to the total energy $E$ inside the shell that also includes the
thermal energy of the nonrelativistic particles and the negative gravitational
potential energy (but not the rest mass energies), so I shall also include
this radiation energy. However, I shall ignore energy losses to neutrinos.

The radial structure of a nonrelativistic self-gravitating spherically
symmetric ball of perfect fluid is given in dimensionless form by the Emden-
Chandrasekhar equation [12, 13]

$\frac{1}{\xi^{2}}\frac{d}{d\xi}\left(\xi^{2}\frac{d\psi}{d\xi}\right)=e^{-\psi},$
(9)

where

$\xi=r/L$ (10)

is a dimensionless radial variable, the physical radius $r$ (which ranges from
0 to the radius $R$ of the perfectly reflecting shell) divided by the length
scale (which should not be confused with the solar luminosity $L_{\odot}$)

$L=\left(\frac{kT}{4\pi Gm\rho_{c}}\right)^{1/2},$ (11)

where $T$ is the (constant in space) temperature, $m$ is the mean particle
mass, and $\rho_{c}=\rho(r=0)$ is the central density of the fluid.
Furthermore,

$\psi(\xi)=\ln{[\rho_{c}/\rho(\xi)]}$ (12)

is a logarithmic dimensionless measure of the inverse density as a function of
the dimensionless radial variable $\xi$, so $\rho=\rho_{c}e^{-\psi}$.

The boundary conditions at the center, $\xi=0$, with no point mass present
there but just a smooth fluid density configuration, are that there $\psi=0$
and $d\psi/d\xi=0$. This leads to a unique solution for $\psi(\xi)$, which one
can readily calculate to have the following series form for small $\xi$:

$\psi(\xi)=\frac{1}{6}\xi^{2}-\frac{1}{120}\xi^{4}+\frac{1}{1\,890}\xi^{6}-\frac{61}{1\,632\,960}\xi^{8}+\frac{629}{224\,532\,000}\xi^{10}+O(\xi^{12}).$
(13)

Series for other useful functions of $\xi$ and $\psi$, using $\psi^{\prime}$
for $d\psi/d\xi$, are

$\frac{\rho}{\rho_{c}}=e^{-\psi}=1-\frac{1}{6}\xi^{2}+\frac{1}{45}\xi^{4}-\frac{61}{22\,680}\xi^{6}+\frac{629}{2\,041\,200}\xi^{8}+O(\xi^{10}),$
(14)
$\frac{\rho_{c}}{\rho}=e^{\psi}=1+\frac{1}{6}\xi^{2}+\frac{1}{180}\xi^{4}-\frac{1}{11\,340}\xi^{6}+\frac{1}{510\,300}\xi^{8}+O(\xi^{10}),$
(15)
$\psi^{\prime}\equiv\frac{d\psi}{d\xi}=\frac{1}{3}\xi-\frac{1}{30}\xi^{3}+\frac{1}{315}\xi^{5}-\frac{61}{204\,120}\xi^{7}+\frac{629}{22\,453\,200}\xi^{9}+O(\xi^{11}),$
(16) $u\equiv\frac{\xi
e^{-\psi}}{\psi^{\prime}}=3-\frac{1}{5}\xi^{2}+\frac{19}{1\,050}\xi^{4}-\frac{118}{70\,875}\xi^{6}+\frac{33\,661}{218\,295\,000}\xi^{8}+O(\xi^{10}),$
(17)
$v\equiv\xi\psi^{\prime}=\frac{1}{3}\xi^{2}-\frac{1}{30}\xi^{4}+\frac{1}{315}\xi^{6}-\frac{61}{204\,120}\xi^{8}+\frac{629}{22\,453\,200}\xi^{10}+O(\xi^{12}),$
(18) $w\equiv
uv\equiv\xi^{2}e^{-\psi}=\xi^{2}-\frac{1}{6}\xi^{4}+\frac{1}{45}\xi^{6}-\frac{61}{22\,680}\xi^{8}+\frac{629}{2\,041\,200}\xi^{10}+O(\xi^{12}),$
(19)
$\frac{\rho_{c}}{\bar{\rho}}=\frac{\xi^{2}}{3v}=1+\frac{1}{10}\xi^{2}+\frac{1}{2\,100}\xi^{4}-\frac{1}{121\,500}\xi^{6}+\frac{83}{327\,442\,500}\xi^{8}+O(\xi^{10}),$
(20)

where $\bar{\rho}$ is the mean density inside $r=L\xi$. At $\xi=1$, the five
terms given for the last series are larger than $\rho_{c}/\bar{\rho}$ at this
$\xi$ by only about one part in a million.

The derivatives of $u$, $v$, and $w$ can be written in terms of these same
quantities as follows:

$\xi\frac{du}{d\xi}=r\frac{du}{dr}=u(3-u-v),$ (21)
$\xi\frac{dv}{d\xi}=r\frac{dv}{dr}=v(u-1),$ (22)
$\xi\frac{dw}{d\xi}=r\frac{dw}{dr}=w(2-v).$ (23)

Therefore, one can eliminate $\xi$ from the Emden-Chandrasekhar Eq. (9) [12,
13] to write it as an autonomous first-order equation between $u$ and $v$, $u$
and $w$, or $v$ and $w$:

$v(u-1)du=u(3-u-v)dv,$ (24) $w(2u-w)du=u(3u-u^{2}-w)dw,$ (25)

or

$w(2-v)dv=(w-v)dw.$ (26)

One can see that as $\xi$ becomes large, $u$ oscillates around 1, whereas $v$
and $w$ oscillate around 2. Indeed, asymptotically

$\frac{\rho}{\rho_{c}}=e^{-\psi}=\frac{2}{\xi^{2}}\left[1+\frac{a}{\sqrt{\xi}}\cos{\left(\sqrt{7/4}\ln{\xi}+b\right)}+O\left(\frac{1}{\xi}\right)\right]$
(27)

with constants $a$ and $b$ that can be determined by numerically integrating
the Emden-Chandrasekhar Eq. (9) [12, 13] but which will not be important to us
here. One can see that, leaving aside the decaying oscillatory term, the
density $\rho$ asymptotically decreases inversely proportional to the square
of the radial distance $r=L\xi$, so the mass inside $r$ asymptotically
increases linearly with $r$ and diverges as $r$ is taken to infinity.
Therefore, an isothermal self-gravitating fluid with no outer boundary does
not have a finite mass, and hence is unphysical (and beyond a certain $\xi$ it
would also be unstable [14, 15, 16, 17]). However, here we are postulating a
perfectly reflecting sphere at some radius $r=R$ to confine the isothermal
plasma and keep the total mass inside at the fixed value of the solar mass
$M_{\odot}$.

If we say that the value of $\xi=r/L$ at the surface (the reflecting sphere of
radius $R$) is $X$ (henceforth not to be confused with the previous use of $X$
as the mass fraction of hydrogen in the solar composition), then, using the
Emden-Chandrasekhar Eq. (9) and $\rho_{c}=kT/(4\pi GmL^{2})$,

$\displaystyle M(X)$ $\displaystyle=$ $\displaystyle\int_{0}^{R}4\pi r^{2}\rho
dr=\int_{0}^{X}4\pi L^{3}\xi^{2}d\xi\rho_{c}e^{-\psi}=4\pi
L^{3}\rho_{c}\int_{0}^{X}\frac{d}{d\xi}\left(\xi^{2}\frac{d\psi}{d\xi}\right)d\xi$
(28) $\displaystyle=$ $\displaystyle 4\pi L^{3}\rho_{c}Xv(X)=\frac{RkTv}{Gm}.$

Therefore, with $v$ the value of $v(\xi)$ at the surface, $\xi=X$, the
temperature of the isothermal plasma fluid ball of solar mass $M_{\odot}$ is

$T=\frac{GM_{\odot}m}{kRv}=\left(\frac{R_{\odot}}{R}\right)\frac{2T_{\odot}}{v}.$
(29)

The Chandrasekhar-Wares [18] tabulated values of $\psi$, $e^{-\psi}$,
$\psi^{\prime}\equiv d\psi/d\xi$, $u\equiv\xi e^{-\psi}/\psi^{\prime}$,
$v\equiv\xi\psi^{\prime}$, $\xi v=\xi^{2}\psi^{\prime}$, and
$\rho_{c}/\bar{\rho}=\xi^{2}/(3v)=\xi/(3\psi^{\prime})$ show that $v$ has a
maximum value near $\xi=9$ of $v_{\mathrm{max}}\approx 2.5176$, so as the
confined sun burns hydrogen to helium and increases $E$, which decreases $X$,
the temperature $T$ initially drops until it reaches a minimum value of

$T_{\mathrm{min}}=\left(\frac{R_{\odot}}{R}\right)\frac{2T_{\odot}}{v_{\mathrm{max}}}\approx(5.665\times
10^{6}\ \mathrm{K})\left(\frac{R_{\odot}}{R}\right).$ (30)

when $X=R/L$, the surface value of $\xi=r/L$, drops to approximately 9. At
this minimum in the temperature, the ratio of the central density to the
average density is

$\frac{\rho_{c}}{\bar{\rho}}=\frac{\xi^{2}}{3v}\approx 10.71$ (31)

Since the average density of a solar mass $M_{\odot}$ in a sphere of radius
$R$ is

$\bar{\rho}=\frac{3M_{\odot}}{4\pi
R^{3}}=\rho_{\odot}\left(\frac{R_{\odot}}{R}\right)^{3}\approx(1\,410\
\mathrm{kg/m}^{3})\left(\frac{R_{\odot}}{R}\right)^{3},$ (32)

the central density at the temperature minimum will be about 10.71 times this,
or

$\rho_{c}\approx(15\,100\
\mathrm{kg/m}^{3})\left(\frac{R_{\odot}}{R}\right)^{3}.$ (33)

The kinetic energy of the $N_{\odot}=M_{\odot}/m$ nonrelativistic particles in
the isothermal plasma ball of the mass and composition of the sun and inside a
perfectly reflecting shell of radius $R$ and with $v=-(r/\rho)d\rho/dr$ at
this shell at the outer surface of the ball is

$K=\frac{3}{2}N_{\odot}kT=\frac{3GM_{\odot}^{2}}{2Rv}=E_{\odot}\left(\frac{R_{\odot}}{R}\right)\frac{3}{v},$
(34)

The gravitational potential energy is, with $M(r)$ being the mass inside
radius $r=L\xi$,

$\displaystyle U$ $\displaystyle=$
$\displaystyle-\int_{0}^{R}\frac{GM(r)}{r}\frac{dM(r)}{dr}dr=-\frac{L}{G}\left(\frac{kT}{m}\right)\int_{0}^{X}uv^{2}d\xi=-\frac{L}{G}\left(\frac{GM_{\odot}}{Rv}\right)^{2}Xv(3-u)$
(35) $\displaystyle=$
$\displaystyle-\frac{GM_{\odot}^{2}}{R}\left(\frac{3-u}{v}\right)=E_{\odot}\left(\frac{R_{\odot}}{R}\right)\left(\frac{2u-6}{v}\right).$

Here the integral can be evaluated by using the fact that

$\frac{d}{d\xi}[\xi v(3-u)]=uv^{2},$ (36)

which one can easily see follows from Eqs. (21) and (22).

One also has the energy density of the electromagnetic radiation, which is

$E_{r}=\frac{4\pi}{3}R^{3}aT^{4}=\frac{R^{3}(kT)^{4}}{45(hc)^{3}}=E_{0}\left(\frac{R_{\odot}}{R}\right)\frac{\lambda}{v^{4}}$
(37)

with

$\lambda=\frac{8\pi^{3}}{45N_{\odot}^{4}}\left(\frac{GM_{\odot}^{2}}{\hbar
c}\right)^{3}=\frac{8\pi^{3}}{45}\frac{M_{\odot}^{2}m^{4}}{m_{\mathrm{Pl}}^{6}}\approx
0.2328,$ (38)

where

$m_{\mathrm{Pl}}\equiv\sqrt{\frac{\hbar c}{G}}=\sqrt{\frac{hc}{2\pi
G}}=2.176\,434(24)\times 10^{-8}\ \mathrm{kg}$ (39)

is the Planck mass.

Therefore, the total energy of the isothermal plasma fluid ball, not counting
the rest mass energies of the electrons and nuclei, is

$E=K+U+E_{r}=E_{0}\left(\frac{R_{\odot}}{R}\right)\left(\frac{2u-3}{v}+\frac{\lambda}{v^{4}}\right)=E_{0}\left(\frac{R_{\odot}}{R}\right)P,$
(40)

where

$P=\frac{2u-3}{v}+\frac{\lambda}{v^{4}}$ (41)

just depends on $X=\xi=R/L$ at the perfectly reflecting sphere at the surface
of the isothermal ball of plasma, through the solution $\psi(\xi)$ of the
Emden-Chandrasekhar Eq. (9) and the auxiliary dimensionless functions $u=\xi
e^{-\psi}/(d\psi/d\xi)$ and $v=\xi d\psi/d\xi$ evaluated at the surface.

