# Stability of equilibrium solutions of a double power reaction-diffusion
equation with a Dirac interaction

César A. Hernández M.111Email<EMAIL_ADDRESS>Tel. (55)-44-3011-5358, Fax:
(55)-11-3011-3873
Department of Mathematics, DMA-UEM
Av. Colombo, 5790 Jd. Universitário,,
CEP 87020-900, Maringá, PR, Brazil
Edgar Mayorga L.222Email<EMAIL_ADDRESS>Tel. (571)-861
5555-25041
University of Sabana
Department of Mathematics, Physics and Statistics
Chía, Colombia

(July 24, 2017)

###### Abstract

In this paper we provide detailed information about the instability of
equilibrium solutions of a nonlinear family of localized reaction-difussion
equations in dimensione one. Beyond we provide explicit formulas to the
equilibrium solutions, via perturbation method and we calculate the exact
number of positive eigenvalues of the linear operator associated to the
stability problem, which allow us to compute the dimension of the unstable
manifold.

Mathematics Subject Classification (2010). Primary 35K05, 35B10; 35B35; 35B38.
Key words. Reaction-difussion equation, Dirac interaction, Stability of
equilibrium solutions, Blow-up of solutions, Analytic perturbation.

##  1 Introduction

In this paper, we study the stability/instability of equilibrium solutions
associated to the following generalized Huxley equation with a point defect
interaction (henceforth GDH),

$u_{t}-u_{xx}=Z\delta(x)u+wu+au^{p}+bu^{2p-1},\hskip 19.91684pt\text{for
}t,x\in\mathbb{R},$ (1)

where $u=u(x,t)$, $\delta:H^{1}(\mathbb{R})\rightarrow\mathbb{R}$ defined by
$\delta(g)=g(0)$ the Dirac distribution localized at zero, and $a,b,p,Z$ are
real parameters with $p>1$.

The GDH equation has many applications. For instance, when $Z=0$,
$w=-\gamma_{1}\beta_{1}$, $a=\beta_{1}(1+\gamma_{1})$ and $b=-\beta_{1}$, with
$\beta_{1}>0$ and $\gamma_{1}\in(0,1)$, the GDH equation is reduced to the
genearalized Huxley equation, namely

$u_{t}-u_{xx}=\beta_{1}u(1-u^{p-1})(u^{p-1}-\gamma_{1}),$ (2)

which describes nerve pulse propagation in nerve fibres [12], wall motion in
liquid crystals [18], genetic population [16] and combustion [13]. In
chemistry, when $Z\neq 0$, the GDH equation can be considered as an specific
model describing the concentration $u$ of a substance distributed one
dimension space under the influence of local reaction site, bulk reaction and
trasport, see equation (II.1) in [3] and the references therein for details.

From the mathematical point of view, exact travelling solitary wave solutions,
exact equilibrium solutions, and numerical solutions of the equation (2) have
been discussed in the last years, see ([14], [18], [19], [20], [21], [17],
[23], [24], [15]). The stability/instability of travelling wave solutions and
equilibrium solutions of more general equations than the equation (2) have
been discussed widely in §5.4 of [4], and more recently in [25]. The problem
of blow-up of solutions of semilinear parabolic equations have been also
discussed in the las decades, for a survey on this subject, we refer the
reader to [11].

However, the existence and stability of equilibrium solutions as well as the
blow up of solutions of the GDH equation, when $Z\neq 0$, have not been
studied yet.

A- Equilibrium solutions of the GDH equation.

By an equilibrium solution of the equation (1), we mean a function $\phi$ in
the domain of the operator $\partial_{xx}+Z\delta(x)$, that is to say

$\phi\in{\cal D}(\partial_{xx}+Z\delta(x))=\left\\{g\in H^{1}(\mathbb{R})\cap
H^{2}(\mathbb{R}\setminus\left\\{0\right\\})|g^{\prime}(0+)-g^{\prime}(0-)=-Zg(0)\right\\}$

satisfying the differential equation

$(\partial_{xx}+Z\delta(x))\phi+w\phi+a\phi^{p}+b\phi^{2p-1}=0.$ (3)

Now, if we set $\alpha=\frac{a}{p+1}$, $\beta=\frac{b}{p}$, and

$\phi_{w,Z}(x)=\left[\frac{\alpha}{-w}+\frac{\sqrt{\alpha^{2}-\beta
w}}{-w}\text{
cosh}\left((p-1)\sqrt{-w}\left(|x|+R^{-1}\left(\frac{Z}{2\sqrt{-w}}\right)\right)\right)\right]^{\frac{-1}{p-1}},$
(4)

with $R:(-\infty,\infty)\rightarrow(-1,1)$ being the diffeomorphism defined by

$R(s)=\frac{\sqrt{\alpha^{2}-\beta w}\text{
senh}((p-1)\sqrt{-w}s)}{\alpha+\sqrt{\alpha^{2}-\beta w}\text{
cosh}((p-1)\sqrt{-w}s)},$ (5)

then, in the section 3 below, we will show that

  1. 1.

If $a>0$, and $b>0$, then the family of functions (4) are equilibrium
solutions of the GDH equation, providing

$\frac{Z^{2}}{4}<-w.$

  2. 2.

If $a>0$, and $b<0$, then the family of functions $\phi_{w,Z}$ are equilibrium
solutions of the GDH equation, providing

$\frac{Z^{2}}{4}<-w<-\frac{pa^{2}}{(p+1)^{2}b}.$

######  Remark 1.

When $Z=0$, $p=2$, $w=-\gamma_{1}$, $a=1+\gamma_{1}$, $b=-1$, the solution
$\phi_{w,Z}$ is the equilibrium solution of the Nagumo equation, see formula
$(7)$ in [22]. When $Z=0$, $p=3$, $a=1$, $b=1$, $\phi_{w,Z}$ is the
equilibrium solution appearing in the formula $(32)$ of [5].

B- Stability/Instability of the equilibrium solutions of GDH equation.

The equilibrium solution $\phi_{w,Z}$ given in (4), (5) is _stable_ in
$H^{1}(\mathbb{R})$ by the flow of the GDH equation (1), if for every
$\epsilon>0$ there exists $\delta>0$ such that,

$\text{if}\hskip 8.5359ptg\in H^{1}(\mathbb{R})\hskip
8.5359pt\text{with}\hskip 8.5359pt||g-\phi_{w,Z}||_{1}<\delta,\hskip
8.5359pt\text{then}\hskip 8.5359pt||u(t)-\phi_{w,Z}||_{1}<\epsilon,$

for all $t>0$, here $u(t)$ denotes the solution of the equation (1) generated
by the initial data $u(0)=g$. Otherwise, $\phi_{w,Z}$ is _unstable_.

We are now in position to establish the principal result of this note.

######  Theorem 1.

Let $a,b,p,Z,w$ parameters satisfying the conditions 1-2 above, then the
equilibrium solutions $\phi_{w,Z}$ are unstable. In addition,

  1. 1.

For $Z<0$, the unstable manifold associated to $\phi_{w,Z}$ has dimension $2$.

  2. 2.

For $Z>0$, the unstable manifold associated to $\phi_{w,Z}$ has dimension $1$.

The proof of the Theorem 1 can be obtained in the classical way, by analysing
the spectrum of the linear self-adjoint operator

$\left\\{\begin{aligned} &{\cal L}_{Z}:{\cal
D}(\partial_{xx}+Z\delta(x))\rightarrow L^{2}(\mathbb{R})\\\ &{\cal
L}_{Z}g=(\partial_{xx}+Z\delta(x))g+wg+ap\phi_{w,Z}^{p-1}g+b(2p-1)\phi_{w,Z}^{2p-2}g,\end{aligned}\right.$
(6)

which is the linear approximation of the function

$G(u)=u_{xx}+Z\delta(x)u+wu+au^{p}+bu^{2p-1},$

at $\phi_{w,Z}$, i.e., $G^{\prime}(\phi_{w,Z})g={\cal L}_{Z}g$, see Theorem
5.1.1, 5.1.3, and 5.2.1 in [4]. In general, to count the number of positive
eigenvalues of a linear operator is a delicate issue. In the case of the self-
adjoint operator $\mathcal{L}_{Z}$ in (6) our strategy is based in two basic
facts. First, if one is $Z=0$, the spectrum of the self-adjoint operator
$\mathcal{L}_{0}$ defined by

$\mathcal{L}_{0}=\frac{d^{2}}{dx^{2}}+w+ap\phi^{p-1}_{w,0}+b(2p-1)\phi^{2p-2}_{w,0}$

with domain $H^{2}(\mathbb{R})$ is well-known: there is only one positive
eigenvalue which is simple, zero is a simple eigenvalue with eigenfunction
$\frac{d}{dx}\phi_{\omega,0}$. The rest of the spectrum is negative and away
from zero. Second, if $Z$ is small, $\mathcal{L}_{Z}$ can be considered as a
real-holomorphic perturbation of $\mathcal{L}_{0}$. So, we have that the
spectrum of $\mathcal{L}_{Z}$ depends holomorphically on the spectrum of
$\mathcal{L}_{0}$. Then we obtain that for $Z<0$ there are exactly two
positive eigenvalues of $\mathcal{L}_{Z}$ and exactly one for $Z>0$. We refer
the reader to Section 4 for the precise details on these statements.

######  Remark 2.

When $Z=0$, the proof of the instability of the equilibrium solution
$\phi_{w,0}$ can be obtained as a consequence of the Theorem 5.1.3 in [4] and
from the spectral properties of the operator ${\cal L}_{0}$ presented for
instance in Theorem B.61 in [2].