Making a quadratic interpolation of the Chandrasekhar-Wares tabulated values
[18] for $u$ and $v$ for $\xi=32$, $\xi=34$, and $\xi=36$ to calculate
$P(\xi)$ for intermediate values shows that the minimum value for $P(\xi)$ is

$P_{m}\approx-0.6556$ (42)

at $\xi\approx 33.36$, giving the minimum energy for an isothermal plasma ball
of the solar mass and composition and radius $R$ as

$E_{m}\approx-0.6556\,E_{0}\left(\frac{R_{\odot}}{R}\right)\approx-(1.243\times
10^{41}\ \mathrm{J})\left(\frac{R_{\odot}}{R}\right).$ (43)

Because of the positive contribution of the thermal radiation, this minimum
value for $P$ is slightly less negative than the minimum value without
including the thermal radiation, which Padmanabhan [16] calculated as
approximately $-0.670.$

In contrast, the work required to dissipate the present solar matter to
infinity is about $6.6\times 10^{41}$ J [19], which if one sets $R=R_{\odot}$
corresponds to $P\approx-3.48$. Therefore, the present sun is too tightly
bound gravitationally to be able to form an isothermal ball of plasma, which
if enclosed by a perfectly reflecting sphere of radius $R=R_{\odot}$ would
require the sun to generate additional energy

$\Delta E\approx 2.8E_{\odot}\approx 5.4\times 10^{41}\ \mathrm{J}$ (44)

which at the present solar luminosity would require about 44 million years.
However, since this time is short compared with the solar lifetime of the
present sun and with the lifetime for the sun to be confined inside a
perfectly reflecting sphere of radius $R\geq R_{\odot}$, I shall ignore the
time needed for the sun to produce enough energy to become an isothermal
sphere at the minimum energy $E_{m}$ given above. Huston and Wright [6, 7]
raise the possibility that even after reaching the energy of an isothermal
sphere, thermalization might take a long time for stars with radiative
exteriors such as the sun, and I am also ignoring that time which I have not
calculated.

##  4 Formulas for the Isothermal Hydrogen Burning Time Inside a Perfectly
Reflecting Sphere

Now I shall estimate the lifetime of the sun once it has been put inside a
perfectly reflecting sphere of radius $R$ and has reached an isothermal
configuration at the minimum value of $P$, $P_{m}\approx-0.6556$. I shall
assume that most of the time will be spent burning hydrogen to helium by the
PPI chain, and I shall use Eq. (5-30) on page 378 of Clayton [20] to give the
rate. In cgs units with hydrogen mass fraction $X_{H}$ and $T_{6}$ being the
temperature in units of $10^{6}$ K, the energy production rate per time and
per mass is given as

$\displaystyle\epsilon_{\mathrm{PPI}}=2.32\times 10^{6}\rho
X_{H}^{2}T_{6}^{-\frac{2}{3}}\exp{(-33.81T_{6}^{-\frac{1}{3}})}(1+0.0123T_{6}^{\frac{1}{3}}\\!\\!\\!\\!$
$\displaystyle+$
$\displaystyle\\!\\!\\!\\!0.0109T_{6}^{\frac{2}{3}}+0.00095T_{6})$ (45)
$\displaystyle\mathrm{erg\ g^{-1}\ sec^{-1}}.$

Most of the time needed for the hydrogen burning inside the perfectly
reflecting sphere will occur when $T_{6}$ is not large enough for the series
in $T_{6}^{\frac{1}{3}}$ to be much larger than 1. In particular, the minimum
temperature when $R=R_{\odot}$ from Eq. (30) gives $T_{6}\approx 5.665$ and
$T_{6}^{1/3}\approx 1.783$, which makes the series have the value

$1+0.0123T_{6}^{\frac{1}{3}}+0.0109T_{6}^{\frac{2}{3}}+0.00095T_{6}\approx
1.0619,$ (46)

which is within about 6% of unity. Of course, for larger $R/R_{\odot}$ and
hence smaller minimum $T_{6}\approx 5.665(R_{\odot}/R)$, the series will be
even closer to unity. Therefore, I shall neglect all but the first term in the
series in $T_{6}^{\frac{1}{3}}$. Then setting the hydrogen fraction by mass at
$X_{H}=0.70$ as I have done, and expressing quantities in terms of the solar
parameters rather than in terms of cgs units, the energy generation rate, the
rate for converting part of the rest mass energy of the nuclei to
$E=K+U+E_{r}$ (kinetic, gravitational potential, and radiation energies) is

$\frac{dE}{dt}=\int\epsilon_{PPI}dM=\frac{\epsilon_{\odot}}{\rho_{\odot}}F(\tau)\int\rho
dM=\frac{\epsilon_{\odot}}{\rho_{\odot}}F(\tau)J,$ (47)

where

$\tau\equiv\frac{T}{2T_{\odot}}=\frac{R_{\odot}}{Rv},$ (48)

is a dimensionless temperature that for an isothermal fluid ball varies
inversely with the radius $R$ of the reflecting sphere at the surface and also
inversely with the function $v(\xi)=\xi d\psi/d\xi=-d\ln{\rho}/d\ln{r}$
evaluated at the surface of the ball, and Eq. (45) with the series truncated
to the first term (1) leads to

$F(\tau)\approx 142\,000\,\tau^{-2/3}\exp{(-13.94\tau^{-1/3})}.$ (49)

I am also defining the dimensionful

$J\equiv\int\rho dM=4\pi
L^{3}\rho_{c}^{2}\int_{0}^{X}\xi^{2}e^{-2\psi}d\xi=\frac{M_{\odot}^{2}Xj}{4\pi
R^{3}v^{2}},$ (50)

where $X$ and $v$ are the values of $\xi$ and of $v(\xi)$ at the surface, and
where the dimensionless analogue of $J$ is

$j(X)\equiv\int_{0}^{X}\xi^{2}e^{-2\psi}d\xi.$ (51)

I could not find any explicit exact formula for $j(X)$ in terms of $X$,
$u(X)$, and $v(X)$, but to avoid doing a numerical integration for it, I used
the approximation [21] that

$e^{-\psi}\approx e^{-\psi_{PE}}=\frac{50}{10+\xi^{2}}-\frac{48}{12+\xi^{2}},$
(52)

which leads to the following explicit approximation for $j(X)$:

$j(X)\approx
j_{PE}(X)=2525\sqrt{10}\tan^{-1}{\frac{X}{\sqrt{10}}}-\frac{1250X}{10+X^{2}}-4608\sqrt{3}\tan^{-1}{\frac{X}{\sqrt{12}}}-\frac{1152X}{12+X^{2}}.$
(53)

Estimates showed that up to $X\approx 33.36$, where the energy is minimized,
$j_{PE}(X)$ is only a few percent larger than $j(X)$, and the error of using
it rather than $j(X)$ has the opposite sign as the error of dropping the
higher powers of $T_{6}^{1/3}$ in the series in Eq. (45), with both errors
small and of the same general order of magnitude.

Now, after some algebra, one can show that the approximate formulas above lead
to the time (in units of the characteristic solar time $t_{\odot}\equiv
E_{\odot}/L_{\odot}\equiv(GM_{\odot}^{2})/(2R_{\odot}L_{\odot})\approx
1.570\times 10^{7}\ \mathrm{years}$) for the hydrogen burning stage (which is
expected to take up the majority of the lifetime of sun when enclosed in a
perfectly reflecting sphere of radius $R$, at least if the thermalization time
is not enormously larger than $t_{\odot}$) to be

$\frac{\Delta
t}{t_{\odot}}\approx\alpha(R/R_{\odot})^{4/3}\int_{34}^{1}\frac{v^{4/3}}{Xj_{PE}(X)}[\exp{(\beta
v^{1/3})}]^{(R/R_{\odot})^{1/3}}\frac{dP}{dX}dX$ (54)

where the numerical constants are $\alpha\approx 2\times 10^{-5}$ and
$\beta\approx 13.94$, $j_{PE}(X)$ is given by Eq. (53), and
$P=(2u-3)/v+\lambda/v^{4}$, as given by Eq. (41). Remember that $X=R/L$ is the
value of the dimensionless radial variable $\xi=r/L$ at the surface of the
isothermal plasma ball (the location of the perfectly reflecting shell), and
$v$ is the value of $\xi d\psi/d\xi=-d\ln{\rho}/d\ln{r}$ at the surface. I am
taking the integration in the direction of the time evolution, from $X=34$
that is near where the energy $E$ is the minimum for an isothermal fluid ball
(ignoring the relatively short time of the evolution for enough energy to be
emitted to get to this point and to thermalize), down to a small value of $X$,
chosen here to be 1, beyond which the evolution would become much more rapid
because of the neglected terms in $\epsilon_{PPI}$ and because of helium
burning. (The value of the integral depends only weakly on where this cutoff
is located, especially for $R$ several times $R_{\odot}$.) In the direction of
the time evolution, $P$ increases, as both $dP/dX$ and $dX$ are negative.

##  5 Results of the Numerical Integrations

Rather than spend time programming a computer to do the numerical
integrations, and also to highlight the fact that the results can fairly
easily be obtained from simple calculations using published results such as
Clayton’s formula for the PPI chain hydrogen burning energy production rate
[20] and the Chandrasekhar-Wares tables [18] of the solution of the Emden-
Chandrasekhar equation for the structure of a self-gravitating isothermal
perfect fluid ball, I have done the numerical integrations of Eq. (54) mainly
by hand with an HP 35s Scientific Calculator, using its programmable features
only to calculate the functions in that equation, including $P(X)$ and
$j_{PE}(X)$ for integer values of $X$ ($\xi$ at the plasma fluid ball surface
where it is contained by a perfectly reflecting spherical shell) between 1 and
34 (except for 31 and 33, which were not given in the tables). I did the
calculations for $R/R_{\odot}=$ 1, 2, 3, 4, 5, 10, 20, 40, 83.24 (the
semimajor axis of the orbit of Mercury), 155.54 (the semimajor axis of the
orbit of Venus), and 215.03 (one astronomical unit, essentially the semimajor
axis of the orbit of the earth).

For each step, I used the trapezoid rule by multiplying the change in $P$ (the
integral of $(dP/dX)dX$ from the beginning to the end of the step) by the
average of the rest of the integrand in Eq. (54) at the two endpoints and
added the result for all steps. For $R/R_{\odot}=$ 1, 2, and 3, the sum of the
contributions for the steps between $X=16$ and $X=34$ were less than 1.7% of
the total, so for larger values of $R/R_{\odot}$ I just calculated the 15
steps between $X=1$ and $X=16$ and approximated the uncalculated steps by a
geometric series matched to the last two calculated steps, which also always
contributed less than 2% to the total.

I then tried various fitting functions for the resulting estimates of the
lifetimes of the hydrogen burning phases while the solar material was an
isothermal plasma ball of radii given by the 11 values of $R$ above. For
$R/R_{\odot}\leq 5$, I found an excellent fit was given by the following
2-parameter simple exponential function

$\Delta t_{2}=(5.4\ \mathrm{yr})\exp{[20.825(R/R_{\odot})^{1/3}]},$ (55)

which gave a relative discrepancy of less than 0.5% for all five values of
$R/R_{\odot}=$ 1, 2, 3, 4, and 5. (The exponent 1/3 of the $(R/R_{\odot})$
inside the exponential came from the $T_{6}^{1/3}$ factor in the exponential
in the Clayton formula Eq. (45) for the PPI chain energy production rate, so
it was not one of the 2 parameters I used to fit the results.)

For fitting all 11 values, I did not find any simple 2-parameter formula that
fit well over the whole range of $R/R_{\odot}$ from 1 to 215.03, but after
trying about five different forms, I found the following function with 3 free
parameters that I fit to the numerical results for $R/R_{\odot}=$ 1, 10, and
215.03 to match all 11 numerical results to within 9%, which is itself roughly
my crude estimate of the error in the calculated lifetimes inside the 11
different values of $R/R_{\odot}$:

$\Delta t_{3}=(50\
\mathrm{yr})\exp{[18.6(R/R_{\odot})^{1/3}]}\left(\frac{R}{R_{\odot}}\right)^{\frac{4}{3}\\{1+0.2\ln{[(R/R_{\odot})^{1/3}/6]}\\}}.$
(56)

Here the 3rd fitting parameter (besides the 50 and the 18.6 that are analogous
to the 5.4 and 20.825 in $\Delta t_{2}$) is the 0.2 coefficient of the
logarithm in the exponent of $R/R_{\odot}$; the denominator of 6 in the
argument of that logarithm was chosen so that the logarithm would be nearly 0
when the perfectly reflecting sphere is at one astronomical radius, $R\approx
215.03\sim 6^{3}$, to simplify the fitting procedure.

Table 1 gives the calculated lifetimes for hydrogen burning for the solar
composition confined inside perfectly reflecting spheres of various radii $R$,
in units of the present solar radius $R_{\odot}$, along with the values given
by the 2-parameter and 3-parameter approximate fitting formulas $\Delta t_{2}$
and $\Delta t_{3}$ of Eqs. (55) and (56) respectively, as well as the ratios
of those approximations to the numerical calculations of the lifetimes. Note
that the 2-parameter fitting formula $\Delta t_{2}$ works better for $1\leq
R/R_{\odot}\leq 5$, but the 3-parameter fitting formula $\Delta t_{3}$ works
better overall for $1\leq R/R_{\odot}\leq 215.03$.

Figure 1 gives a log-log plot the fitting function $\Delta t_{3}$ of Eq. (56),
in years, as a function of $R/R_{\odot}$, the radius of the confining
perfectly reflecting sphere at the surface divided by the current solar
radius. The horizontal axis is marked at the values of $R/R_{\odot}$ that were
used in the simple numerical calculations done on a pocket calculator, using
the PPI energy production rate formula from [20] and the numerical tables [18]
of the Emden-Chandrasekhar isothermal function [12, 13].

$R/R_{\odot}$ | $\Delta t$ | $\Delta t_{2}$ | $\Delta t_{2}/\Delta t$ | $\Delta t_{3}$ | $\Delta t_{3}/\Delta t$
---|---|---|---|---|---
1.00 |  $5.90\times 10^{9}\;$ yr |  $5.92\times 10^{9}\;$ yr | 1.003 |  $5.98\times 10^{9}\;$ yr | 1.014
2.00 |  $1.31\times 10^{12}$ yr |  $1.32\times 10^{12}$ yr | 1.008 |  $1.42\times 10^{12}$ yr | 1.081
3.00 |  $5.90\times 10^{13}$ yr |  $5.89\times 10^{13}$ yr | 0.998 |  $6.37\times 10^{13}$ yr | 1.079
4.00 |  $1.21\times 10^{15}$ yr |  $1.21\times 10^{15}$ yr | 0.995 |  $1.29\times 10^{15}$ yr | 1.064
5.00 |  $1.55\times 10^{16}$ yr |  $1.55\times 10^{16}$ yr | 1.003 |  $1.62\times 10^{16}$ yr | 1.049
10.00 |  $1.46\times 10^{20}$ yr |  $1.61\times 10^{20}$ yr | 1.108 |  $1.45\times 10^{20}$ yr | 0.997
20.00 |  $1.25\times 10^{25}$ yr |  $1.86\times 10^{25}$ yr | 1.493 |  $1.22\times 10^{25}$ yr | 0.975
40.00 |  $1.77\times 10^{31}$ yr |  $4.45\times 10^{31}$ yr | 2.517 |  $1.66\times 10^{31}$ yr | 0.941
83.24 |  $2.52\times 10^{39}$ yr |  $1.60\times 10^{40}$ yr | 6.341 |  $2.33\times 10^{39}$ yr | 0.925
155.54 |  $1.09\times 10^{48}$ yr |  $2.23\times 10^{49}$ yr | 20.390 |  $9.99\times 10^{47}$ yr | 0.914
215.03 |  $1.66\times 10^{53}$ yr |  $7.76\times 10^{54}$ yr | 46.669 |  $1.59\times 10^{53}$ yr | 0.958
Table 1: Lifetime $\Delta t$ of the sun confined to the interior of a
perfectly reflecting sphere of radius $R$, along with fitting formulas $\Delta
t_{2}=(5.4\ \mathrm{yr})\exp{[20.825(R/R_{\odot})^{1/3}]}$ and $\Delta
t_{3}=(50\
\mathrm{yr})\exp{[18.6(R/R_{\odot})^{1/3}]}(R/R_{\odot})^{\frac{4}{3}\\{1+0.2\ln{[(R/R_{\odot})^{1/3}/6]}\\}}$.
Note that $R=83.24R_{\odot}$ is the semimajor axis of the orbit of Mercury,
$R=155.54R_{\odot}$ is that of Venus, and $R=215.03R_{\odot}$ is that of
Earth. Figure 1: The confined solar lifetime in years versus the radius of the
perfectly reflecting sphere at the surface in units of the present solar
radius, $R/R_{\odot}$.

##  6 Discussion

Enclosing the sun inside a perfectly reflecting shell of the same radius as
that of the present sun, $R=R_{\odot}$, does not increase the lifetime
significantly. (And in this case the corrections from the series in
$T_{6}^{1/3}$ are not negligible, so I would expect the lifetime to be
significantly less than my estimate.) However, enclosing the sun inside a
sphere of radius just twice as large, $R=2R_{\odot}$, would increase the
lifetime by several orders of magnitudes (even if in this case the corrections
might be rather more than 10%). Making the reflecting shell radius even larger
increases the lifetime greatly. By the time one gets $R$ up to the orbital
radii of the inner planets, the calculated lifetime is so long that it is
likely to be invalidated by baryon decay, which would seem to put a very firm
upper limit on the lifetime of the sun that cannot be circumvented just by
putting it into a perfectly reflecting shell.

Therefore, it seems that by putting the sun inside a perfectly reflecting
shell of a suitable radius between the present radius of the sun and that of
the orbits of the inner planets, one can get any lifetime between that of the
unconfined sun and that of baryon decay.

It is tempting to speculate whether some very advanced civilization might be
able to circumvent the extremely severe practical constraints I have ignored
here and construct such a shell or Dyson sphere around a star to extend the
stellar lifetime as long as desired, up to the time of baryon decay. In
principle one could construct an aperture in the shell to withdraw as much
energy as the civilization needs without significantly reducing the lifetime
(unless the civilization requires far more power than ours does, as it might
if it needs to construct and maintain the Dyson sphere). However, I shall
leave it as a challenge for the future for how even in principle an advanced
civilization might be enabled to survive past the time of baryon decay.

After finishing all the calculations and the bulk of the writing of this
paper, I found that J. D. Bernal [5] had written a foretaste of what I have
calculated could be the case if stars were surrounded by perfectly reflecting
spheres to extend their lifetimes by millions of millions of times:

“A star is essentially an immense reservoir of energy which is being
dissipated as rapidly as its bulk will allow. It may be that, in the future,
man will have no use for energy and be indifferent to stars except as
spectacles, but if (and this seems more probable) energy is still needed, the
stars cannot be allowed to continue to in their old way, but will be turned
into efficient heat engines. The second law of thermodynamics, as Jeans
delights in pointing out to us, will ultimately bring this universe to an
inglorious close, may perhaps always remain the final factor. But by
intelligent organization the life of the universe could probably be prolonged
to many millions of millions of times what it would be without organization.
Besides, we are still too close to the birth of the universe to be certain
about its death.”

##  7 Acknowledgments

This research was funded by the Natural Sciences and Engineering Research
Council of Canada. After I shared a nearly-final version of this paper with
Macy Huston and Jason Wright, Jason Wright raised several helpful questions,
some of which (such as whether a star such as the sun with radiative energy
transfer can thermalize inside a perfectly reflecting shell before hydrogen
burning is completed) will have to await future research.

## References

  * [1] Freeman J. Dyson, “Search for artificial stellar sources of infrared radiation.” Science 131 (3414), 1667-1668 (1960).
  * [2] Freeman J. Dyson. Disturbing the Universe (Basic Books, New York, 1979), p. 211.
  * [3] Freeman J. Dyson and Richard Carrigan, “Dyson sphere,” Scholarpedia 4 (5), 6647. doi:10.4249/scholarpedia.6647, http://www.scholarpedia.org/article/Dyson_sphere.
  * [4] Olaf Stapledon, Starmaker (Methune, London, 1987).
  * [5] J. D. Bernal, The World, the Flesh & the Devil: An Enquiry into the Future of the Three Enemies of the Rational Soul (Foyle, London, 1929), p. 12.
  * [6] Macy Huston, “Evolutionary and observational consequences of Dyson sphere feedback.” Zenodo. https://doi.org/10.5281/zenodo.5160235 (2021, July 30).
  * [7] Macy Huston and Jason Wright, “Evolutionary and observational consequences of Dyson sphere feedback,” Ap. J. 924, 78 (13 pp.) (2022).
  * [8] Giora Shaviv, “Why the Kelvin–Helmholtz timescale is not really their timescale.” New Astronomy Reviews 51 (10-12), 803-813 (2008).
  * [9] K. P. Schroder and R. C. Smith, “Distant future of the Sun and Earth revisited,” Mon. Not. Roy. Astron. Soc. 386, 155 (2008) doi:10.1111/j.1365-2966.2008.13022.x [arXiv:0801.4031 [astro-ph]].
  * [10] R. L. Workman et al. (Particle Data Group), “Review of particle physics,” to be published in Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  * [11] James N. Connelly et al, “The absolute chronology and thermal processing of solids in the solar protoplanetary disk,” Science 338 (6107), 651-655 (2012).
  * [12] Robert Emden, Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf kosmologische und meteorologische Probleme (B. Teubner, 1907).
  * [13] S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, 1958), p. 156.
  * [14] V. A. Antonov, Vest. Leningrad Gros. Univ. 7, 135 (1962); English translation in Dynamics of Star Clusters, IAU Symposium Vol. 113, eds. J. Goodman and P. Hut (Reidel, Dordrecht, 1985), pp. 525-540.
  * [15] D. Lynden-Bell and Roger Wood, “The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems,” Mon. Not. R. astr. Soc. 138, 495-525 (1968).
  * [16] T. Padmanabhan, “Antonov instability and gravothermal catastrophe—revisited” Ap. J. Suppl. 71. 651-664 (1989).
  * [17] Joseph Katz, “Thermodynamics of self-gravitating systems,” Found. Phys. 33 (2), 223-269 (2003).
  * [18] S. Chandrasekhar and Gordon W. Wares. “The isothermal function,” Ap. J. 109, 551-554 (1949).
  * [19] William C. Livingston, “Sun,” in Arthur N. Cox, ed., Allen’s Astrophysical Quantities, 4th edition (AIP and Springer-Verlag, New York, 2000), p. 340.
  * [20] Donald D. Clayton, Principles of Stellar Evolution and Nucleosynthesis
(University of Chicago Press, Chicago, 1983), p. 378.

  * [21] José Luis G. Pestaña and Donald H. Eckhardt, “An approximate analytic solution of the three-dimensional Poisson-Boltzmann equation,” J. Phys. A: Math. Theor. 40. 12001-12006 (2007).


# The one-dimensional Schrödinger operators with singular periodic potentials

Vladimir Mikhailets  Institute of Mathematics of NAS of Ukraine
<EMAIL_ADDRESS>and  Volodymyr Molyboga  Institute of Mathematics of
NAS of Ukraine<EMAIL_ADDRESS>
###### Abstract.

The one-dimensional Schrödinger operators

$S(q)u:=-u^{\prime\prime}+q(x)u,\quad u\in\mathrm{Dom}\left(S(q)\right)$

with real-valued 1-periodic singular potentials $q(x)\in
H_{per}^{-1}(\mathbb{R})$ are studied on the Hilbert space
$L_{2}\left(\mathbb{R}\right)$. An equivalence of five basic definitions for
the operators $S(q)$ and their self-adjointness are established. A new proof
of spectral continuity of the operators $S(q)$ is found. Endpoints of spectral
gaps are precisely described.

###### Key words and phrases:

Hill equations, Schrödinger operators, singular potentials, spectral gaps,
periodic eigenvalues

###### 2000 Mathematics Subject Classification:

Primary 34L05, 47A05; Secondary 34L40, 47A10, 47B25

##  1\. Introduction

On the complex Hilbert space $L_{2}(\mathbb{R})$ we consider the one-
dimensional Schrödinger operators

(1) $S(q)u:=-u^{\prime\prime}+q(x)u,\quad u\in\mathrm{Dom}\,\left(S(q)\right)$

with real-valued 1-periodic distributional potentials $q(x)$, so called the
Hill-Schrödinger operators.

Under the assumption

(2) $q(x)=\sum_{k\in 2\mathbb{Z}}\widehat{q}(k)e^{ik\pi x}\in
H_{per}^{-1}(\mathbb{R},\mathbb{R}),$

i.e., when

$\sum_{k\in
2\mathbb{Z}}(1+|k|)^{-2}|\widehat{q}(k)|^{2}<\infty,\quad\text{and}\quad\widehat{q}(k)=\overline{\widehat{q}(-k)}\quad\forall
k\in 2\mathbb{Z},$

the Hill-Schrödinger operators $S(q)$ can be well defined on the Hilbert space
$L_{2}(\mathbb{R})$ in the following different ways:

  * $\bullet$

as minimal/maximal quasi-differential operators $S_{min}(q)$/$S_{max}(q)$;

  * $\bullet$

as Friedrichs extensions $S_{F}(q)$ of quasi-differential operators
$S_{min}(q)$;

  * $\bullet$

as form-sum operators $S_{form}(q)$;

  * $\bullet$

as a sequence limits $S_{lim}(q)$ of the Hill-Schrödinger operators with
smooth periodic potentials in the norm resolvent sense.