This paper is organized as follows. In section 2, we establish a local and
global well-posedness theory for the GDH equation, in addition we establish
the existence of solutions that blow up in finite time. Section 3 describes
the construction of the profile $\phi_{\omega,Z}$ in (4) for $w,Z,a,b$
satisfying the conditions in the Theorem 5 below. Section 4 describes the
spectral theory for the operators $\mathcal{L}_{Z}$ in (6).

##  2 Local and global well-posedness for the GDH equation

In this section we discuss some results about the local and global well-
posedness problem associated to the GDH equation in $H^{1}(\mathbb{R})$

$\left\\{\begin{array}[]{lll}\displaystyle
u_{t}+(A_{Z}-w)u=au^{p}+bu^{2p-1},\\\ u(0)=g\in
H^{1}(\mathbb{R}),\end{array}\right.$ (7)

where

$A_{Z}:=-\frac{d^{2}}{dx^{2}}-Z\delta(x).$ (8)

Since our approach will be based in the abstract results in §3 of [4], we will
establish the necessary framework. Initially, we recall that the formal
expression in (8) can be understood as the family of self-adjoint operators
with domain

$D(A_{Z})=\left\\{g\in H^{1}(\mathbb{R})\cap
H^{2}(\mathbb{R}-\left\\{0\right\\})|g^{\prime}(0+)-g^{\prime}(0-)=-Zg(0)\right\\},$

which represent all the self-adjoint extensions associated to the following
closed, symmetric, densely defined linear operator (see [1]):

$\left\\{\begin{aligned} A_{0}&=-\frac{d^{2}}{dx^{2}}\\\ D(A_{0})&=\\{g\in
H^{2}(\mathbb{R}):g(0)=0\\}.\end{aligned}\right.$

Moreover, for $Z\in\mathbb{R}$ we have that the essential spectrum of $A_{Z}$
is the nonnegative real axis, $\Sigma_{ess}(A_{Z})=[0,+\infty)$. For $Z>0$,
$A_{Z}$ has exactly one negative, simple eigenvalue, i.e., its discrete
spectrum $\Sigma_{dis}(A_{Z})$ is $\Sigma_{dis}(A_{Z})=\\{{-Z^{2}/4}\\}$, with
a strictly (normalized) eigenfunction
$\Psi_{Z}(x)=\sqrt{\frac{Z}{2}}e^{-\frac{Z}{2}|x|}$. For $Z\leqq 0$, $A_{Z}$
has not discrete spectrum, $\Sigma_{dis}(A_{Z})=\emptyset$. Therefore the
operators $A_{Z}$ are bounded from below,

$\left\\{\begin{aligned} &A_{Z}\geq-Z^{2}/4,\hskip 14.22636pt&Z>0\ \\\
&A_{Z}\geq 0&Z\leq 0.\end{aligned}\right.$ (9)

######  Theorem 2.

For any $u_{0}\in H^{1}(\mathbb{R})$, there exists $T>0$ and a unique solution
$u$ of $(\ref{cachy12})$ such that $u\in C^{1}([0,T);H^{1}(\mathbb{R}))$ and
$u(0)=u_{0}$. For each $T_{0}\in(0,T)$ the mapping

$u_{0}\in H^{1}(\mathbb{R})\to u\in C([0,T);H^{1}(\mathbb{R}))$

is continuous. If an initial data $u_{0}$ is even the solution $u(t)$ is also
even.

###### Proof.

The proof of this theorem is an application of Theorem 3.3.3 in [4]. First of
all, from (9), we have the self-adjoint operator ${\cal A}\equiv
A_{Z}-w+a_{1}$ on the space $X=L^{2}(\mathbb{R})$, with $a_{1}>Z^{2}/4+w$ for
$Z>0$ and $a_{1}>w$ for $Z\leqq 0$, and domain $D({\cal A})=D(A_{Z})$,
satisfies $\Sigma({\cal A})>0$. Secondly, in our case it is possible to
consider the space $X^{1/2}=H^{1}(\mathbb{R})$ with norm

$||u||^{2}_{1/2}=||u_{x}||_{L^{2}}^{2}+(-w+a_{1})||u||_{L^{2}}^{2}-Z|u(0)|^{2},$

which is equivalent to the usual norm in $H^{1}(\mathbb{R})$. Lastly, it is
well known that the function $u\in H^{1}(\mathbb{R})\rightarrow
f(u)=au^{p}+bu^{2p-1}$ is locally Lipschitzian. ∎

Now, for the case $Z=0$ in (7) is well-known that the double-power
nonlinearity induce restrictions on the existence of global solutions. The
following theorem shows that a similar picture happens for $Z\neq 0$.

######  Theorem 3.

i) For $Z>0$, and $\frac{Z^{2}}{4}<-w$. The solution of the Cauchy problem
$(\ref{cachy12})$ is globally well defined in $H^{1}(\mathbb{R})$ providing
the norm of the initial data $u(0)=g$ small in $H^{1}(\mathbb{R})$.
ii) For $Z\leq 0$, and $0<-w$. The solution of the Cauchy problem
$(\ref{cachy12})$ is globally well defined in $H^{1}(\mathbb{R})$ providing
the norm of the initial data $u(0)=g$ small in $H^{1}(\mathbb{R})$.

###### Proof.

This result is a consequence of the stability of the equilibrium solution
$u\equiv 0$ in $H^{1}(\mathbb{R})$ that can be obtained via spectral analysis
of the linear operator $A\equiv-A_{Z}+w$, and via Theorem 5.1.1 in [4].
Indeed, from (9), we deduce that

  1. i)

For $Z>0$, $\Sigma(-A_{Z}+w)=(-\infty,w]\cup\\{w+\frac{Z^{2}}{4}\\}$.

  2. ii)

For $Z\leq 0$, $\Sigma(-A_{Z}+w)=(-\infty,w]$.

Then the spectrum of the operator $A$ is negative and away from zero. The rest
of the hypothesis of the Theorem 5.1.1 were discussed above. ∎

Now, let $u$ be the solution of the Cauchy problem (7), let $Z>0$, and
consider $t\to R(t)\in\mathbb{R}$ the function defined by the expression

$R(t)=\int_{-\infty}^{\infty}u(x,t)e^{\frac{-Z|x|}{2}}dx.$ (10)

In addition, consider

$\lambda=w+Z^{2}/4<0,\hskip 11.38092pt\beta=a(Z/4)^{\frac{p-1}{p}},\hskip
11.38092pt\gamma=b(Z/4)^{\frac{2(p-1)}{2p-1}},\hskip
11.38092pt\text{and}\hskip
11.38092ptz_{1}=\frac{-\beta\lambda^{-1}-\sqrt{(\beta\lambda^{-1})^{2}-4\gamma\lambda^{-1}}}{2}.$

Then, the result below establishes the blow up of solutions of the Cauchy
problem $(\ref{cachy12})$ for specific values of the parameters.

######  Theorem 4.

For $Z>0$, $a,b>0$ and $\frac{Z^{2}}{4}<-w$. The solution of the Cauchy
problem (7) with initial positive data $u(0)=g$ blows up in finite time
providing

$\int_{-\infty}^{\infty}g(x)e^{-\frac{Z}{2}|x|}dx>R_{1},\hskip
28.45274pt\text{with}\hskip
28.45274ptR_{1}=\left[\frac{z_{1}\lambda}{\gamma}\right]^{\frac{1}{p-1}}.$

Moreover, if $T$ denotes the time where the solution blows up, we have that

$T\leq\int_{R(0)}^{\infty}\frac{1}{\lambda s+\beta s^{p}+\gamma s^{2p-1}}ds.$

###### Proof.

Let $\phi(x)=e^{-\frac{Z|x|}{2}}$, from (10), we obtain

$\displaystyle R^{\prime}(t)$
$\displaystyle=\int_{-\infty}^{\infty}u_{t}\phi(x)dx=\int_{-\infty}^{\infty}[u_{xx}+Z\delta(x)+wu+au^{p}+bu^{2p-1}]\phi(x)dx.$

Now, since $Z^{2}/4$ is an eigenvalue of the self-adjoint operator $-A_{Z}$,
with associated eigenfunction $\phi$, then we have

$\int_{-\infty}^{\infty}[-A_{Z}u+wu]\phi(x)dx=\left(\frac{Z^{2}}{4}+w\right)\int_{-\infty}^{\infty}u\phi(x)dx.$
(11)

On the other hand, Holder inequality and the positivity of the solution $u$
imply that

$\int_{-\infty}^{\infty}[au^{p}+bu^{2p-1}]\phi(x)dx\geq
a||\phi||_{L^{1}}^{\frac{1-p}{p}}R^{p}+b||\phi||_{L^{1}}^{\frac{2(1-p)}{2p-1}}R^{2p-1}.$
(12)

Since $||\phi||_{L^{1}}=4/Z$, from (11), (12) we get the following
differential inequality

$\lambda R(t)+\beta R^{p}(t)+\gamma R^{2p-1}(t)\leq R^{\prime}(t).$ (13)

By standard arguments, it is possible to prove that if $R(0)>R_{1}$, with
$R_{1}$ being the unique positive constant solution of (13), then, there
exists $T>0$, such that,

$T\leq\int_{R(0)}^{\infty}\frac{1}{\lambda s+\beta s^{p}+\gamma
s^{2p-1}}ds\hskip 28.45274pt\text{and}\hskip 28.45274pt\lim_{t\to
T}R(t)=\infty.$ (14)

where, $h(s)=\lambda s+\beta s^{p}+\gamma s^{2p-1}$. Now, since

$R(t)\leq||u(t)||_{L^{2}}||\phi||_{L^{2}}=||u(t)||_{L^{2}}\left[\frac{2}{Z}\right]^{\frac{1}{2}},$
(15)

from (14) and (15), we conclude

$\lim_{t\to T}||u(t)||_{L^{2}}=\infty.$

∎

######  Remark 3.

Following the ideas and notations of Theorem 4, it is possible to prove that
for $Z>0$, $a,b>0$, and $\frac{Z^{2}}{4}\geq-w$, the solution of the Cauchy
problem (7) with initial positive data $u(0)=g$ blows up in finite time.

######  Remark 4.

Let $Z>0$, $a,b>0$, $\frac{Z^{2}}{4}<-w$, and $\phi_{w,Z}$ given in (4) the
positive equilibrium solution of the Cauchy problem (7) with
$g\equiv\phi_{w,Z}$. Then, as a consequence of the Theorem 4, we conclude that

$\int_{-\infty}^{\infty}\phi_{w,Z}(x)e^{-\frac{Z}{2}|x|}dx\leq R_{1},\hskip
28.45274pt\text{with}\hskip
28.45274ptR_{1}=\left[\frac{z_{1}\lambda}{\gamma}\right]^{\frac{1}{p-1}}.$

##  3 Equilibrium solutions of the GDH equation

In this section, we deal with the deduction of the explicit equilibrium
solutions given in (4), (5) for the GDH equation (1) with $a>0$, $b\neq 0$. We
consider the cases, $Z=0$ and $Z\neq 0$. For $Z=0$ in (3), we have that
$\phi\equiv\phi_{w}$ satisfies the nonlinear elliptic equation

$\phi^{\prime\prime}+w\phi+a\phi^{p}+b\phi^{2p-1}=0.$ (16)

The quadrature method and the boundary condition of the function
$\phi\rightarrow 0$ as $|x|\rightarrow\infty$, imply that

$[\phi^{\prime}]^{2}+w\phi^{2}+2\alpha\phi^{p+1}+\beta\phi^{2p}=0,$ (17)

where, $\alpha=a/(p+1)$, $\beta=b/p$. In order to obtain an explicit solution
of the equation (16), we will assume that $w<0$ and $0<\phi$. Replacing

$y=\phi^{p-1},$

in (17), we obtain

$\frac{1}{(p-1)\sqrt{-w}}\bigintssss{\frac{dy}{y\sqrt{1+2\alpha w^{-1}y+\beta
w^{-1}y^{2}}}}=x$ (18)

so, by assuming $c$ a positive constant, we have the formula

$\bigintssss{\frac{dy}{y\sqrt{1+2\alpha w^{-1}y+\beta
w^{-1}y^{2}}}}=-\ln\left[c\left(\frac{\alpha y+w}{-y}+\frac{\sqrt{\beta
wy^{2}+2w\alpha y+w^{2}}}{y}\right)\right].$

Now, for $c=1/\sqrt{\alpha^{2}-\beta w}$, and recalling that

$\text{arccosh}(x)=\ln(x-\sqrt{x^{2}-1}),\hskip 14.22636pt\text{for all}\hskip
14.22636ptx\geq 1,$

we can rewrite the integral in (18) as

$\bigintssss{\frac{dy}{y\sqrt{1+2\alpha w^{-1}y+\beta
w^{-1}y^{2}}}}=-\text{arccosh}\left(\frac{\alpha y+w}{-y\sqrt{\alpha^{2}-\beta
w}}\right).$

Note that a solution of the equation (16) is given implicitly by the formula

$-\text{arccosh}\left(\frac{\alpha\phi^{p-1}+w}{-\phi^{p-1}\sqrt{\alpha^{2}-\beta
w}}\right)=(p-1)\sqrt{-w}x,$

or

$\phi(x)=\left[\frac{-w}{\alpha+\sqrt{\alpha^{2}-\beta w}\text{
cosh}((p-1)\sqrt{-w}x}\right]^{\frac{1}{p-1}}=\left[\frac{\alpha}{-w}+\frac{\sqrt{\alpha^{2}-\beta
w}}{-w}\text{ cosh}\left((p-1)\sqrt{-w}x\right)\right]^{\frac{-1}{p-1}}$ (19)

with $-w>0$ and $\alpha^{2}-\beta w>0$. Now, we notice that if $b>0$, the
solution $\phi$ in (19) is well defined for all $w<0$. In contrast, if $b<0$,
the solution $\phi$ in (19) is well defined for $w$ satisfying the following
condition

$0<-w<-\frac{pa^{2}}{(p+1)^{2}b}.$

Now, we proceed to calculate solutions of the equation (3) when $Z\neq 0$. The
following lemma shows us some of the properties that a solution $\phi\in
H^{1}(\mathbb{R})$ of the equation (3) must satisfy.

######  Lemma 1.

Let $\phi\in H^{1}(\mathbb{R})$ be a solution of (3), then $\phi$ satisfies
the following properties,

$\displaystyle\phi\in C^{j}(\mathbb{R}-\\{0\\})\cap C(\mathbb{R}),\text{
}\text{ }j=1,2.$ (20a)
$\displaystyle\phi^{\prime\prime}(x)+w\phi(x)+a\phi^{p}(x)+b\phi^{2p-1}(x)=0,\hskip
17.07182pt\text{for all}\hskip 17.07182ptx\neq 0.$ (20b)
$\displaystyle\phi^{\prime}(0+)-\phi^{\prime}(0-)=-Z\phi(0).$ (20c)
$\displaystyle\phi^{\prime}(x),\phi(x)\rightarrow 0,\hskip
17.07182pt\text{if}\hskip 8.5359pt|x|\rightarrow\infty.$ (20d)

###### Proof.

The proof of this lemma follows the ideas of the proof of Lemma 3.1 in [6].
The properties (20a) and (20d) are proved by a standard boostrap argument,
namely, for all $\xi\in C_{0}^{\infty}(\mathbb{R}\setminus\\{0\\})$, the
function $\xi\phi$ satisfies

$(\xi\phi)^{\prime\prime}+w(\xi\phi)=\xi^{\prime\prime}\phi+2\xi^{\prime}\phi^{\prime}-a\xi\phi^{p}-b\xi\phi^{2p-1}$

in the sense of distributions. Since the right hand side of the previous
identity is in $L^{2}(\mathbb{R})$, then $\xi\phi\in H^{2}(\mathbb{R})$, that
is to say, $\phi\in H^{2}(\mathbb{R}\setminus\\{0\\})\cap
C^{1}(\mathbb{R}\setminus\\{0\\})$. The equation (20b) follows from the fact
that $C_{0}^{\infty}(\mathbb{R}\setminus\\{0\\})$ is dense in
$L^{2}(\mathbb{R})$. In relation to (20c), it is enough to “integrate” (3)
from $-\epsilon$ to $\epsilon$,

$\int_{-\epsilon}^{\epsilon}\phi^{\prime\prime}dx+w\int_{-\epsilon}^{\epsilon}\phi
dx+Z\int_{-\epsilon}^{\epsilon}\delta(x)\phi
dx+\int_{-\epsilon}^{\epsilon}a\phi^{p}+b\phi^{2p-1}dx=0.$

If $\epsilon\to 0$, we obtain that
$\phi^{\prime}(0+)-\phi^{\prime}(0-)=-Z\phi(0)$. ∎

We notice that the function

$\phi_{s}(x):=\phi(|x|-s),\hskip 14.22636pts\in\mathbb{R},$ (21)

where $\phi$ given in (19), satisfies all of the properties of the previous
lemma except possibly the jump condition (20c). Since $\phi_{s}$ is an even
function the condition (20c) can be rewritten as

$\phi_{s}^{\prime}(0+)=-\frac{Z}{2}\phi_{s}(0),\hskip 14.22636pt\text{or
equivalently,}\hskip 14.22636pt\phi^{\prime}(s)=\frac{Z}{2}\phi(s).$

From (19), we obtain that

$\frac{\sqrt{\alpha^{2}-\beta w}\text{
senh}((p-1)\sqrt{-w}s)}{\alpha+\sqrt{\alpha^{2}-\beta w}\text{
cosh}((p-1)\sqrt{-w}s)}=\frac{-Z}{2\sqrt{-w}}.$ (22)

If we define $R:(-\infty,\infty)\rightarrow(-1,1)$ by

$R(s)=\frac{\sqrt{\alpha^{2}-\beta w}\text{
senh}((p-1)\sqrt{-w}s)}{\alpha+\sqrt{\alpha^{2}-\beta w}\text{
cosh}((p-1)\sqrt{-w}s)}.$ (23)

We have that $R$ is an odd, increassing diffeomosphism between the intervals
$(-\infty,\infty)$ and (-1,1). In particular, from the expression (22), we
conclude that

$\frac{Z^{2}}{4}<-w,$

and furthermore, that

$s=R^{-1}\left(\frac{-Z}{2\sqrt{-w}}\right).$ (24)

Finally, from (19), (21) and (24), we can conclude that for
$\alpha=\frac{a}{p+1}$ and $\beta=\frac{b}{p}$, the function $\phi_{w,Z}$
given by

$\phi_{w,Z}(x)=\left[\frac{\alpha}{-w}+\frac{\sqrt{\alpha^{2}-\beta
w}}{-w}\text{
cosh}\left((p-1)\sqrt{-w}\left(|x|+R^{-1}\left(\frac{Z}{2\sqrt{-w}}\right)\right)\right)\right]^{\frac{-1}{p-1}}$
(25)

is a solution of the equation (3) providing

  1. i)

For $a>0$, $b>0$: $\frac{Z^{2}}{4}<-w$.

  2. ii)

For $a>0$, $b<0$: $\frac{Z^{2}}{4}<-w<-\frac{pa^{2}}{(p+1)^{2}b}$.

We observe that if $Z=0$ in the previous formula, we recover the function
$\phi$ given in (19), namely $\phi_{w,0}=\phi$. Thus, we can establish an
existence result of equilibrium solutions for (3).