Hryniv and Mykytyuk [HrMk], Djakov and Mityagin [DjMt] studied Friedrichs
extensions $S_{F}(q)$, and Korotyaev [Krt] treated form-sum operators
$S_{form}(q)$. We propose join together these results showing an equivalence
of all definitions.

More precisely, we will prove the following statements.

######  Theorem A (Theorem 14).

The Hill-Schrödinger quasi-differential operators $S_{max}(q)$ with
distributional potentials $q(x)\in
H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$ are self-adjoint.

######  Theorem B (Corollary 15, Corollary 16, Theorem 18).

Quasi-differential operators $S_{min}(q)$ and $S_{max}(q)$, Friedrichs
extensions $S_{F}(q)$, form-sum operators $S_{form}(q)$ and operators
$S_{lim}(q)$ coincide.

In the paper [HrMk, Theorem 3.5] the authors tried to show that the operators
$S_{max}(q)$ and $S_{F}(q)$ coincide. But proof of this assertion is
incorrect. Our proofs of Theorem A and Theorem B base on a different idea (see
Lemma 5).

The equality $S(q)=S_{lim}(q)$ together with the classical Birkhoff-Lyapunov
Theorem allow to prove the following statement.

######  Theorem C (Theorem 19).

The Hill-Schrödinger operators $S(q)$ with distributional potentials $q(x)\in
H_{per}^{-1}(\mathbb{R},\mathbb{R})$ have continuous spectra with a band and
gap structure such that the endpoints
$\\{\lambda_{0}(q),\lambda_{k}^{\pm}(q)\\}_{k=1}^{\infty}$ of spectral gaps
satisfy the inequalities:

$-\infty<\lambda_{0}(q)<\lambda_{1}^{-}(q)\leq\lambda_{1}^{+}(q)<\lambda_{2}^{-}(q)\leq\lambda_{2}^{+}(q)<\ldots\,.$

Moreover, the endpoints of spectral gaps for even/odd numbers
$k\in\mathbb{Z}_{+}$ are periodic/semiperiodic eigenvalues of the problems on
the interval $[0,1]$,

$S_{\pm}(q)u=-u^{\prime\prime}+q(x)u=\lambda u,\quad
u\in\mathrm{Dom}\left(S_{\pm}(q)\right).$

It is interested to remark that the last assertion is nontrivial and for the
more singular $\delta^{\prime}$-interactions,

$q(x)=\sum_{k\in\mathbb{Z}}\beta\,\delta^{\prime}(x-k)\notin
H_{per}^{-1}(\mathbb{R}),\quad\beta<0,$

an unusual situation, when the endpoints of spectral gaps for even/odd numbers
$k\in\mathbb{Z}_{+}$ are semiperiodic/periodic eigenvalues of the problems on
the interval $[0,1]$, is possible [Alb, Theorem III.3.6].

In the closely related paper [HrMk] Hryniv and Mykytyuk established that
spectra of the operators $S(q)$ are absolutely continuous.

##  2\. Preliminaries

###  2.1. Sobolev spaces

Let us denote by $\mathfrak{D}^{\prime}_{1}\left(\mathbb{R}\right)$ the
Schwartz space of 1-periodic distributions defined on the whole real axis
$\mathbb{R}$ (see, for an example, [Vld]). For a detail characteristic of
1-periodic distributions we introduce Sobolev spaces.

So, Sobolev spaces $H_{per}^{s}(\mathbb{R})$, $s\in\mathbb{R}$ of 1-periodic
functions/distributions are defined by means of their Fourier coefficients:

$\displaystyle H_{per}^{s}(\mathbb{R})$ $\displaystyle:=\left\\{f=\sum_{k\in
2\mathbb{Z}}\widehat{f}(k)e^{ik\pi x}\left|\;\parallel
f\parallel_{H_{per}^{s}(\mathbb{R})}<\infty\right.\right\\},$
$\displaystyle\parallel f\parallel_{H_{per}^{s}(\mathbb{R})}$
$\displaystyle:=\left(\sum_{k\in 2\mathbb{Z}}\langle
k\rangle^{2s}|\widehat{f}(k)|^{2}\right)^{1/2},\quad\langle k\rangle:=1+|k|,$
$\displaystyle\widehat{f}(k)$ $\displaystyle:=\langle f,e^{ik\pi
x}\rangle_{L_{2,per}(\mathbb{R})},\quad k\in 2\mathbb{Z},$ $\displaystyle
2\mathbb{Z}$ $\displaystyle:=\left\\{k\in\mathbb{Z}\;\left|\;k\equiv
0\;(\mathrm{mod}\,2)\right.\right\\}.$

Sesqui-linear form $\langle\cdot,\cdot\rangle_{L_{2,per}(\mathbb{R})}$ pairs
the dual, respectively $L_{2,per}(\mathbb{R})$, spaces
$H_{per}^{s}(\mathbb{R})$ and $H_{per}^{-s}(\mathbb{R})$, and it is an
extension by continuity the $L_{2,per}(\mathbb{R})$-inner product [Brz, GrGr],

$\langle
f,g\rangle_{L_{2,per}(\mathbb{R})}:=\int_{0}^{1}f(x)\overline{g(x)}\,dx\quad\forall
f,g\in L_{2,per}(\mathbb{R}).$

It should be noted that

$H_{per}^{0}(\mathbb{R})=L_{2,per}(\mathbb{R}),$

and that by $\mathfrak{D}^{\prime}_{1}\left(\mathbb{R},\mathbb{R}\right)$ and
$H_{per}^{s}(\mathbb{R},\mathbb{R})$, $s\in\mathbb{R}$ are denoted real-valued
1-periodic distributions from the correspondent spaces,

$\displaystyle\mathfrak{D}^{\prime}_{1}\left(\mathbb{R},\mathbb{R}\right)$
$\displaystyle:=\left\\{f(x)\in\mathfrak{D}^{\prime}_{1}(\mathbb{R})\left|\,\mathrm{Im}f(x)=0\right.\right\\},$
$\displaystyle H_{per}^{s}(\mathbb{R},\mathbb{R})$
$\displaystyle:=\left\\{f(x)\in
H_{per}^{s}(\mathbb{R})\left|\,\mathrm{Im}f(x)=0\right.\right\\}.$

Note that $\mathrm{Im}f(x)=0$ for a 1-periodic distribution
$f(x)\in\mathfrak{D}^{\prime}_{1}\left(\mathbb{R}\right)$ means that

$\widehat{f}(2k)=\overline{\widehat{f}(-2k)}\quad\forall k\in\mathbb{Z}.$

###  2.2. Quasi-differential equations

The differential expressions in the right-hand of the (1) by introducing
quasi-derivatives:

$u^{[1]}(x):=u^{\prime}(x)-Q(x)u(x),\qquad\langle q,\varphi\rangle=-\langle
Q,\varphi^{\prime}\rangle\quad\forall\varphi\in
C_{comp}^{\infty}(\mathbb{R}),$

may be re-written as quasi-differential ones [SvSh1, SvSh2],

$l_{Q}[u]:=-(u^{\prime}-Qu)^{\prime}-Q(u^{\prime}-Qu)-Q^{2}u,$

which are well defined if $u,u^{[1]}\in W_{1,loc}^{1}(\mathbb{R})$ [Nai].

######  Proposition 1 (Existence and Uniqueness Theorem).

Let $\lambda\in\mathbb{C}$, and $f(x)\in L_{1,loc}(\mathbb{R})$. Then for any
complex numbers $c_{0},\;c_{1}\in\mathbb{C}$ and arbitrary
$x_{0}\in\mathbb{R}$ the quasi-differential equation

(3) $l_{Q}[u]=\lambda u+f,\quad\lambda\in\mathbb{C},\;f\in
L_{1,loc}(\mathbb{R})$

has one and only one solution $u\in W_{1,loc}^{1}(\mathbb{R})$ with the
initial conditions

$u(x)\left|\right._{x=x_{0}}=c_{0},\quad
u^{[1]}(x)\left|\right._{x=x_{0}}=c_{1}.$

With the quasi-differential equation (3) it is related the normal
2-dimensional system of the first order differential equations with the
locally integrable coefficients,

$\begin{pmatrix}u_{1}\\\ u_{2}\ \end{pmatrix}^{\prime}=\begin{pmatrix}Q&1\\\
-\lambda-Q^{2}&-Q\ \end{pmatrix}\begin{pmatrix}u_{1}\\\ u_{2}\
\end{pmatrix}+\begin{pmatrix}0\\\ -f\ \end{pmatrix},$

where $u_{1}(x):=u(x)$, $u_{2}(x):=u^{[1]}(x)$.

Then Proposition 1 follows from [Nai, Theorem 1, §16], also see [AhGl].

######  Lemma 2 (Lagrange Formula).

Let $u(x)$ and $v(x)$ be functions such that quasi-differential expressions
$l_{Q}[\cdot]$ are well defined. Then the Lagrange formula

$l_{Q}[u]\overline{v}-ul_{Q}[\overline{v}]=\frac{d}{dx}[u,v]_{x}$

holds, where the sesqui-linear forms $[u,v]_{x}$ are defined in the following
fashion:

$[u,v]_{x}:=u(x)\overline{\left(v^{\prime}(x)-Q(x)v(x)\right)}-\left(u^{\prime}(x)-Q(x)u(x)\right)\overline{v(x)}.$

###### Proof.

Under the assumption $u(x)$ and $v(x)$ are such that

$u,u^{\prime}-Qu\in W_{1,loc}^{1}(\mathbb{R})\quad\text{and}\quad
v,v^{\prime}-Qv\in W_{1,loc}^{1}(\mathbb{R}).$

Then we have

$\displaystyle\frac{d}{dx}[u,v]_{x}$
$\displaystyle\equiv\frac{d}{dx}\left(u\overline{\left(v^{\prime}-Qv\right)}-\left(u^{\prime}-Qu\right)\overline{v}\right)$
$\displaystyle=u^{\prime}\overline{\left(v^{\prime}-Qv\right)}+u\overline{\left(v^{\prime}-Qv\right)^{\prime}}-\left(u^{\prime}-Qu\right)^{\prime}\overline{v}-\left(u^{\prime}-Qu\right)\overline{v^{\prime}}$
$\displaystyle=l_{Q}[u]\overline{v}-ul_{Q}[\overline{v}]+Qu^{\prime}\overline{v}-Qu\overline{v^{\prime}}+u^{\prime}\overline{\left(v^{\prime}-Qv\right)}-\left(u^{\prime}-Qu\right)\overline{v^{\prime}}$
$\displaystyle=l_{Q}[u]\overline{v}-ul_{Q}[\overline{v}]$

taking into account that under the made assumptions

$u^{\prime}\overline{v^{\prime}},\,Q^{2}u\overline{v},\,Qu^{\prime}\overline{v},\,Qu\overline{v^{\prime}}\in
L_{1,loc}(\mathbb{R}).$

The proof is complete. ∎

Integrating both parts of the Lagrange Formula over the compact interval
$[\alpha,\beta]\Subset\mathbb{R}$ we obtain the Lagrange Identity in an
integral form,

(4)
$\int_{\alpha}^{\beta}l_{Q}[u]\overline{v}\,dx-\int_{\alpha}^{\beta}ul_{Q}[\overline{v}]\,dx=[u,v]_{\alpha}^{\beta},$

where

$[u,v]_{\alpha}^{\beta}:=[u,v]_{\beta}-[u,v]_{\alpha}.$

###  2.3. Quasi-differential operators on a finite interval

Here, we due to Savchuk and Shkalikov [SvSh1] give a short review of results
related with Sturm-Liouville operators with distributional potentials defined
on a finite interval.

On the Hilbert space $L_{2}(0,1)$ we consider the Sturm-Liouville operators

$L(q)u:=-u^{\prime\prime}+q(x)u,\quad u\in\mathrm{Dom}\,\left(L(q)\right)$

with real-valued distributional potentials $q(x)\in
H^{-1}\left([0,1],\mathbb{R}\right)$, i.e., when

$Q(x)=\int q(\xi)\,d\xi\in L_{2}\left((0,1),\mathbb{R}\right).$

Set

$\displaystyle L_{max}(q)u$ $\displaystyle:=l_{Q}[u],$
$\displaystyle\mathrm{Dom}(L_{max}(q))$ $\displaystyle:=\left\\{u\in
L_{2}(0,1)\left|\right.u,u^{\prime}-Qu\in W_{1}^{1}[0,1],\,l_{Q}[u]\in
L_{2}(0,1)\right\\},$

and

$\displaystyle\dot{L}_{min}(q)u$ $\displaystyle:=l_{Q}[u],$
$\displaystyle\mathrm{Dom}(\dot{L}_{min}(q))$
$\displaystyle:=\left\\{u\in\mathrm{Dom}(L_{max}(q))\left|\right.\mathrm{supp}\,u\Subset[0,1]\right\\}.$

We also consider the operators

$\displaystyle L_{min}(q)u$ $\displaystyle:=l_{Q}[u],$
$\displaystyle\mathrm{Dom}(L_{min}(q))$
$\displaystyle:=\left\\{u\in\mathrm{Dom}(L_{max}(q))\left|\right.u^{[j]}(0)=u^{[j]}(1)=0,\;j=0,1\right\\}.$

######  Proposition 3 ([SvSh1]).

Let suppose that $q(x)\in H^{-1}\left([0,1],\mathbb{R}\right)$. Then the
following statements are fulfilled:

  * (``I”)

Operators $L_{min}(q)$ are densely defined on the Hilbert space $L_{2}(0,1)$.