######  Theorem 5.

i) For $a>0$, $b>0$ and $\frac{Z^{2}}{4}<-w$, the family of functions
$\phi_{w,Z}$ given in $(\ref{numeratorwithZ})$ are equilibrium solutions of
the equation (1). Moreover, the mapping
$Z\in(-2\sqrt{-w},2\sqrt{-w})\rightarrow\phi_{w,Z}$ is a real analytic
function.

ii) For $a>0$, $b<0$ and $\frac{Z^{2}}{4}<-w<-\frac{pa^{2}}{(p+1)^{2}b}$, the
family of functions $\phi_{w,Z}$ given in (25) are equilibrium solutions of
the equation (1). Moreover, the mapping
$Z\in(-2\sqrt{-w},2\sqrt{-w})\rightarrow\phi_{w,Z}$ is a real analytic
function.

Figure 1 shows the profile of $\phi_{\omega,Z}$ in (25) in the case $p=5$,
$a=6$ $b=-1$.

(a) $\phi_{w,Z}$: $w=-4$, $Z=-2$.

(b) $\phi_{w,Z}$: $w=-4$, $Z=2$.

(c) $\phi_{w,Z}$: $w=-4$, $Z=0$.

Figure 1: The equilibrium solutions $\phi_{w,Z}$ for $Z<0$, $Z=0$ and $Z>0$.

######  Remark 5.

Considering $b=0$, $a>0$, and $\frac{Z^{2}}{4}<-w$, the function $\phi_{w,Z}$
in (25) can be written in the following form:

$h_{w,Z}(x)=\left[\frac{(p+1)(-w)}{2a}{\rm
sech}^{2}\left(\frac{(p-1)\sqrt{-w}}{2}|x|+\tanh^{-1}\left(\frac{Z}{2\sqrt{-w}}\right)\right)\right]^{\frac{1}{p-1}},$

it is an equilibrium solution of the GDH equation, for $b=0$.

##  4 Spectral properties of ${\cal L}_{Z}$

This section is devoted to the study of the spectral properties of the
operators $\mathcal{L}_{Z}$ given in (6). We establish the relation between
the second variation of the functional
$S_{w,Z}:H^{1}(\mathbb{R})\to\mathbb{R}$ defined by

$S_{w,Z}(u)=\int\limits_{\mathbb{R}}\left(\frac{(u^{\prime})^{2}}{2}-\frac{wu^{2}}{2}-\frac{au^{p+1}}{p+1}-\frac{bu^{2p}}{2p}\right)dx-\frac{Zu^{2}(0)}{2},$

at $\phi_{\omega,Z}$ and the self-adjoint operators $\mathcal{L}_{Z}$. It can
be easily verified that the equilibrium solution $\phi_{w}=\phi_{w,Z}$ is a
critical point of $S_{w,Z}$. Indeed, for $u,v\in H^{1}(\mathbb{R})$,

$S_{w,Z}^{\prime}(u)v=\left.\frac{d}{dt}S_{\omega,Z}(u+tv)\right|_{t=0}=\int\limits_{\mathbb{R}}u^{\prime}v^{\prime}\,dx-\int\limits_{\mathbb{R}}(wu+au^{p}+bu^{2p-1})v\,dx-
Zu(0)v(0).$

Since $\phi_{w}$ satisfies (3), $S_{w,Z}^{\prime}(\phi_{\omega})=0$. In the
same way, we get

$S_{w,Z}^{\prime\prime}(\phi_{w})(u_{1},v_{1})=\int\limits_{\mathbb{R}}u_{1}^{\prime}v_{1}^{\prime}dx-\int\limits_{\mathbb{R}}(w+ap\phi^{p-1}_{w}+b(2p-1)\phi^{2p-2}_{w})u_{1}v_{1}\,dx-
Zu_{1}(0)v_{1}(0)$ (26)

with $D(S_{w,Z}^{\prime\prime}(\phi_{w}))=H^{1}(\mathbb{R})\times
H^{1}(\mathbb{R})$. Note that the form $S_{w,Z}^{\prime\prime}(\phi_{w})$ is
bilinear bounded from below and closed. Therefore, by the First Representation
Theorem (see [7, Chapter VI, §2.1]), it defines an operator $-\mathcal{L}_{Z}$
such that

$\left\\{\begin{aligned} &D(\mathcal{-L}_{Z})=\\{v_{1}\in
H^{1}(\mathbb{R}):\exists v_{2}\in L^{2}(\mathbb{R})\;s.t.\;\forall z\in
H^{1}(\mathbb{R}),\;S_{w,Z}^{\prime\prime}(\phi_{w})(v_{1},z)=(v_{2},z)\\},\\\
&\mathcal{-L}_{Z}v_{1}=v_{2}.\end{aligned}\right.$ (27)

######  Theorem 6.

The operator $-\mathcal{L}_{Z}$ determined in (27) is given by

$-\mathcal{L}_{Z}=-\frac{d^{2}}{dx^{2}}-\omega-
ap\phi^{p-1}_{w}-b(2p-1)\phi^{2p-2}_{w},$

on the domain $D_{Z}:=D\left(-{\cal L}_{Z}\right)=\left\\{g\in
H^{1}(\mathbb{R})\cap
H^{2}(\mathbb{R}-\left\\{0\right\\})|g^{\prime}(0+)-g^{\prime}(0-)=-Zg(0)\right\\}$.

###### Proof.

The proof of this theorem follows the same lines as in Le Coz et al. [8]. ∎

Now, we proceed to establish some specific spectral properties of the operator
$-{\cal L}_{Z}$. Here we will consider the parameters $a,b$, $w$ and $Z$ such
that satisfy the relations in Theorem 5.

######  Theorem 7.

For $w<0$ and $\phi_{w,0}$ being the equilibrium solution $\phi_{w,Z}$ with
$Z=0$, we consider the self-adjoint linear operator $-{\cal
L}_{0}:H^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$

$-{\cal
L}_{0}=-\frac{d^{2}}{dx^{2}}-w-ap\phi^{p-1}_{w,0}-b(2p-1)\phi^{2p-2}_{w,0}.$

Then $-{\cal L}_{0}$ has a unique negative simple eigenvalue $-\lambda$, with
$\lambda>0$. Zero is a simple eigenvalue with eigenfunction $\phi^{\prime}$.
The rest of the espectrum is away from zero and the essential spectrum is
given by the interval $[-w,\infty)$.

###### Proof.

This result is a consequence of the classical Sturm Liouville oscillation
theory (see [10]). ∎

Now, we proceed to study the spectral properties of the operator ${\cal
L}_{Z}$ for $Z\neq 0$.

######  Lemma 2.

For $Z$ satisfying the conditions in Theorem 5 and $Z\neq 0$, the kernel of
${\cal L}_{Z}$ is trivial.

###### Proof.

Let $v\in Ker({\cal L}_{Z})$. It is clear that all elements in the kernel of
the operator ${\cal L}_{Z}$ are solutions of

$\left\\{\begin{aligned} &{\cal L}_{Z}\phi(x)=0,\hskip 14.22636ptx>0.\\\
&\phi\in L^{2}((0,\infty)).\end{aligned}\right.$ (28)

It is well known that the linear problem (28) has dimension one. Since
$\phi_{w,Z}^{\prime}$ satisfies the problem (28) then we conclude that there
exists $\alpha\in\mathbb{R}$ such that $v(x)=\alpha\phi^{\prime}_{w,Z}(x)$,
for all $x>0$. A similar argument can be applied to the problem

$\left\\{\begin{aligned} &{\cal L}_{Z}\phi(x)=0,\hskip 14.22636ptx<0.\\\
&\phi\in L^{2}((-\infty,0)).\end{aligned}\right.$

Thus, there exists a real number $\beta$ such that
$v(x)=\beta\phi^{\prime}_{w,Z}(x)$, for $x<0$. From the continuity of $v$ and
the parity of the function $\phi_{w,Z}$, we deduce that $\alpha=-\beta$ so
that $v$ is writen as

$v(x)=\begin{cases}\alpha\phi^{\prime}_{w,Z}(x),&\text{if}\hskip 8.5359ptx\geq
0,\\\ -\alpha\phi^{\prime}_{w,Z}(x)&\text{if}\hskip 8.5359ptx<0.\end{cases}$
(29)

Since $v\in{\cal D}({\cal L}_{Z})$, it follows that
$v^{\prime}(0+)-v^{\prime}(0-)=-Zv(0)$. From (29)

$v^{\prime}(0+)-v^{\prime}(0-)=\alpha\phi_{w,Z}^{\prime\prime}(0+)+\alpha\phi_{w,Z}^{\prime\prime}(0-)=-Z\alpha\phi_{w,Z}^{\prime}(0+).$
(30)

Suppose that $\alpha\neq 0$, from (20b) and (30), we obtain that
$\phi_{w,Z}^{\prime\prime}(0+)=-Z/2\phi_{w,Z}^{\prime}(0+)$. Multiplying the
equation (20b) by $g^{\prime}$ and integrating on the interval $(0,R)$, we get

$-\frac{1}{2}(g^{\prime}(R))^{2}+\frac{1}{2}(g^{\prime}(0+))^{2}-F(g(R))+F(g(0+))=0$

where $F(s)=ws^{2}/2+as^{p+1}/(p+1)+bs^{2p}/(2p)$. If $R\rightarrow\infty$
equation (20d) imply that

$\frac{1}{2}(g^{\prime}(0+))^{2}+F(g(0+))=0.$ (31)

Since $\phi_{w,Z}$ satisfies the equation (20b), from (31), we get that
$\frac{1}{2}(\phi_{w,Z}^{\prime}(0+))^{2}=-F(\phi_{w,Z}(0))$. In addition, as
$\phi_{w,Z}$ is an even function, we obtain that
$\phi_{w,Z}^{\prime}(0+)=\frac{-Z}{2}\phi_{w,Z}(0)$. Therefore, we deduce that
$\phi_{w,Z}(0)$ is a positive zero of the following function

$P(s)=\frac{Z^{2}}{8}s^{2}+w\frac{s^{2}}{2}+a\frac{s^{p+1}}{p+1}+b\frac{s^{2p}}{2p}.$
(32)

On the other hand, from the equation (3), we have that

$\lim_{x\rightarrow
0+}\phi_{w,Z}^{\prime\prime}(x)=\phi_{w,Z}^{\prime\prime}(0+)=-w\phi_{w,Z}(0)-a\phi^{p}_{w,Z}(0)-b\phi^{2p-1}_{w,Z}(0).$

Since,

$\phi_{w,Z}^{\prime\prime}(0+)=\frac{Z^{2}}{4}\phi_{w,Z}(0),$

it follows that $\phi_{w,Z}(0)$ is a positive zero of the function

$R(s)=\frac{Z^{2}}{4}s+ws+as^{p}+bs^{2p-1}.$ (33)

Since $s_{0}=\phi_{w,Z}(0)$ is a zero of both (32) and (33), we deduce that

$s_{0}^{p-1}=-\frac{ap}{b(p+1)}.$ (34)

Notice that $s_{0}=\phi_{w,Z}(0)$ is positive, then (34) gives us a
contradiction if $a,b>0$. Now, in the case $a>0$, $b<0$, from (19) and (25)
the relation $\phi^{p-1}_{w,Z}(0)\leq\phi^{p-1}(0)$ is obtained. Since
$-w<-\frac{pa^{2}}{(p+1)^{2}b}$ from (19), we get

$\phi^{p-1}(0)=\frac{-w}{\alpha}\frac{1}{1+\sqrt{1-\beta
w\alpha^{-2}}}<-\frac{ap}{b(p+1)}\frac{1}{1+\sqrt{1-\beta w\alpha^{-2}}}$

which is also a contradiction. Therefore, we conclude that $\alpha=0$ and then
$v\equiv 0$. ∎

Now we show that the family of operators $-{\cal L}_{Z}$ depends analytically
of the variable $Z$, where $Z$ satisfies the conditions of Theorem 5.

######  Lemma 3.

As a function of the variable $Z$, $\left\\{-{\cal
L}_{Z}\right\\}_{Z\in\mathbb{R}}$ is a real analytic family of self-adjoint
operators of type (B) in the sense of Kato.

###### Proof.

By the Theorem VII-4.2 in [7] and [9], it is enough to show that the family of
bilinear forms $S_{w,Z}^{\prime\prime}(\phi_{w})$ defined in (26) are real
analytic of type (B). Namely

  1. 1.

The domain $D\left(S_{w,Z}^{\prime\prime}(\phi_{w})\right)$ of the forms
$S_{w,Z}^{\prime\prime}(\phi_{w})$ is independent of the parameter $Z$. In our
case, $D\left(S_{w,Z}^{\prime\prime}(\phi_{w})\right)=H^{1}(\mathbb{R})$ for
all $Z\in\mathbb{R}$.

  2. 2.

For each $Z\in\mathbb{R}$, $S_{w,Z}^{\prime\prime}(\phi_{w})$ is closed and
bounded from below.

  3. 3.

Since $\phi$ and $R$ given in (19) and (23) respectively are analytic
functions then $\phi_{w,Z}$ is an analytic function. Thus, for each $v\in{\cal
D}$ the function $Z\rightarrow S_{w,Z}^{\prime\prime}(\phi_{w})(v,v)$ is
analytic.

∎

Next, we use the Kato-Rellich Theorem to prove some specific properties of the
second eigenvalue and eigenfunction of the operator $-{\cal L}_{Z}$.

######  Lemma 4.

There exist $Z_{0}>0$ and analytic functions
$\Pi_{2}:(-Z_{0},Z_{0})\rightarrow\mathbb{R}$,
$\Omega_{2}:(-Z_{0},Z_{0})\rightarrow L^{2}(\mathbb{R})$, such that

  1. $(i)$

For each $Z\in(-Z_{0},Z_{0})$, $\Pi_{2}(Z)$ is the second eigenvalue of
$-{\cal L}_{Z}$, which is simple and $\Omega_{2}(Z)$ its corresponding
eigenfunction.

  2. $(ii)$

$\Pi_{2}(0)=0$ and $\Omega_{2}(0)=\phi^{\prime}$, where $\phi$ is given in
(19).

  3. $(iii)$

$Z_{0}$ can be chosen small enough such that the spectrum of $-{\cal L}_{Z}$
with $Z\in(-Z_{0},Z_{0})$ is greater than $0$ except by the first $2$
eigenvalues.

###### Proof.

The proof proceeds in several steps:

  1. (1)

There is a positive $M$ such that $\sigma\left(-{\cal
L}_{Z}\right)\cap(-\infty,-M)=\emptyset$ for $Z\in[-a,a]$, and $a$ small
enough.

  2. (2)

From the Theorem 7, defining $\lambda_{1,0}=-\lambda$ and $\lambda_{2,0}=0$,
we can separete the spectrum $\sigma\left(-{\cal L}_{0}\right)$ of $-{\cal
L}_{0}$ into two parts $\sigma_{0}=\\{-\lambda,0\\}$ and
$\sigma_{1}=[-w,\infty)$ by a simple closed curve
$\Gamma\subset\rho\left(-{\cal L}_{0}\right)$ such that
$\sigma_{0}\subset\text{Int}(\Gamma)$ and $\sigma_{1}$ in its exterior; where,
$\text{Int}(\Gamma)$ denotes the interior of $\Gamma$.

  3. (3)

From Lemma 3, we have that $-{\cal L}_{Z}$ converges to $-{\cal L}_{0}$ as
$Z\to 0$ in the generalized sense. So, from Theorem IV, §3.16 in [7], we have
that $\Gamma\subset\rho\left(-{\cal L}_{Z}\right)$ for
$Z\in(-\epsilon_{1},\epsilon_{1})$, $\epsilon_{1}$ small enough, and
$\sigma\left(-{\cal L}_{Z}\right)$ is also separated by $\Gamma$ into two
parts such that the part of $\sigma\left(-{\cal L}_{Z}\right)$ inside $\Gamma$
consists of a finite set of eigenvalues with total algebraic multiplicity $2$.

  4. (4)

For $i=1,2$, and $\gamma$ small enough we define circles
$\Gamma_{i}=\\{z\in\mathbb{C}:|z-\lambda_{i,0}|=\gamma\\}$ such that
$\Gamma_{1}\cap\Gamma_{2}=\emptyset$ and $\Gamma_{i}$ is in the interior of
$\Gamma$. Thus from the nondegeneracy of the eigenvalues $-\lambda_{i,0}$, we
obtain that there exists $0<Z_{1}<\epsilon_{1}$ such that for
$Z\in(-Z_{1},Z_{1})$, $\sigma\left(-{\cal
L}_{Z}\right)\cap\text{Int}(\Gamma_{i})=\left\\{\lambda_{i,Z}\right\\}$, where
$\lambda_{i,Z}$ are simple eigenvalues for $-{\cal L}_{Z}$, furthermore,
$\lambda_{i,Z}\to\lambda_{i,0}$, as $Z\to 0$.

  5. (5)

Applying the Kato-Rellich’s Theorem (Theorem XII.8 in [9]) for each one of the
simple eigenvalues $\lambda_{i,0}$, $i=1,2,$ we obtain the existence of a
positive $Z_{0}<Z_{1}$, and analytic functions $\Pi_{2},\Omega_{2}$ defined on
the intervals $(-Z_{0},Z_{0})$ satisfying the items $(i),(ii)$ and $(iii)$ of
the theorem.

∎

Now, we proceed to count the number of negative eigenvalues of the operator
$-{\cal L}_{Z}$. First, we assume that $Z$ is small.

######  Lemma 5.

There exists $r\in\mathcal{R}$, $0<r<Z_{0}$ such that $\Pi_{2}(Z)<0$, for any
$Z\in(-r,0)$ and $\Pi_{2}(Z)>0$ for any $Z\in(0,r)$. Therefore, $-{\cal
L}_{Z}$ has exactly two negative eigenvalues para $Z$ negative and small and
$-{\cal L}_{Z}$ has exactly one negative eigenvalue for $Z$ positive and
small.

###### Proof.

By Taylor’s Theorem around $Z=0$, the functions $\Pi_{2}$ and $\Omega_{2}$ in
lemma 4 can be written as

$\displaystyle\Pi_{2}(Z)$ $\displaystyle=\beta Z+O\left(Z^{2}\right),$ (35)
$\displaystyle\Omega_{2}(Z)$
$\displaystyle=\phi^{\prime}_{w,0}+Z\psi_{0}+O\left(Z^{2}\right),$

where $\phi^{\prime}_{w,0}=\frac{d}{dx}\phi_{w,0}$, $\beta\in\mathbb{R}$,
($\beta=\Pi_{2}^{\prime}(0)$) and $\psi_{0}\in L^{2}(\mathbb{R})$
$(\psi_{0}=\Omega_{2}^{\prime}(0))$. To show the result, it is enough to show
that $\beta>0$ or equivalently that $\Pi_{2}(Z)$ is an increasing function of
the variable $Z$ around $Z=0$. Since the function $Z\rightarrow\phi_{w,Z}$ is
analytic, then for $Z$ close to zero, we have that

$\phi_{w,Z}=\phi_{w,0}+Z\chi_{0}+O\left(Z^{2}\right),$

where

$\chi_{0}=\left.\frac{d}{dZ}\phi_{w,Z}\right|_{Z=0}.$ (36)

Now, from the equation (3), we have that for all $\psi\in H^{1}(\mathbb{R})$,

$\left\langle-\phi^{\prime\prime}_{w,Z}-w\phi_{w,Z}-a\phi^{p}_{w,Z}-b\phi^{2p-1}_{w,Z},\psi\right\rangle=Z\phi_{w,Z}(0)\psi(0).$
(37)

Taking derivative with respect to the variable $Z$ in (37) and evaluating in
$Z=0$, we get that

$\left\langle-{\cal L}_{0}\chi_{0},\psi\right\rangle=\phi_{w,0}(0)\psi(0).$
(38)

In order to obtain $\beta$ as a function of the variable $Z$. We compute the
quantity $\left\langle-{\cal
L}_{Z}\Omega_{2}(Z),\phi^{\prime}_{w,0}\right\rangle$ in two different ways.