  * (``II”)

Operators $L_{min}(q)$ and $L_{max}(q)$ are mutually adjoint,

$L_{min}^{\ast}(q)=L_{max}(q),\qquad L_{max}^{\ast}(q)=L_{min}(q).$

In particular, operators $L_{min}(q)$ and $L_{max}(q)$ are closed.

In Statement 4, which proof is given in Appendix A.1, we establish
relationships between operators $\dot{L}_{min}(q)$ and $L_{min}(q)$.

######  Statement 4.

Operators $L_{min}(q)$ are closures of operators $\dot{L}_{min}(q)$,

$L_{min}(q)=(\dot{L}_{min}(q))^{\sim}=\dot{L}_{min}^{\ast\ast}(q).$

##  3\. Main results

###  3.1. Principal lemma

The following operator-theory statement is an essential point of our approach.
It has two important applications in this section.

######  Lemma 5.

Let $A$ be a densely defined and closed on a complex Banach space $X$ linear
operator, and let $B$ be a bounded on $X$ linear operator, such that

  * (a)

$BA\subset AB$ ($A$ and $B$ commute);

  * (b)

$\sigma_{p}(B)=\varnothing$ (point spectrum $\sigma_{p}(B)$ of operator $B$ is
empty).

Then the operator $A$ has no eigenvalues of a finite multiplicity.

###### Proof.

Let suppose that the operator $A$ has eigenvalue $\lambda\in\sigma_{p}(A)$ of
a finite multiplicity, and let $G_{\lambda}$ be a correspondent eigenspace.

Further, let $f$ be an eigenvector of the operator $A$,

$Af=\lambda f,\qquad f\in G_{\lambda},$

Then we have

$A(Bf)=B(Af)=\lambda(Bf),\qquad f\in G_{\lambda},$

from where one may conclude that

$BG_{\lambda}\subset G_{\lambda}.$

Taking into account that under the assumption
$\mathrm{dim}(G_{\lambda})\in\mathbb{N}$ from the latter we obtain that point
spectrum $\sigma_{p}(B)$ of the operator $B$ is not empty. This contradicts to
the condition (b).

The proof is complete. ∎

######  Remark 6.

The condition (b) is valid if the space $X=L_{p}(\mathbb{R},\mathbb{C})$,
$1\leq p<\infty$, and $B$ be a shift operator,

$B:\,y(x)\mapsto y(x+T),\,T>0.$

Indeed, the operator $B$ is an unitary one on the space
$X=L_{p}(\mathbb{R},\mathbb{C})$. Therefore

$\sigma_{p}(B)\subset\sigma(B)=\left\\{\lambda\in\mathbb{C}\left|\right.|\lambda|=1\right\\},$

and the equality

$By(x)=\lambda y(x)=y(x+T),\quad y(x)\not\equiv 0,\quad|\lambda|=1$

implies that the function $|y(x)|$ is $T$-periodic. Then $y(x)\not\in
L_{p}(\mathbb{R},\mathbb{C})$, and one may conclude that
$\sigma_{p}(B)=\varnothing$.

The condition (a) means in this case that the operator $A$ is $T$-periodic on
the line.

###  3.2. Self-adjointness of the Hill-Schrödinger operators with
distributional potentials

Under the assumption (2) the distributional potentials $q(x)$ have got the
representations

$q(x)=C+Q^{\prime}(x)$

with

$C=\widehat{q}(0)$

and

$Q(x)=\sum_{k\in
2\mathbb{Z}\setminus\\{0\\}}\frac{1}{ik\pi}\widehat{q}(2k)e^{ik\pi x}\in
L_{2,per}(\mathbb{R},\mathbb{R})$

such that

$\langle q,\varphi\rangle=-\langle
Q,\varphi^{\prime}\rangle\quad\forall\varphi\in
C_{comp}^{\infty}(\mathbb{R}),$

see [DjMt, Proposition 1], [Vld]. Here, by $\langle f,\cdot\rangle$,
$f\in\mathfrak{D}^{\prime}(\mathbb{R})$ we denote sesqui-linear functionals
over the space $C_{comp}^{\infty}(\mathbb{R})$.

######  Remark 7.

Without loss of any generality throughout of the remainder of the paper we
will assume that

$\widehat{q}(0)=0.$

Then the Hill-Schrödinger operators can be well defined on the Hilbert space
$L_{2}(\mathbb{R})$ as quasi-differential ones [SvSh1, SvSh2] by means of
quasi-expressions,

$l_{Q}[u]=-(u^{\prime}-Qu)^{\prime}-Q(u^{\prime}-Qu)-Q^{2}u.$

Set

$\displaystyle S_{max}(q)u$ $\displaystyle:=l_{Q}[u],$
$\displaystyle\mathrm{Dom}(S_{max}(q))$ $\displaystyle:=\left\\{u\in
L_{2}(\mathbb{R})\left|\right.u,u^{\prime}-Qu\in
W_{1,loc}^{1}(\mathbb{R}),\,l_{Q}[u]\in L_{2}(\mathbb{R})\right\\},$

and

$\displaystyle\dot{S}_{min}(q)u$ $\displaystyle:=l_{Q}[u],$
$\displaystyle\mathrm{Dom}(\dot{S}_{min}(q))$
$\displaystyle:=\left\\{u\in\mathrm{Dom}(S_{max}(q))\left|\right.\mathrm{supp}\,u\Subset\mathbb{R}\right\\}.$

It is obvious that operators $S_{max}(q)$ are defined on the maximally
possible linear manifolds on which quasi-expressions $l_{Q}[\cdot]$ are well
defined.

######  Proposition 8.

Let $q(x)\in H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$. Then the
following statements are fulfilled:

  * (``I”)

Operators $\dot{S}_{min}(q)$ are symmetric and lower semibounded on the
Hilbert space $L_{2}(\mathbb{R})$. In particular, they are closable.

  * (``II”)

Closures $S_{min}(q)$ of operators $\dot{S}_{min}(q)$,
$S_{min}(q):=(\dot{S}_{min}(q))^{\sim}$, are symmetric, lower semibounded
operators on the Hilbert space $L_{2}(\mathbb{R})$ with deficiency numbers of
a view $(m,m)$ where $0\leq m\leq 2$. Operators $S_{max}(q)$ are adjoint to
operators $S_{min}(q)$,

$S_{min}^{\ast}(q)=S_{max}(q).$

In particular, $S_{max}(q)$ are closed operators on the Hilbert space
$L_{2}(\mathbb{R})$, and

$S_{max}^{\ast}(q)=S_{min}(q).$

  * (``III”)

Domains $\mathrm{Dom}(S_{min}(q))$ of operators $S_{min}(q)$ consist of those
and only those functions $u\in\mathrm{Dom}(S_{max}(q))$ which satisfy the
conditions:

$[u,v]_{+\infty}-[u,v]_{-\infty}=0\quad\forall v\in\mathrm{Dom}(S_{max}(q)),$

where the limits

$[u,v]_{+\infty}:=\lim_{x\rightarrow+\infty}[u,v]_{x},\quad\text{and}\quad[u,v]_{-\infty}:=\lim_{x\rightarrow-\infty}[u,v]_{x}$

are well defined and exist.

Proposition 8, which describes properties of operators $\dot{S}_{min}(q)$ and
$S_{max}(q)$, is proved by using methods of a linear quasi-differential
operators theory in Appendix A.2.

In Proposition 10 we define the Friedrichs extensions of minimal operators
$S_{min}(q)$. But firstly, for a convenience, we remind some related facts and
prove the useful Lemma 9.

Let $H$ be a Hilbert space, and $\dot{A}$ is a densely defined, lower
semibounded linear operator on $H$. Hence, $\dot{A}$ is a closable, symmetric
operator. Define by $A$ its closure, $A:=(\dot{A})^{\sim}$.

Set

$\dot{t}[u,v]:=(\dot{A}u,v),\qquad\mathrm{Dom}(\dot{t}):=\mathrm{Dom}(\dot{A}).$

As known [Kt] sesqui-linear form $\dot{t}[u,v]$ is closable, lower semibounded
and symmetric on a Hilbert space $H$ one. Let $t[u,v]$ be its closure,
$t:=(\dot{t})^{\sim}$.

By the operator $\dot{A}$ it is uniquely defined its Friedrichs extension
$A_{F}$ [Kt],

$t[u,v]=(A_{F}u,v),\qquad
u\in\mathrm{Dom}(A_{F})\subset\mathrm{Dom}(t),\,v\in\mathrm{Dom}(t).$

Due to the First Representation Theorem [Kt] operator $A_{F}$ is lower
semibounded and self-adjoint. In Lemma 9 we describe its domain, but at first
note that the following relationships hold:

$\dot{A}\subset A\subset A_{F}\subset A^{\ast}.$

######  Lemma 9.

Let $A_{F}$ be a Friedrichs extension of a densely defined, lower semibounded
operator $\dot{A}$ on a Hilbert space $H$, and let $t[u,v]$ is a densely
defined, closed, symmetric and bounded from below on $H$ sesqui-linear form
built by operator $\dot{A}$. Then the following formula

$\mathrm{Dom}(A_{F})=\mathrm{Dom}(t)\cap\mathrm{Dom}(A^{\ast})$

holds.

###### Proof.

It is obvious that

$\mathrm{Dom}(A_{F})\subset\mathrm{Dom}(t)\cap\mathrm{Dom}(A^{\ast}).$

Let prove the inverse inclusion.

Let $u\in\mathrm{Dom}(t)\cap\mathrm{Dom}(A^{\ast})$, and
$v\in\mathrm{Dom}(\dot{A})\subset\mathrm{Dom}(A_{F})\subset\mathrm{Dom}(t)$.
Remark that $\mathrm{Dom}(\dot{A})$ is a core of the form $t[u,v]$ as well as
$\mathrm{Dom}(t)\cap\mathrm{Dom}(A^{\ast})$ containing
$\mathrm{Dom}(\dot{A})$. Then we have

$(A^{\ast}u,v)=(u,\dot{A}v)=(u,A_{F}v)=\overline{(A_{F}v,u)}=\overline{t[v,u]}=t[u,v],$

i.e.,

$t[u,v]=(A^{\ast}u,v),\qquad
u\in\mathrm{Dom}(t)\cap\mathrm{Dom}(A^{\ast}),\,v\in\mathrm{Dom}(\dot{A}).$

Due to the First Representation Theorem [Kt] we get that
$u\in\mathrm{Dom}(A_{F})$, i.e.,

$\mathrm{Dom}(t)\cap\mathrm{Dom}(A^{\ast})\subset\mathrm{Dom}(A_{F}).$

The proof is complete. ∎

######  Proposition 10.

The Friedrichs extensions $S_{F}(q)$ of the operators $S_{min}(q)$ are defined
in the following fashion:

$\displaystyle S_{F}(q)u$ $\displaystyle:=l_{Q}[u],$
$\displaystyle\mathrm{Dom}(S_{F}(q))$ $\displaystyle:=\left\\{u\in
H^{1}(\mathbb{R})\left|\right.u^{\prime}-Qu\in
W_{1,loc}^{1}(\mathbb{R}),\,l_{Q}[u]\in L_{2}(\mathbb{R})\right\\}.$

###### Proof.

Let us introduce the sesqui-linear forms,

$\dot{t}[u,v]:=(\dot{S}_{min}(q)u,v),\quad\mathrm{Dom}(\dot{t}):=\mathrm{Dom}(\dot{S}_{min}(q)).$

As well known [Kt] the sesqui-linear forms $\dot{t}[u,v]$ are densely defined,
closable, symmetric and bounded from below on the Hilbert space
$L_{2}(\mathbb{R})$. Taking into account that
$\mathrm{Dom}(\dot{S}_{min}(q))\subset H_{comp}^{1}(\mathbb{R})$ the forms
$\dot{t}[u,v]$ have got a view

$\dot{t}[u,v]=(u^{\prime},v^{\prime})-(Qu,v^{\prime})-(Qu^{\prime},v),\quad\mathrm{Dom}(\dot{t})\subset
H_{comp}^{1}(\mathbb{R}).$

Set

$\displaystyle\dot{t}_{1}[u,v]$
$\displaystyle:=(u^{\prime},v^{\prime})+(u,v),\quad$
$\displaystyle\mathrm{Dom}(\dot{t}_{1}):=\mathrm{Dom}(\dot{S}_{min}(q))\subset
H_{comp}^{1}(\mathbb{R}),$ $\displaystyle\dot{t}_{2}[u,v]$
$\displaystyle:=-(Qu,v^{\prime})-(Qu^{\prime},v)-(u,v),\quad$
$\displaystyle\mathrm{Dom}(\dot{t}_{2}):=\mathrm{Dom}(\dot{S}_{min}(q))\subset
H_{comp}^{1}(\mathbb{R}),$

i.e.,

$\dot{t}=\dot{t}_{1}+\dot{t}_{2}.$

As well known the form $\dot{t}_{1}[u,v]$ is closable, and its closure
$t_{1}[u,v]$, $t_{1}:=(\dot{t}_{1})^{\sim}$, has the representation

$t_{1}[u,v]=(u^{\prime},v^{\prime})+(u,v),\quad\mathrm{Dom}(t_{1})=H^{1}(\mathbb{R}).$

As it was shown in the [HrMk] the forms $\dot{t}_{2}[u,v]$ are $t_{1}$-bounded
with relative boundary 0. So, we finally obtain that the forms $\dot{t}[u,v]$
closures $t[u,v]$, $t:=(\dot{t})^{\sim}$, are defined as following,

$t[u,v]=(u^{\prime},v^{\prime})-(Qu,v^{\prime})-(Qu^{\prime},v),\quad\mathrm{Dom}(t)=H^{1}(\mathbb{R}).$

And the sesqui-linear forms $t[u,v]$ are densely defined, closed, symmetric
and lower semibounded on the Hilbert space $L_{2}(\mathbb{R})$.