  1. (1)

Since $-{\cal L}_{Z}\Omega_{2}(Z)=\Pi_{2}(Z)\Omega_{2}(Z)$, then from (35)

$\left\langle-{\cal L}_{Z}\Omega_{2}(Z),\phi^{\prime}_{w,0}\right\rangle=\beta
Z||\phi^{\prime}_{w,0}||^{2}+O\left(Z^{2}\right).$ (39)

  2. (2)

Since $-{\cal L}_{Z}$ is selfadjoint, then $\left\langle-{\cal
L}_{Z}\Omega_{2}(Z),\phi^{\prime}_{w,0}\right\rangle=\left\langle\Omega_{2}(Z),-{\cal
L}_{Z}\phi^{\prime}_{w,0}\right\rangle$. Now, since $-{\cal
L}_{0}\phi^{\prime}_{w,0}=0$, it follows from (36) that

$\displaystyle-{\cal L}_{Z}\phi^{\prime}_{w,0}$ $\displaystyle=-{\cal
L}_{0}\phi^{\prime}_{w,0}+\left[f^{\prime}\left(\phi_{w,0}\right)-f^{\prime}\left(\phi_{w,Z}\right)\right]\phi^{\prime}_{w,0}$
(40)
$\displaystyle=\left[f^{\prime}\left(\phi_{w,0}\right)-f^{\prime}\left(\phi_{w,0}+Z\chi_{0}+O(Z^{2})\right)\right]\phi^{\prime}_{w,0}$
$\displaystyle=-f^{\prime\prime}\left(\phi_{w,0}\right)Z\chi_{0}\phi^{\prime}_{w,0}+O\left(Z^{2}\right),$

where $f(x)=ax^{p}+bx^{2p-1}$. Thus, from (35) and (40), we obtain that

$\displaystyle\left\langle-{\cal
L}_{Z}\Omega_{2}(Z),\phi^{\prime}_{w,0}\right\rangle$
$\displaystyle=-Z\left\langle\phi^{\prime}_{w,0},f^{\prime\prime}\left(\phi_{w,0}\right)\chi_{0}\phi^{\prime}_{w,0}\right\rangle+O(Z^{2}).$
(41)

On the other hand, by direct computation, we see that

$-{\cal
L}_{0}(-w\phi_{w,0}-f\left(\phi_{w,0}\right))=f^{\prime\prime}(\phi_{w,0})[\phi^{\prime}_{w,0}]^{2}.$
(42)

Since $\phi$ satisfies the equation (16), it follows from (38), (41), (42)

$\displaystyle\left\langle-{\cal
L}_{Z}\Omega_{2}(Z),\phi^{\prime}_{w,0}\right\rangle$
$\displaystyle=-Z\left\langle\chi_{0},f^{\prime\prime}(\phi_{w,0})[\phi^{\prime}_{w,0}]^{2}\right\rangle+O\left(Z^{2}\right)$
(43) $\displaystyle=-Z\left\langle-{\cal
L}_{0}\chi_{0},-w\phi_{w,0}-f\left(\phi_{w,0}\right)\right\rangle+O\left(Z^{2}\right)$
$\displaystyle=-Z\phi_{w,0}(0)(-w\phi_{w,0}(0)-f(\phi_{w,0}(0)))+O\left(Z^{2}\right)$
$\displaystyle=-Z\phi_{w,0}(0)(\phi^{\prime\prime}_{w,0}(0))+O\left(Z^{2}\right).$

Finally, from (39) and (43), we conclude that

$\beta=\frac{\phi_{w,0}(0)(-\phi^{\prime\prime}_{w,0}(0))}{||\phi^{\prime}_{w,0}||^{2}}+O(Z).$

Hence $\Pi_{2}^{\prime}[0]=\beta>0$. This completes the proof of the lemma. ∎

Next, we extend the results in the Lemma 5 for all the values of $Z$.

######  Lemma 6.

Let $Z\in\mathbb{R}$ and set $n\left(-{\cal L}_{Z}\right)$ the number of
negative eigenvalues of $-{\cal L}_{Z}$, then we have that

  1. 1.

For $Z<0$, $n\left(-{\cal L}_{Z}\right)=2$.

  2. 2.

For $Z\geq 0$, $n\left(-{\cal L}_{Z}\right)=1$.

###### Proof.

The proof is based in Lemma 2 above and the ideas in Le Coz et al. [8]. ∎

Finally, the proof of the Theorem 1 follows from the Theorem 5.1.3 and 5.2.1
in [4].

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11institutetext: INFN – Section of Milano Bicocca, Milan, Italy
22institutetext: Department of Physics, Università degli Studi di Milano -
Bicocca, Milan, Italy 33institutetext: INFN - Section of Roma1, Roma, Italy

# An acrylic assembly for low temperature detectors

M. Biassoni 11 C. Brofferio 1122 M. Faverzani 1122 E. Ferri 1122 S.
Ghislandi111e-mail<EMAIL_ADDRESS>(corresponding author).222Gran
Sasso Science Institute, L’Aquila, Italy (present affiliation). 22 S. Milana
33 I. Nutini 1122 V. Pettinacci 33 S. Pozzi 1122 S. Quitadamo††footnotemark:
22

(Received: date / Revised version: date)

###### Abstract

Thermal detectors are a powerful instrument for the search of rare particle
physics events. Inorganic crystals are classically used as thermal detectors
held in supporting frames made of copper. In this work a novel approach to the
operation of thermal detectors is presented, where TeO2 crystals are cooled
down to $\sim$10\text{\,}\mathrm{m}\mathrm{K}$$ in a light structure built
with plastic materials. The advantages of this approach are discussed.

##  1 Introduction

Thermal detectors Enns (1, 2) are largely used in rare event particle physics
in the search of neutrinoless double beta decay ($0\nu\beta\beta$) FIORINI (3,
4) and dark matter interactions reviewDM (5). They are employed as
calorimeters, thus measuring the energy released by particles, in a setup
cooled down to $\sim$10\text{\,}\mathrm{m}\mathrm{K}$$.

One of the main experimental challenges of this technique is the capability of
cooling down several crystals to extreme temperatures, maintaining them
thermally stable. Detector holders characteristics are crucial to achieve
satisfying performances. For this reason, copper holders have always been used
for their known mechanical response at milli-Kelvin scale and high thermal
conductivity. This allows to cool the material more efficiently and to provide
an ideal thermal bath in which the detector discharges the absorbed energy.
The CUORE experiment CUORE (6) succeeded in cooling and operating a
$1\text{\,}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}$ detector made
of TeO_2 crystals, by building the most powerful cryostat ever made Criostato
(7).

In experiments employing ^130Te as $\beta\beta$ emitter, the largest gamma-
background component comes from high-energy photons undergoing Compton
scattering in passive elements of the system BackgroundBudget (8). Copper is
therefore not an optimal choice from this point of view as it has a large
Compton scattering cross-section, on top of being heavy and expensive when
produced with high radio-purity standards.

We decided to test another material to be used as holder in the close vicinity
of thermal detectors. We selected an organic compound, commercially available
as Stratasys VeroClear™. It is a transparent PolyJet photopolymer for clear
acrylic simulation. It is also inexpensive, light (density
$\rho=$1.18\text{\,}\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$$) and, being
organic, characterized by a low $Z_{\text{eff}}=6.5$333The effective atomic
number has been computed as $Z_{eff}=\sqrt[2.94]{\sum_{i}^{elements}f_{i}\cdot
Z_{i}^{2.94}}$ as in MURTY1965 (9).. These features could passively lower the
background induced by Compton interaction with the holder itself, meeting the
requirements of rare events physics.

From a mechanical point of view, at room temperature, the VeroClear™ is
characterized by good rigidity and strength444The VeroClear™ specifics are
available at https://www.stratasys.com/-/media/files/material-spec-
sheets/mds_pj_veroclear_0320a.pdf., not far from the ones in copper.
Conversely the acrylic has lower elasticity. These characteristics makes the
polymer suitable to build structural frames. If copper properties are well
known at cryogenic temperatures, VeroClear™ has to be tested in such extreme
conditions.

VeroClear™ was born to simulate the optical properties of polymethyl
methacrylate (PMMA), largely used as base for many organic scintillators
PMMAScintillator (10). This opens the possibility to activate a component of
the setup which is passive in classical designs. An active holder would veto
its own $\alpha$ surface contamination and part of the crystals one, together
with the $\gamma$ component that Compton scatters with the holder itself (See
Figure 2).

Figure 1: Sketch of the event types that contribute to the $0\nu\beta\beta$
background in TeO_2 and that would be vetoed by an active holder. The green
straight arrows represent $\alpha$ particles generated in the holder surface,
later reaching the crystals and vice-versa. The curled red arrow shows an
energetic photon undergoing a Compton scattering with the holder and being
absorbed by the detector

Figure 2: CAD section of the VeroClear™ holder together with a crystal and the
relative spacer. The top face of one of the columns is highlighted in order to
appreciate its shape

The main difference of VeroClear™ with respect to PMMA is the possibility to
be 3D printed, a useful opportunity for building structures with complex
geometries, if required. Right now, we are not yet considering the impact of
such a build-up technique from the point of view of induced contamination.

However, there is poor literature about properties of acrylic and, more
generally, organic compounds, at the milli-Kelvin scale, even if some can
still be found Ventura (11). We don’t know how the VeroClear™ characteristics,
such as low thermal conductance and large heat capacity, would modify the
cool-down as well as the detector response’s shape and performance. For these
reasons, an initial proof of principle is of primary importance. It is
essential to check detector behaviour at cryogenic temperature, to be compared
to the one coming from copper holders.

##  2 Experimental Setup

The measurements have been performed using $1\times 1\times 1$
$\mathrm{c}\mathrm{m}^{3}$ TeO_2 crystals. They have been operated in a
cryogenic facility located at Milano - Bicocca University (Italy). The
crystals were equipped with $3\times 3\times 1$ $\mathrm{m}\mathrm{m}^{3}$
Neutron Transmutation Doped Ge (NTD) employed as thermistors Haller (12). The
NTDs were mechanically and thermally coupled to the crystals with six spots of
epoxy glue. The NTD consists of a Ge chip with a doping level slightly below
the Mott transition MottTransition (13). Since the NTDs are naturally well-
compensated semiconductors (the uniformly-distributed dopants are both donors
and acceptors), the Ge lattice electrons behavior can be well described by the
Shklovskii’s and Efros’ Variable Range Hopping (VRH) model
VariableRangeHopping (14). In this regime, the NTD’s conductivity strongly
depends on the temperature $T$. It can be shown that the resistance $R$ can be
written as:

$R=R_{0}e^{\sqrt{\frac{T_{0}}{T}}}.$ (1)

The employed NTDs had already been characterized ArticoloSimone (15) and their
parameters, $R_{0}$ and $T_{0}$, were respectively evaluated as
$\sim$$0.5\text{\,}\mathrm{\SIUnitSymbolOhm}$ and
$\sim$$4.75\text{\,}\mathrm{K}$. A biasing and read-out circuit provided the
polarization current to the thermistors, using either
$20\text{\,}\mathrm{M\SIUnitSymbolOhm}$ or
$15\text{\,}\mathrm{G\SIUnitSymbolOhm}$ load resistors, depending on the NTD’s
resistance value at the chosen working point. The latter is defined by the
position in the NTD’s I-V curve, usually referred as _load curve_. The working
point mainly determines the signal-to-noise ratio and the pulse shape of the
detector.

Figure 3: Picture of the setup containing the two towers wired and installed
below the cold plate through S-shaped copper strips (_left_). Detail of the
acrylic holder (_right_). It is possible to see the NTDs glued on the crystals
and their gold wire link to the Cu strips of the pins. These strips are not
thermally coupled to the crystals. Only two channels for acrylic holder and
one reference channel in the copper were used for this paper purposes

Two similar holding structures have been built for our purposes. One is the
acrylic tower, composed of a $5\text{\,}\mathrm{m}\mathrm{m}$ thick base and
four columns which accommodate three TeO_2 stacked crystals. To reduce the
risk of damage to the detectors, due to the distinct differential contractions
of acrylic and TeO_2, the holder was designed to minimize stresses while
maintaining the needed stiffness. This has been accomplished by shaping the
VeroClear™ columns as in Figure 2. The holder fits the crystals’ faces so to
avoid vibration but leaving a mechanical unloading at the edges. Moreover, on
top of the crystal stack, some space has been left to account for contractions
at cryogenic temperatures. Lastly, $0.5\text{\,}\mathrm{m}\mathrm{m}$ thick
VeroClear™ spacers were interposed between crystals mainly to avoid thermal
cross-talk among detectors. The VeroClear™ holder and spacers have been
produced through a Stratasys ObjetPRO machine, a 3D-Printer working by means
of polyjet technology which guarantees high accuracy (order of hundredths of a
mm) and quality of the parts produced through a process of instant cure by UV
light of photosensitive resins (such as VeroClear™). Soluble support material
(Stratasys SUP706-B™) was used for the construction of these parts, in order
to optimize the cleaning and consequently the surface finishing. The printed
substrates can be also treated for a few hours inside a lighting chamber
equipped with fluorescent lamps, selected both in power and color temperature,
in order to improve the final transparency. The production took place at INFN
Rome mechanical workshop, exploiting the HAMMER555For more information consult
https://hammer.lngs.infn.it/. facilities (Hub for Additive Manufacturing
Materials Engineering and Research) instituted by INFN.

The second holder, made of copper, includes a $5\text{\,}\mathrm{m}\mathrm{m}$
thick base, $2$ columns and a thin plate. It can house two crystals glued, by
means of Araldite® Rapid glue spots, on the base and on the plate.

Both the towers were hosted in an Oxford TL-200 ^3He-^4He dilution
refrigerator. Each holder was fixed through brass screws to a copper S-shaped
strip, in turn thermally coupled to a copper plate. The latter is directly
connected to the cold finger and the mixing chamber. The cryostat operating
temperature was maintained at $\sim$$15\text{\,}\mathrm{m}\mathrm{K}$ and the
heat bath was monitored through a MAGNICON MFFT SQUID thermometer, positioned
under the mixing chamber, and thermally linked to the same copper plate the
detectors were connected to. The entire detector setup is shown in fig. 3. The
NTDs readout was performed through
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ gold wires ball-bonded to
the Ge thermistors and soldered to metallic pins on the other side. In the
case of acrylic holder, these gold wires provided the detector main thermal
coupling, as demonstrated in ArticoloSimone (15). The electric signal was
carried out of the cryostat through twisted constantan wires, lastly reaching
a Faraday cage where it underwent amplification and anti-aliasing Bessel
filtering. Eventually, the detector signal was digitized and stored using two
$18\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$ ADC boards.

A 2 cm thick Cu layer surrounded the setup to reduce the environmental
$\gamma$ radiation reaching the two towers, together with a
$5\text{\,}\mathrm{c}\mathrm{m}$ thick Pb disk on top of the cold plate. A
^232Th source was employed to calibrate the energy scale and was put under the
cryostat during calibration runs.

##  3 Data analysis

When a particle passes through the TeO_2 crystals, it releases energy and,
consequently, the detector temperature raises. This variation is detected
through the NTD, operated as a thermometer.

The NTDs output was digitized and stored as a continuous
$2\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ waveform. During the data-taking, a
derivative trigger flagged events from the continuous stream. The selected
pulses were stored as windows of $400\text{\,}\mathrm{m}\mathrm{s}$ length
(Figure 5). The pulse window contains a $100\text{\,}\mathrm{m}\mathrm{s}$
pre-trigger region used to extract information about the specific event
baseline. Moreover, noise windows were sampled randomly. They have been used
later to construct a noise power spectrum for each channel, run by run.

The first part of the offline analysis identified periods of time in which the
detectors were misbehaving. The check was done by looking for intervals with
unstable baseline or high baseline RMS. These “bad intervals” were excluded
from the subsequent analysis steps. For each channel the _average pulse_ (AP)
and the _average noise power spectrum_ (ANPS) have been computed. They were
used to evaluate the pulse amplitude with the Optimum Filter technique
OptimumFilter (16), which allows to maximize the signal-to-noise ratio.

Since the thermal gain depends on temperature, it was essential to correct the
detector response compensating for temperature variations among events. For
this reason, we applied a thermal gain stabilization by using the detector
baseline level, that is the pretrigger average baseline, as a temperature
proxy. The temperature dependence of the gain was corrected by correlating the
amplitude of pulses belonging to a peak of known energy
($2615\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ from ^208Tl) to the baseline
level ThermalStabilization (17).

The spectrum was calibrated using recognizable $\gamma$ peaks, mainly coming
from the ^232Th calibration source. These lines have been fitted using a
Gaussian function added to a linear background. For the calibrating function a
$2^{nd}$ order polynomial with intercept at zero has been employed. An example
of the obtained energy spectra is reported in fig. 5.

All considered, we analyzed one reference channel in the copper assembly (here
named channel 1) and $2$ channels from the acrylic tower (channel 2 and
channel 3).

Figure 4: Typical $400\text{\,}\mathrm{ms}$ event window containing a pulse.
Rise time ($t_{\text{rise}}$) and decay time ($t_{\text{decay}}$) are defined
as the time elapsed between $10\%$ and $90\%$ of the pulse amplitude

Figure 5: Example of calibration energy spectrum. The energy of lines used for
the calibration is reported on top of the relative peaks and expressed in
$\mathrm{k}\mathrm{e}\mathrm{V}$. The
$511\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ peak is due to $e^{+}$-$e^{-}$
annihilation, $583\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ and
$2615\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ belong to ^208Tl,
$911\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ and
$969\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ come from ^228Ac decay, the
$1461\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ peak is due to ^40K and the
$1593\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ is the double escape peak of the
highest energy ^208Tl line

##  4 Results

During the cool-down some high-statistics calibration runs were collected.
They all lasted more than $20\text{\,}\mathrm{h}\mathrm{r}$ and they contain a
total of $~{}10^{5}-10^{6}$ physical pulses each. Before the data acquisition,
test runs were needed to optimize NTDs’ working point and trigger parameters.

The first obtained result is related to the detector stability over time.
Acrylic is amorphous and it has low thermal conductance. The disordered
molecule structure could in principle relax and, unpredictably, release
energy, acting as a residual heat load on the holder. Therefore, the presence
of acrylic could induce temperature shifts on long time scales or fast and
large spikes. The latter, in contrast to the usual temperature variations,
could not be corrected with the stabilization described in the previous
chapter. However, the acrylic setup baseline level, proxy for the crystals
temperature in the holder, was as stable as the one in the copper holder for
very long times. The stability has been checked for all channels and all
acquired runs. An example is shown in fig. 6.