Further, as

$\displaystyle S_{min}^{\ast}(q)u$ $\displaystyle=l_{Q}[u],$
$\displaystyle\mathrm{Dom}(S_{min}^{\ast}(q))$ $\displaystyle=\left\\{u\in
L_{2}(\mathbb{R})\left|\right.u,u^{\prime}-Qu\in
W_{1,loc}^{1}(\mathbb{R}),\,l_{Q}[u]\in L_{2}(\mathbb{R})\right\\},$

applying Lemma 9 we get the need representations for Friedrichs extensions of
operators $\dot{S}_{min}(q)$.

The proof is complete. ∎

######  Statement 11.

It is valid the following envelops:

$\dot{S}_{min}(q)\subset S_{min}(q)\subset S_{F}(q)\subset S_{max}(q)$

and

$\displaystyle\mathrm{Dom}(\dot{S}_{min}(q))$ $\displaystyle\subset
H_{comp}^{1}(\mathbb{R}),$ $\displaystyle\mathrm{Dom}(S_{min}(q))\subset
H^{1}(\mathbb{R}),$ $\displaystyle\quad\mathrm{Dom}(S_{F}(q))\subset
H^{1}(\mathbb{R}),$ $\displaystyle\mathrm{Dom}(S_{max}(q))$
$\displaystyle\subset L_{2}(\mathbb{R})\cap H_{loc}^{1}(\mathbb{R}).$

Statement 11 immediately follows from the correspondent definitions and not
very complicate computations.

Now, our aim is to prove a self-adjointness of maximal quasi-differential
operators $S_{max}(q)$.

######  Proposition 12.

Let $q(x)\in H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$. The following
statements are equivalent:

  * (a)

Operators $S_{max}(q)$ are self-adjoint.

  * (b)

$\mathrm{Dom}(S_{max}(q))\subset H^{1}(\mathbb{R})$.

  * (c)

$u^{\prime}-Qu\in L_{2}(\mathbb{R})\cap W_{1,loc}^{1}(\mathbb{R})\quad\forall
u\in\mathrm{Dom}(S_{max}(q))$.

###### Proof.

$(a)$ Let $S_{max}(q)$ are self-adjoint. Then as it follows from Proposition
8.II and Statement 11 we obtain

$\displaystyle S_{min}(q)=S_{F}(q)=S_{max}(q),$
$\displaystyle\mathrm{Dom}(S_{min}(q))=\mathrm{Dom}(S_{F}(q))=\mathrm{Dom}(S_{max}(q))\subset
H^{1}(\mathbb{R}),$

and $(b)$ is true.

Further, under the assumptions $Q\in L_{2,per}(\mathbb{R})$ and $u\in
H^{1}(\mathbb{R})$ we have got $Qu\in L_{2}(\mathbb{R})$ [HrMk] that yields
$(c)$.

$(b)$ Let suppose that $\mathrm{Dom}(S_{max}(q))\subset H^{1}(\mathbb{R})$. As
above we get $Qu\in L_{2}(\mathbb{R})$, and as a consequence we obtain $(c)$.
Then the statement $(a)$ follows from the Lagrange Identity (4), taking into
account that

$[u,v]_{+\infty}=0\quad\text{and}\quad[u,v]_{-\infty}=0$

for $u,v\in L_{2}(\mathbb{R})$ and $u^{\prime}-Qu,v^{\prime}-Qv\in
L_{2}(\mathbb{R})\cap W_{1,loc}^{1}(\mathbb{R})$.

$(c)$ Let given $(c)$, i.e., $u^{\prime}-Qu\in L_{2}(\mathbb{R})\cap
W_{1,loc}^{1}(\mathbb{R})\quad\forall u\in\mathrm{Dom}(S_{max}(q))$. Then
applying the Lagrange Identity (4) as above we have got $(a)$, and as a
consequence $(b)$.

The proof is complete. ∎

Hryniv and Mykytyuk [HrMk] studied operators associated due to the First
Representation Theorem [Kt] with the sesqui-linear forms

$t[u,v]=(u^{\prime},v^{\prime})-(Qu,v^{\prime})-(Qu^{\prime},v),\quad\mathrm{Dom}(t)=H^{1}(\mathbb{R}).$

I.e., they studied namely Friedrichs extensions $S_{F}(q)$.

Djakov and Mityagin [DjMt] also treated Friedrichs extensions $S_{F}(q)$ a
priori considering operators on the domains

$\mathrm{Dom}(S_{F}(q))=\left\\{u\in
H^{1}(\mathbb{R})\left|\right.u^{\prime}-Qu\in
W_{1,loc}^{1}(\mathbb{R}),\,l_{Q}[u]\in L_{2}(\mathbb{R})\right\\},$

see Proposition 10 and Proposition 12.

So, due to Proposition 8.II we have

$S_{max}(q)\supset S_{max}^{\ast}(q),$

and therefore it remains to show a symmetry of the operators $S_{max}(q)$,

$S_{max}(q)\subset S_{max}^{\ast}(q).$

We do it applying Lemma 5.

Let consider on the Hilbert space $L_{2}(\mathbb{R})$ a shift operator

$(Uf)(x):=f(x+1),\qquad\mathrm{Dom}(U):=L_{2}(\mathbb{R}),$

then $\sigma_{p}(U)=\varnothing$.

Further, let $f\in\mathrm{Dom}(S_{max}(q))$. It is obvious that
$Uf\in\mathrm{Dom}(S_{max}(q))$ also, and

$U(S_{max}(q)f)=Ul_{Q}[f(x)]=l_{Q}[f(x+1)]=l_{Q}[(Uf)(x)]=S_{max}(q)(Uf),$

i.e., operators $S_{max}(q)$ and $U$ commute.

Taking into account that $S_{max}(q)$ are the second order quasi-differential
operators, i.e., their possible eigenvalues cannot have multiplicities more
than two, and applying Lemma 5 to the operators $S_{max}(q)$ and $U$ we obtain
the next proposition.

######  Proposition 13.

Point spectra $\sigma_{p}(S_{max}(q))$ of the quasi-differential operators
$S_{max}(q)$ are empty.

######  Theorem 14.

The quasi-differential operators $S_{max}(q)$ are self-adjoint.

###### Proof.

From Proposition 8.II and Proposition 13 we get that the minimal symmetric
operators $S_{min}(q)$ have deficiency index of a view $(0,0)$, i.e., they are
self-adjoint. Due to Proposition 8.II this implies that the operators
$S_{max}(q)$ are self-adjoint also.

The proof is complete. ∎

######  Corollary 15.

Minimal operators $S_{min}(q)$, Friedrichs extensions $S_{F}(q)$ and maximal
operators $S_{max}(q)$ coincide. In particular, they are self-adjoint and
lower semibounded.

######  Corollary 16.

Let $q(x)\in H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$, and $q_{n}(x)\in
H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$, $n\in\mathbb{N}$ such that

$q_{n}(x)\overset{H_{per}^{-1}\left(\mathbb{R}\right)}{\longrightarrow}q(x)\quad\text{as}\quad
n\rightarrow\infty.$

Then the Hill-Schrödinger operators $S(q_{n})$, $n\in\mathbb{N}$ converge to
the operators $S(q)$ in the norm resolvent sense,

$\left\|\left(S(q_{n})-\lambda I\right)^{-1}-\left(S(q)-\lambda
I\right)^{-1}\right\|\rightarrow 0\quad\text{as}\quad n\rightarrow\infty,$

for any $\lambda$ belonging to resolvent sets of $S(q)$ and $S(q_{n})$,
$n\in\mathbb{N}$.

###### Proof.

It immediately follows from [HrMk, Theorem 4.1] and Corollary 15. ∎

In particular, the Hill-Schrödinger operators $S(q)$ with distributional
potentials $q(x)\in H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$ are a
sequence limits $S_{lim}(q)$ of operators $S(q_{n})$, $n\in\mathbb{N}$ with
smooth potentials $q_{n}(x)\in L_{2,per}\left(\mathbb{R},\mathbb{R}\right)$.
For an instance, for

$q(x)=\sum_{k\in\mathbb{Z}}\widehat{q}(2k)\,e^{i\,2k\pi x}\in
H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$

we may choose

$q_{n}(x):=\sum_{|k|\leq n}\widehat{q}(2k)\,e^{i\,2k\pi x}\in
C_{per}^{\infty}\left(\mathbb{R},\mathbb{R}\right),\quad n\in\mathbb{N}.$

Now, we are going to define the Hill-Schrödinger operators with distributional
potentials as form-sum operators [Krt]. We will show that this definition
coincides with the given above ones.

Let consider on the Hilbert space $L_{2}(\mathbb{R})$ the sesqui-linear forms

$\tau[u,v]:=\left\langle-\frac{d^{2}}{dx^{2}}u,v\right\rangle_{L_{2}(\mathbb{R})}+\left\langle
q(x)u,v\right\rangle_{L_{2}(\mathbb{R})},\quad\mathrm{Dom}(\tau)=H^{1}(\mathbb{R}),$

generated by the one-dimensional Schrödinger operators with $q(x)\in
H_{per}^{-1}(\mathbb{R},\mathbb{R})$.

Here, by $\langle\cdot,\cdot\rangle_{L_{2}(\mathbb{R})}$ is denoted sesqui-
linear form pairing dual, respectively $L_{2}(\mathbb{R})$, spaces
$H^{s}(\mathbb{R})$ and $H^{-s}(\mathbb{R})$ for $s\in\mathbb{R}$, which
(sesqui-linear form) is an extension by continuity the
$L_{2}(\mathbb{R})$-inner product [Brz, GrGr],

$\langle
f,g\rangle_{L_{2}(\mathbb{R})}:=\int_{\mathbb{R}}f(x)\overline{g(x)}\,dx\quad\forall
f,g\in L_{2}(\mathbb{R}).$

As known [Krt] the sesqui-linear forms $\tau[u,v]$ are densely defined,
closed, bounded from below, defined on the Hilbert space $L_{2}(\mathbb{R})$
ones. Due to the First Representation Theorem [Kt] with them it is associated
uniquely defined on the Hilbert space $L_{2}(\mathbb{R})$ self-adjoint, lower
semibounded operators $S_{form}(q)$ such that

  * i)

$\mathrm{Dom}\left(S_{form}(q)\right)\subset\mathrm{Dom}\left(\tau\right)$,
and

$\tau[u,v]=(S_{form}(q)u,v)\quad\forall
u\in\mathrm{Dom}\left(S_{form}(q)\right),\,\forall
v\in\mathrm{Dom}\left(\tau\right);$

  * ii)

$\mathrm{Dom}\left(S_{form}(q)\right)$ are cores of the forms $\tau[u,v]$;

  * iii)

if $u\in\mathrm{Dom}\left(\tau\right)$, $w\in L_{2}(\mathbb{R})$, and

$\tau[u,v]=(w,v)$

hold for every $v$ in cores of the forms $\tau[u,v]$, then
$u\in\mathrm{Dom}\left(S_{form}(q)\right)$ and

$S_{form}(q)u=w.$

Operators $S_{form}(q)$ are called form-sum operators associated with the
forms $\tau[u,v]$, and denoted as following

$S_{form}(q):=-\frac{d^{2}}{dx^{2}}\dotplus q(x),$

and also it is convenient to use the denotations

$\tau_{S_{form}(q)}[u,v]\equiv\tau[u,v].$

######  Proposition 17 ([Krt]).

The Hill-Schrödinger operators with distributional potentials from the
negative Sobolev space $H_{per}^{-1}(\mathbb{R},\mathbb{R})$ are well defined
on the Hilbert space $L_{2}(\mathbb{R})$ as self-adjoint, lower semibounded
form-sum operators $S_{form}(q)$,

$S_{form}(q)=-\frac{d^{2}}{dx^{2}}\dotplus q(x),$

associated with the sesqui-linear forms

$\tau_{S_{form}(q)}[u,v]=\left\langle-\frac{d^{2}}{dx^{2}}u,v\right\rangle_{L_{2}(\mathbb{R})}+\left\langle
q(x)u,v\right\rangle_{L_{2}(\mathbb{R})},\quad\mathrm{Dom}(\tau)=H^{1}(\mathbb{R}),$

acting on the dense domains

$\mathrm{Dom}\left(S_{form}(q)\right):=\left\\{u\in
H^{1}(\mathbb{R})\left|-\frac{d^{2}}{dx^{2}}u+q(x)u\in
L_{2}(\mathbb{R})\right.\right\\}$

as

$S_{form}(q)u:=-\frac{d^{2}}{dx^{2}}u+q(x)u\in L_{2}(\mathbb{R}),\quad
u\in\mathrm{Dom}\left(S_{form}(q)\right).$

######  Theorem 18.

The quasi-differential operators $S(q)$ and form-sum operators $S_{form}(q)$
coincide.

###### Proof.

Let $u\in\mathrm{Dom}\left(S(q)\right)$. Let us remember that

$\mathrm{Dom}(S(q))=\left\\{u\in
H^{1}(\mathbb{R})\left|\right.u^{\prime}-Qu\in
W_{1,loc}^{1}(\mathbb{R}),\,l_{Q}[u]\in L_{2}(\mathbb{R})\right\\},$

i.e.,

$\mathrm{Dom}(S(q))\subset\mathrm{Dom}(\tau_{S_{form}(q)})=H^{1}(\mathbb{R}).$

Then we have,

$\displaystyle\tau_{S_{form}(q)}[u,v]$
$\displaystyle=\langle-u^{\prime\prime},v\rangle_{L_{2}(\mathbb{R})}+\langle
q(x)u,v\rangle_{L_{2}(\mathbb{R})}=\langle
u^{\prime},v^{\prime}\rangle_{L_{2}(\mathbb{R})}-\langle
Q(x),\overline{u^{\prime}}v+\overline{u}v^{\prime}\rangle_{L_{2}(\mathbb{R})}$
$\displaystyle=(u^{\prime},v^{\prime})-(Qu,v^{\prime})-(Qu^{\prime},v)=(l_{Q}[u],v)\quad\forall
v\in C_{comp}^{\infty}(\mathbb{R}).$

And due to the First Representation Theorem [Kt] we conclude that

$u\in\mathrm{Dom}(S_{form}(q)),\quad\text{and}\quad S_{form}(q)u=l_{Q}[u],$

i.e.,

$S(q)\subset S_{form}(q).$

Taking into account that operators $S(q)$ and $S_{form}(q)$ are self-adjoint
from the latter we have got also the inverse inclusions

$S(q)\supset S_{form}(q).$

The proof is complete. ∎

###  3.3. Spectra of the Hill-Schrödinger operators with distributional
potentials

Here, we establish characteristic properties of a spectrum structure of the
Hill-Schrödinger operators $S(q)$ with distributional potentials $q(x)\in
H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$. By means of a limit process
in generalized sense applied to the Hill-Schrödinger operators $S(q_{n})$,
$n\in\mathbb{N}$ with smooth potentials $q_{n}(x)\in
L_{2,per}(\mathbb{R},\mathbb{R})$ (see Corollary 16) we show that the Hill-
Schrödinger operators $S(q)$ with distributions $q(x)\in
H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$ as potentials have continuous
spectra with a band and gap structure.

For different approaches see [HrMk, Krt, DjMt].

At first let us remind well known results related with the classical case of
$L_{2,per}(\mathbb{R},\mathbb{R})$-potentials $q(x)$,

(5) $q(x)\in L_{2,per}(\mathbb{R},\mathbb{R}),$

see, for an example, [DnSch2, ReSi4]. Under the assumption (5) the Hill-
Schrödinger operators $S(q)$ are lower semibounded and self-adjoint on the
Hilbert space $L_{2}(\mathbb{R})$, they have absolutely continuous spectra
with a band and gap structure.

Spectra of the Hill-Schrödinger operators are defined well by a location of
the spectral gap endpoints. It is known that for the endpoints
$\\{\lambda_{0}(q),\lambda_{k}^{\pm}(q)\\}_{k=1}^{\infty}$ of spectral gaps it
is valid the following inequalities:

(6)
$-\infty<\lambda_{0}(q)<\lambda_{1}^{-}(q)\leq\lambda_{1}^{+}(q)<\lambda_{2}^{-}(q)\leq\lambda_{2}^{+}(q)<\ldots\,.$

Spectral bands (or stability zones),

$\mathcal{B}_{0}(q):=[\lambda_{0}(q),\lambda_{1}^{-}(q)],\qquad\mathcal{B}_{k}(q):=[\lambda_{k}^{+}(q),\lambda_{k+1}^{-}(q)],\quad
k\in\mathbb{N},$

are characterized as a locus of those real $\lambda\in\mathbb{R}$ for which
all solutions of the equation

(7) $S(q)u=\lambda u$

are bounded. On the other hand, spectral gaps (or instability zones),

$\mathcal{G}_{0}(q):=(-\infty,\lambda_{0}(q)),\qquad\mathcal{G}_{k}(q):=(\lambda_{k}^{-}(q),\lambda_{k}^{+}(q)),\quad
k\in\mathbb{N},$

are a locus of those real $\lambda\in\mathbb{R}$ for which any nontrivial
solution of the equation (7) is unbounded.

As we see from the (6) it is possible situation when

$\lambda_{k}^{-}(q)=\lambda_{k}^{+}(q)$

for some $k\in\mathbb{N}$. In this case one say that the correspondent
spectral gap $\mathcal{G}_{k}(q)$ is collapsed, or closed. Note, that for
spectral bands it cannot happen.

Further, it happens that the endpoints of spectral gaps for even numbers
$k\in\mathbb{Z}_{+}$ are periodic eigenvalues of the problems on the interval
$[0,1]$:

$S_{+}(q)u:=-u^{\prime\prime}+q(x)u=\lambda u,\quad
u\in\mathrm{Dom}\left(S_{+}(q)\right),$

and the endpoints of spectral gaps for odd numbers $k\in\mathbb{N}$ are
semiperiodic eigenvalues of the problems on the interval $[0,1]$:

$S_{-}(q)u:=-u^{\prime\prime}+q(x)u=\lambda u,\quad
u\in\mathrm{Dom}\left(S_{-}(q)\right).$

Under the assumption (5) domains of operators $S_{+}(q)$ and $S_{-}(q)$ have a
view

$\mathrm{Dom}(S_{\pm}(q))=\left\\{u\in H^{2}[0,1]\left|\,u^{(j)}(0)=\pm
u^{(j)}(1),\,j=0,1\right.\right\\}.$

Now, applying a limit process in generalized sense (see Corollary 16) to the
Hill-Schrödinger operators $S(q_{n})$, $n\in\mathbb{N}$ with
$L_{2,per}(\mathbb{R},\mathbb{R})$-potentials $q_{n}(x)$ we establish the
following statement.

######  Theorem 19.

Suppose that $q(x)\in H_{per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$. Then
the Hill-Schrödinger operators $S(q)$ have continuous spectra with a band and
gap structure such that the endpoints
$\\{\lambda_{0}(q),\lambda_{k}^{\pm}(q)\\}_{k=1}^{\infty}$ of spectral gaps
satisfy the inequalities:

$-\infty<\lambda_{0}(q)<\lambda_{1}^{-}(q)\leq\lambda_{1}^{+}(q)<\lambda_{2}^{-}(q)\leq\lambda_{2}^{+}(q)<\ldots\,.$

Moreover, the endpoints of spectral gaps for even/odd numbers
$k\in\mathbb{Z}_{+}$ are periodic/semiperiodic eigenvalues of the problems on
the interval $[0,1]$,

$S_{\pm}(q)u=-u^{\prime\prime}+q(x)u=\lambda u,\quad
u\in\mathrm{Dom}\left(S_{\pm}(q)\right).$

######  Remark 20.

Operators $S_{+}(q)$ and $S_{-}(q)$ are well defined on the Hilbert space
$L_{2}(0,1)$ as lower semi-bounded, self-adjoint form-sum operators,

$S_{\pm}(q)=\left(-\frac{d^{2}}{dx^{2}}\right)_{\pm}\dotplus q(x).$

Also they can be well defined in alternative equivalent ways: as quasi-
differential ones and as a sequence limits in the norm resolvent sense of
operators with smooth potentials.

In the papers [MiMl3, MiMl4, MiMl5] the authors meticulously treated the form-
sum operators

$S_{\pm}(V)=\left((-1)^{m}\frac{d^{2m}}{dx^{2m}}\right)_{\pm}\dotplus
V(x),\quad V(x)\in H_{per}^{-m}[0,1],\,m\in\mathbb{N}$

defined on $L_{2}(0,1)$.

And in the [Mlb, MiMl1, MiMl2] it is studied two terms differential operators
of an even order defined on the negative Sobolev spaces.

###### Proof.

Let $\left\\{q_{n}(x)\right\\}_{n\in\mathbb{N}}$ is a sequence of real-valued
trigonometric polynomials which converge to the singular potential $q(x)$ by
the norm of the space $H_{per}^{-1}\left(\mathbb{R}\right)$. With this
sequence one may associate a sequence of self-adjoint operators
$\left\\{S_{\pm}(q_{n})\right\\}_{n\in\mathbb{N}}$ defined on $L_{2}(0,1)$ and
a sequence of Hill operators $\left\\{S(q_{n})\right\\}_{n\in\mathbb{N}}$
defined on $L_{2}(\mathbb{R})$. As it was proved by the authors in the [MiMl3,
MiMl5] the sequences $\left\\{S_{\pm}(q_{n})\right\\}_{n\in\mathbb{N}}$
converge to operators $S_{\pm}(q)$ in the norm resolvent sense. Hence,
eigenvalues of these operators
$\left\\{S_{\pm}(q_{n})\right\\}_{n\in\mathbb{N}}$ converge to correspondent
eigenvalues of limiting operators $S_{\pm}(q)$ [ReSi1, Theorem VIII.23 and
Theorem VIII.24] (also see [Kt]). Further, as well known [CdLv, DnSch2] for
the operators $\left\\{S_{\pm}(q_{n})\right\\}_{n\in\mathbb{N}}$ it holds the
assertions of theorem, i.e.,

(8)
$-\infty<\lambda_{0}(q_{n})<\lambda_{1}^{-}(q_{n})\leq\lambda_{1}^{+}(q_{n})<\lambda_{2}^{-}(q_{n})\leq\lambda_{2}^{+}(q_{n})<\ldots\,.$

Moreover, as we already have proved (see Corollary 16) the sequence
$\left\\{S(q_{n})\right\\}_{n\in\mathbb{N}}$ converge to the operator $S(q)$
in the norm resolvent sense. Therefore, from the (8) we get

$-\infty<\lambda_{0}(q)\leq\lambda_{1}^{-}(q)\leq\lambda_{1}^{+}(q)\leq\lambda_{2}^{-}(q)\leq\lambda_{2}^{+}(q)\leq\ldots\,,$

where $\lambda_{0}(q),\,\lambda_{2k}^{\pm}(q)\in\sigma(S_{+}(q))$ and
$\lambda_{2k-1}^{\pm}(q)\in\sigma(S_{-}(q))$, $k\in\mathbb{N}$.

Now, it remains to show that the strong inequalities

$\lambda_{k}^{+}(q_{n})<\lambda_{k+1}^{-}(q_{n}),\quad k\in\mathbb{Z}_{+}$

can not become into equalities. Really, let suppose contrary. Then, one of the
operator $S(q)$ spectral zones degenerates into the point:

$\lambda_{k_{0}}^{+}(q)=\lambda_{k_{0}+1}^{-}(q),\quad
k_{0}\in\mathbb{Z}_{+}.$

As it is isolate the operator $S(q)$ spectrum point, therefore, it cannot
belong to continuous spectrum $\sigma_{c}\left(S(q)\right)$ of the one. On the
other hand, it cannot belong to the operator $S(q)$ point spectrum as
$\sigma_{p}\left(S(q)\right)=\varnothing$. And obtained contradiction proves
the inequalities of theorem.

The proof is complete. ∎

##  4\. Concluding remarks

From the direct integral decomposition of the Hill-Schrödinger operators
$S(q)$ [HrMk] and the known Reed-Simon Theorem [ReSi4, Theorem XIII.86] follow
that $\sigma_{sc}(S(q))=\varnothing$. Therefore, the proved in this paper
continuity of the operators $S(q)$ spectra yield their absolutely continuity
[MiSb].