Figure 6: Baseline level-time scatter plot for channels $1$, $2$ and $3$ of a
sample run. Arbitrary offsets have been added to not overlap all the points.
The baseline, proxy of the temperature at the crystal level, is stable over
the $30\text{\,}\mathrm{h}\mathrm{r}$ of data-taking both for acrylic and
copper towers. The detectors in the VeroClear™ don’t show long time shifts or
spikes out of the expected ones. The visible overall baseline modulation,
present in all the three channels, is due to fluctuations of the cryostat
temperature, reported in the lower plot

Since crystal holders have also structural roles, another important result
concerns acrylic mechanical properties. The plastic holder had already
undergone many cool-down cycles from room temperature to
$77\text{\,}\mathrm{K}$ and few cycles from room temperature to
$\sim$15\text{\,}\mathrm{m}\mathrm{K}$$. After each group of tests it was
examined carefully at the microscope and no cracks or visible defects were
found. No spurious thermal noise of the type registered in CRESST CRESST (18)
was observed during data taking, nor cracks on the crystals after warm-up.

We expected the different conductivity of the holding material ArticoloSimone
(15) to influence pulse formation. Indeed, at their respective optimal working
points (_wp1_) the signals from channel in the acrylic holder are faster than
those from channel $1$ (Cu tower). The effect can be qualitatively appreciated
by comparing the pulse shapes plotted in fig. 8. This can be explained by the
weaker thermal coupling provided by the acrylic, leading to an higher crystals
operating temperature of $\sim$20\text{\,}\mathrm{m}\mathrm{K}$$ (with smaller
working resistances $R_{W}$ resulting in smaller intrinsic gain but faster
time response). Conversely, the detectors in the copper tower were at the
cryostat temperature, that was $\sim$15\text{\,}\mathrm{m}\mathrm{K}$$. Since
fast pulses could be useful to reduce pile-up whenever this background is an
issue (as in the proposed CUPID experiment CUPIDpileup (19, 20)), this acrylic
holder additive feature is very promising. To achieve similarly fast pulses on
standard Cu frames, the NTDs can be warmed up by increasing the bias current.
This approach was applied to channel $1$ by operating it at a different
working point (_wp2_), dissipating $\sim$$30\text{\,}\mathrm{p}\mathrm{W}$
through the bias circuit666In the case of _wp1_ the dissipated power is
negligible, in the order of $0.5\text{\,}\mathrm{p}\mathrm{W}$.. The
dissipated power was such that the pulses duration were shortened but the
resolution (and the pulse shape) didn’t suffered the non optimal working point
yet. We characterize the signals by means of their rise and decay time,
defined as the time elapsed for a pulse from $10\%$ to $90\%$ of its amplitude
and vice versa. The obtained average values are reported in table 1, together
with their working resistances $R_{\text{w}}$. Using these results, the pile-
up probability, modelled as a Poissonian distribution, would be lower in
acrylic holder channels by a factor $\sim 5$, when the NTDs are operated at
their optimal working point.

Table 1: Average rise time and decay time, as defined in fig. 5, computed for the two mentioned working points: _wp1_ and _wp2_. In the last column, the NTDs working resistances. Holder material | Channel | Working point | Rise time [$\mathrm{ms}$] | Decay time [$\mathrm{ms}$] | NTD $R_{\text{w}}$
---|---|---|---|---|---
Copper | 1 | _wp1_ | $24.0\pm 0.2$ | $64.8\pm 0.8$ | $12.8\text{\,}\mathrm{M\SIUnitSymbolOhm}$
Acrylic | 2 | _wp1_ | $4.4\pm 0.1$ | $47.6\pm 0.5$ | $139\text{\,}\mathrm{k\SIUnitSymbolOhm}$
Acrylic | 3 | _wp1_ | $2.49\pm 0.05$ | $17.8\pm 0.1$ | $287\text{\,}\mathrm{k\SIUnitSymbolOhm}$
Copper | 1 | _wp2_ | $4.6\pm 0.1$ | $30.4\pm 0.4$ | $1.37\text{\,}\mathrm{M\SIUnitSymbolOhm}$

Besides these thermal considerations, the pulse shape of the detectors housed
in the acrylic holder is not further modified. This fact allowed to process
events from the two setups with the same algorithms without adapting or tuning
the employed software.

An important parameter used to assess the detector performance is the FWHM
energy resolution. The most prominent calibration spectrum peaks, whose
energies are reported in fig. 5, have been fitted with a Gaussian function
plus a linear background at the end of the calibration step. A zero-energy
FWHM, corresponding to the baseline resolution, has been computed taking into
account the baseline distribution spread evaluated after filtering. The
obtained resolution-energy plot is shown in fig. 8. The resolution is
dominated, at low energies, by the baseline noise. Then, it grows
monotonically, following the typical low temperature calorimeters trend. The
FWHM resolutions of baseline and of the highest energy peak of the calibration
are reported in table 2, where channel 1 is operated at _wp2_ and channel 2
and 3 at _wp1_.

Figure 7: Examples of pulses coming from detectors in acrylic holder at _wp1_
, the optimal working point and from copper assembly operated at _wp1_ and
_wp2_ , obtained by dissipating $30\text{\,}\mathrm{p}\mathrm{W}$ on channel
1. Notice that this is not the entire $400\text{\,}\mathrm{m}\mathrm{s}$
acquired time window

Figure 8: Energy resolution expressed as FWHM for the considered channels. The
point at zero energy is the resolution of randomly acquired noise windows

Table 2: Summary of the results, in terms of energy resolution, obtained for the considered channels. In the table, the baseline FWHM resolution and the $2615\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ peak one are reported. Channel | Holder material | FWHM baseline [$\mathrm{k}\mathrm{e}\mathrm{V}$] | FWHM $@$ $2615\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ [$\mathrm{k}\mathrm{e}\mathrm{V}$]
---|---|---|---
$1$ | Copper | $1.31\pm 0.02$ | $6.1\pm 0.5$
$2$ | Acrylic | $1.72\pm 0.02$ | $5.3\pm 0.3$
$3$ | Acrylic | $1.04\pm 0.01$ | $6.1\pm 0.4$

We also compared the noise registered on the different channels by computing,
run by run, their ANPS and normalizing for the electronic gain. As can be seen
from fig. 9, the noise belonging to channels in the acrylic tower doesn’t show
problematic features when operated at the optimal working points. As already
pointed out, the crystals in the acrylic work at slightly higher temperatures,
that is at lower $R_{w}$. As a check, the Johnson contribution to the
continuum is confirmed to be lower.

Figure 9: Normalized ANPS for channel $1$, belonging to the copper tower,
operated at _wp2_ and channels $2$ and $3$, framed in the acrylic holder and
operated at _wp1_. The difference in the amplitude scale and in the RMS can be
ascribed to the different $R_{W}$ of the detectors (see tab 1), contributing
to the Johnson noise on the channels

##  5 Conclusions and Comments

Neutrinoless double beta decay searches, decade after decade, improve the
limits on the $0\nu\beta\beta$ half-life for different emitting isotopes.
Future experiments will ask for more demanding material radio-purity,
background rejection capability, and better energy resolutions. For this
reason, acting on structures close to the detectors will be essential to
fulfill the desired requirements. In the field of low temperature detectors,
copper has always been employed for the construction of the crystal holding
structures. However, copper is expensive and its Compton scattering cross-
section can cause significant background for experiments using ^130Te as
target isotope. A novel approach would consist in using holding structures
made of light organic compounds, which would reduce both the cost and the
impact on the measured background.

In our experimental setup, VeroClear™ has shown good mechanical
characteristics and thermal coupling with our crystals down to the milli-
Kelvin scale. Since it can be 3D printed, even complicated structures can be
realised with a low number of components, simplifying the assembly. From a
mechanical point of view, it would be possible to scale our test holder to
real experiments dimension. A VeroClear™ holder would bear the CUORE detector
strings, while maintaining the same frame volume employed in the experiment
(with great margin, thus leaving room for further optimization). There are
already examples of acrylic laminates produced with low ^238U and ^232Th
contamination, at levels comparable to the CUORE cleanest copper. Therefore,
the bulk contamination introduced by the detectors’ holder would gain a factor
at least 7.5 (density ratio), passively lowering the induced background.

We demonstrated that the material replacement doesn’t compromise the detector
working properties (resolution, stability, etc.). The detectors housed in the
plastic tower are as stable as the ones framed in copper for long times. In
addition, their equilibrium temperature is only a few $\mathrm{m}\mathrm{K}$
higher than in the copper case. As a consequence, the pulses are faster,
allowing to reduce the background induced by $2\nu\beta\beta$ pile-up in
$0\nu\beta\beta$ searches with isotopes characterized by shorter half-life
CUPIDpileup (19, 20). The acrylic holder doesn’t cause any deterioration in
the energy resolution and in the noise power spectrum. Since the acrylic frame
is characterized by low conductance, thermalization issues cannot be excluded,
especially when the detector strings become longer. These kind of tests are
required to check where such effects are negligible and where they become a
severe issue.

A first hint of a real reduction of the Compton background events has been
obtained and validated by Monte Carlo simulations and it is discussed in a
specific paper ArticoloSIF (21). We will therefore further proceed in this
direction in the near future, studying the VeroClear™ scintillation light
properties at the milli-Kelvin scale, together with the associated readout.
These tests will be critical to evaluate the feasibility of an “active”
acrylic holder, useful to reach a further background rejection mainly tagging
and suppressing $\alpha$ induced events and rejecting $\beta$ radiation coming
from the components close to the detectors. These additional features will be
essential for future-generation $0\nu\beta\beta$ experiments, expected to
operate close to the background-free regime.

## Acknowledgments

We thank Marco Iannone from INFN - Section of Roma1 and member of the Hammer
project for helping with the production and preparation of the 3D printed
components of the assembly.

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