From Theorem C and the authors results [MiMl3] one obtain a series of theorems
establishing relationships between spectral gap lengths and a smoothness of
distributional potentials $q(x)\in H_{per}^{-s}(\mathbb{R},\mathbb{R})$,
$s\geq-1$ of the Hill-Schrödinger operators $S(q)$ [MiMl6].

Acknowledgment. The investigation of the first author was partially supported
by Ukrainian Foundation for Fundamental Research, Grant 14.1/003.

## Appendix: Some proofs

A.1. Proof of Statement 4. At first note that from the relationships

$\dot{L}_{min}(q)\subset L_{min}(q)$

follows that

$(\dot{L}_{min}(q))^{\sim}\subset L_{min}(q),$

see Proposition 3.III. Therefore, it suffices to show the inverse inclusions

$(\dot{L}_{min}(q))^{\sim}\supset L_{min}(q).$

Let $\Delta=[\alpha,\beta]$ denotes a fixed, closed interval completely
belonging to the interval $[0,1]$, and let

$\mathfrak{H}_{\Delta}:=L_{2}(\alpha,\beta).$

On the Hilbert space $\mathfrak{H}_{\Delta}$ one may consider operators
$L_{min,\Delta}(q)$ and $L_{max,\Delta}(q)$ generated by $l_{Q}[\cdot]$ on the
interval $\Delta$ which due to Proposition 3.III are mutually adjoint,

$L_{min,\Delta}^{\ast}(q)=L_{max,\Delta}(q),\qquad
L_{max,\Delta}^{\ast}(q)=L_{min,\Delta}(q).$

On the other hand the Hilbert space $\mathfrak{H}_{\Delta}$ can be well inject
into the space $\mathfrak{H}:=L_{2}(0,1)$ assuming that over the interval
$\Delta$ a function $u\in\mathfrak{H}_{\Delta}$ is equal zero. Thus, domains
$\mathrm{Dom}(L_{min,\Delta}(q))$ of operators $L_{min,\Delta}(q)$ become of a
part of domains $\mathrm{Dom}(L_{max}(q))$ of operators $L_{max}(q)$ as under
such extension of function $u\in\mathrm{Dom}(L_{min,\Delta}(q))$ over the
interval $\Delta$ a continuity of its quasi-derivatives $u^{[j]}(x)$, $j=0,1$,
is not destroy. Moreover, extended in such fashion function
$u\in\mathrm{Dom}(L_{min,\Delta}(q))$ then belong to the
$\mathrm{Dom}(\dot{L}_{min}(q))$. Therefore, if
$v\in\mathrm{Dom}(\dot{L}_{min}^{\ast}(q))$ then we have

(9)
$\left(\dot{L}_{min}^{\ast}(q)v,u\right)=\left(v,\dot{L}_{min}(q)u\right)\qquad\forall
u\in\mathrm{Dom}(L_{min,\Delta}(q)).$

As $u(x)=0$ over the interval $\Delta$ scalar product in the (9) is an
$\mathfrak{H}_{\Delta}$-inner product. Denoting these scalar products by index
$\Delta$ we can re-write (9) as following,

$\left((\dot{L}_{min}^{\ast}(q)v)_{\Delta},u\right)_{\Delta}=\left(v_{\Delta},L_{min,\Delta}(q)u\right)_{\Delta}\qquad\forall
u\in\mathrm{Dom}(L_{min,\Delta}(q)).$

Here, by $(\dot{L}_{min}^{\ast}(q)v)_{\Delta}$, $v_{\Delta}$ are denoted
functions $\dot{L}_{min}^{\ast}(q)v$ and $v$ considered only in the interval
$\Delta$. So, from the latter we obtain

$v_{\Delta}\in\mathrm{Dom}(L_{min,\Delta}^{\ast}(q))=\mathrm{Dom}(L_{max,\Delta}(q))$

and

$(\dot{L}_{min}^{\ast}(q)v)_{\Delta}=L_{min,\Delta}^{\ast}(q)v_{\Delta}=L_{max,\Delta}(q)v_{\Delta}=\left(l_{Q}[v]\right)_{\Delta}.$

As these relationships are valid for any interval $\Delta\subset[0,1]$ we
conclude that

$v\in\mathrm{Dom}(L_{max}(q)),\quad\text{and}\quad\dot{L}_{min}^{\ast}(q)v=l_{Q}[v]=L_{max}(q)v.$

Thus, it has been proved

$\dot{L}_{min}^{\ast}(q)\subset L_{max}(q),$

i.e., we have

$\dot{L}_{min}^{\ast\ast}(q)\supset L_{max}^{\ast}(q)=L_{min}(q),$

that implies the required inclusions,

$(\dot{L}_{min}(q))^{\sim}\supset L_{min}(q).$

The proof is complete.

A.2. Proof of Proposition 8. $(\verb"I")$ At first note that

(10) $\mathrm{Dom}(\dot{S}_{min}(q))\subset H_{comp}^{1}(\mathbb{R}).$

Let $u\in\mathrm{Dom}(\dot{S}_{min}(q))$, then we have

$(\dot{S}_{min}(q)u,u)=(l_{Q}[u],u)=(u^{\prime},u^{\prime})-(Qu,u^{\prime})-(Qu^{\prime},u)$

taking into account that due to the (10)

$|u^{\prime}|^{2},\,Quu^{\prime}\in L_{1,comp}(\mathbb{R}).$

Further, $(Qu,u^{\prime})$ and $(Qu^{\prime},u)$ we estimate as in the [HrMk],

$\left|(Qu,u^{\prime})\right|\leq\|Q\|_{L_{2,per}(\mathbb{R})}\left(\varepsilon\|u^{\prime}\|_{L_{2}(\mathbb{R})}+b(\varepsilon^{-1})\|u\|_{L_{2}(\mathbb{R})}\right),\quad\varepsilon\in(0,1],b\geq
0,$

that yields

$(\dot{S}_{min}(q)u,u)\geq-\gamma(\varepsilon^{-1})\|u\|_{L_{2}(\mathbb{R})}\quad\forall
u\in\mathrm{Dom}(\dot{S}_{min}(q)),\,\gamma\geq 0.$

And we can conclude that $\dot{S}_{min}(q)$ are Hermitian lower semibounded on
$L_{2}(\mathbb{R})$ operators.

Now, let show that $\mathrm{Dom}(\dot{S}_{min}(q))$ are dense in the Hilbert
space $L_{2}(\mathbb{R})$.

Obviously, it is sufficient to prove that any element $h\in\mathfrak{H}$,
$\mathfrak{H}:=L_{2}(\mathbb{R})$, which is orthogonal to
$\mathrm{Dom}(\dot{S}_{min}(q))$ is equal zero. Let suppose that $h(x)$ is
namely a such function, i.e.,

$h(x)\perp\mathrm{Dom}(\dot{S}_{min}(q)),$

and let $\Delta=[\alpha,\beta]$ is a fixed, closed interval compactly
belonging to the real axis $\mathbb{R}$ ($\Delta\Subset\mathbb{R}$). Any
element $u\in\mathrm{Dom}(S_{min,\Delta}(q))$ can be viewed as element from
$\mathrm{Dom}(\dot{S}_{min}(q))$ (with respect to the denotations see the
proof of Statement 4), consequently, $h(x)$ is orthogonal to
$\mathrm{Dom}(S_{min,\Delta}(q))$. Due to Proposition 3.II
$\mathrm{Dom}(S_{min,\Delta}(q))$ are dense in
$\mathfrak{H}_{\Delta}=L_{2}(\alpha,\beta)$, hence function $h(x)$ considered
in the interval $\Delta$ has to be equal zero almost everywhere in $\Delta$.

For an arbitrarity of the interval $\Delta\Subset\mathbb{R}$ choice we
conclude that $h(x)=0$ almost everywhere on $\mathbb{R}$.

So, statement $(\verb"I")$ of Proposition 8 has been proved completely.

$(\verb"II")$ It is obvious that operators $S_{min}(q)$ are symmetric, lower
semibouded on the Hilbert space $L_{2}(\mathbb{R})$ ones.

Make sure that operators $S_{min}(q)$ and $S_{max}(q)$ are mutually adjoint.
As $(\dot{S}_{min}(q))^{\sim}=S_{min}(q)$, therefore
$\dot{S}_{min}^{\ast}(q)=S_{min}^{\ast}(q)$, and it suffices to show that

$\dot{S}_{min}^{\ast}(q)=S_{max}(q).$

Applying the Lagrange Identity (4) we have got

$(S_{max}(q)u,v)=(u,\dot{S}_{min}(q)v)\quad\forall
u\in\mathrm{Dom}(S_{max}(q)),\,\forall v\in\mathrm{Dom}(\dot{S}_{min}(q)),$

that implies

$S_{max}(q)\subset\dot{S}_{min}^{\ast}(q).$

So, it remains to prove the inverse inclusions,

$S_{max}(q)\supset\dot{S}_{min}^{\ast}(q).$

We do it in a similar fashion as in the proof of Statement 4.

Let $v(x)$ is an arbitrary element from the domains
$\mathrm{Dom}(\dot{S}_{min}^{\ast}(q))$ of the operators
$\dot{S}_{min}^{\ast}(q)$, and let $\Delta=[\alpha,\beta]$ is a fixed, compact
interval ($\Delta\Subset\mathbb{R}$). As in the proof of Statement 4 we obtain

$\left((\dot{S}_{min}^{\ast}(q)v)_{\Delta},u\right)_{\Delta}=\left(v_{\Delta},S_{min,\Delta}(q)u\right)_{\Delta}\qquad\forall
u\in\mathrm{Dom}(S_{min,\Delta}(q)).$

So, one can conclude that

$v_{\Delta}\in\mathrm{Dom}(S_{max,\Delta}(q))$

and

$(\dot{S}_{min}^{\ast}(q)v)_{\Delta}=S_{min,\Delta}^{\ast}(q)v_{\Delta}=S_{max,\Delta}(q)v_{\Delta}=\left(l_{Q}[v]\right)_{\Delta}.$

Taking into account an arbitrarity of the interval $\Delta\subset\mathbb{R}$
choice we finally get that

$v\in\mathrm{Dom}(S_{max}(q)),\quad\text{and}\quad\dot{S}_{min}^{\ast}(q)v=l_{Q}[v]=S_{max}(q)v,$

i.e, the required inclusions

$S_{max}(q)\supset\dot{S}_{min}^{\ast}(q)$

hold.

Further, let find deficiency index of the operators $S_{min}(q)$. At first it
is necessary note as the operators $S_{min}(q)$ are lower semibounded
therefore their deficiency numbers are equal.

Let $\lambda\in\mathbb{C}$, $\mathrm{Im}\,\lambda\neq 0$. Then the operators
$S_{min}(q)$ deficiency numbers, which we will denote by $m$, are equal to the
number of linear independent solutions of the equation

$S_{min}^{\ast}(q)u=\lambda u,$

i.e., of the equation (Proposition 8.II)

$S_{max}(q)u=\lambda u.$

In other words the deficiency number is a maximal number of linear independent
solutions of the equation

$l_{Q}[u]=\lambda u$

in the Hilbert space $L_{2}(\mathbb{R})$. As a whole number of linear
independent solutions of this equation is equal 2 we conclude that

$0\leq m\leq 2.$

Assertion $(\verb"II")$ has been proved.

$(\verb"III")$ Let $u,v\in\mathrm{Dom}(S_{max}(q))$. Then applying the
Lagrange Identity (4) we conclude that there exist the limits

$[u,v]_{+\infty}:=\lim_{x\rightarrow+\infty}[u,v]_{x},\quad\text{and}\quad[u,v]_{-\infty}:=\lim_{x\rightarrow-\infty}[u,v]_{x},$

and as a consequence the Lagrange Identity (4) has got a view

(11) $(l_{Q}[u],v)-(u,l_{Q}[v])=[u,v]_{-\infty}^{+\infty}\quad\forall
u,v\in\mathrm{Dom}(S_{max}(q)).$

Further, due to Proposition 8.II it holds

$S_{min}(q)=S_{max}^{\ast}(q).$

Therefore, domains $\mathrm{Dom}(S_{min}(q))$ consist of that and only that
functions $u\in\mathrm{Dom}(S_{max}(q))$ which satisfy the relationships

$(u,S_{max}(q)v)=(S_{max}(q)u,v)\quad\forall v\in\mathrm{Dom}(S_{max}(q)).$

Together with the Lagrange Identity (11) the latter implies the required
assertion, i.e.,

$u\in\mathrm{Dom}(S_{min}(q))\Leftrightarrow[u,v]_{+\infty}-[u,v]_{-\infty}=0\quad
u\in\mathrm{Dom}(S_{max}(q))\,,\forall v\in\mathrm{Dom}(S_{max}(q)).$

Proposition 8 has been proved.

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