Abstract
In this paper we study the perturbation \(L=H+V\), where \(H=-\frac{{{d}^{2m}}}{d{{x}^{2m}}}+{{x}^{2m}}\) on \(\mathbb {R}\), \(m\in {{\mathbb {N}}^{*}}\) and V is a decreasing scalar potential. Let \({{\lambda }_{k}}\) be the \(k^{th}\) eigenvalue of H. We suppose that the eigenvalues of L around \({{\lambda }_{k}}\) can be written in the form \({{\lambda }_{k}}+{{\mu }_{k}}\). The main result of the paper is an asymptotic formula for fluctuation \(\{ {{\mu }_{k}} \}\) which is given by a transformation of V. In the case \(m=1\) we recover a result on the harmonic oscillator.
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1 Introduction and main results
We consider in \(\mathbb {R}\) the operator H defined by
We recall that H [1] is essentially self-adjoint in \(C_{0}^{\infty }(\mathbb {R})\) with compact resolvent. Its spectrum is the increasing sequence \({{\{ {{\lambda }_{k}} \}}_{k\ge 0}}\) of eigenvalues of finite multiplicity, such as there exists a positive integer \({{k}_{0}}\), for \(k\ge {{k}_{0}}\), \({{\lambda }_{k}}\) is simple and has the following asymptotic expansion
with
where B is the beta function. Let \( V\in {{\mathbb {C}}^{\infty }}(\mathbb {R},\mathbb {R})\) which satisfies the following estimate
Along this article we set
Remark 1.1
So as not to burden our work, the case “\(s =1\)” will be treated in a different way, we will treat this case in another work.
The operator \(L= H+V\) is essentially self-adjoint with compact resolvent [2]. The Min-Max theorem [3] shows that the spectrum of L around \({{\lambda }_{k}}\) can be written in the form \({{\lambda }_{k}}+{{\mu }_{k}}\). Our goal is to study the asymptotic behavior of the fluctuation \({{\mu }_{k}}\) when \(k\rightarrow +\infty \), by expressing it using a transformation of V. Our main result is
Theorem 1.2
(Main Theorem) \({{\mu }_{k}}\) has the asymptotic expansion
For \(m=1\), the case of the Harmonic Oscillator, the asymptotic behavior of \(\mu _{k}\) is given by
Theorem 1.3
\(\displaystyle {{\mu }_{k}}=\frac{1}{\pi }\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}{V\left( \sqrt{{{\lambda }_{k}}}\sin t \right) }dt+O\left( \lambda _{k}^{-\delta +\eta } \right) \)
where \(\eta \in ] 0,\frac{\delta }{2} [\).
Many authors interested in this kind of problems, especially the case of the Harmonic Oscillator [4, 5]. The case \(m>1\) seems to us as not to have been treated yet. Briefly recall the content of [4], the author studies the perturbation \(L=A+B\), where
and he proved that \({\mu }_{k}\) has the following asymptotic expansion
\(\widetilde{V}\) represents the “Radon Transform” of V. In recent works, we find in [6] a study of
where real functions \(q, q'\) and \( x\rightarrow \int _{0}^{x}{q(s)ds}\) are bounded. \({\mu }_{k}\) has the asymptotic expansion
We notice that, in the case \(s<1\), the expansion (1.6) has the same main part as shown in Theorem 1.3, even though we don’t need to suppose that \( x\rightarrow \int _{0}^{x}{q(s)ds}\) is bounded. In addition, if \(s\in ] \frac{2}{3},1 [\), we get a better estimate and the same goes for \(s>1\) because \(\eta \in ] 0,\frac{1}{2} [\). We can also mention Pushnitski [7], who studied the case where \(q\in C_{0}^{\infty }(\mathbb {R})\). He proved that \({\mu }_{k}\) admits the next development in series
with some coefficients \({{c}_{j}}\in \mathbb {R}\), in particular \({{c}_{1}}=\frac{1}{\pi }\int _{-\infty }^{+\infty }{q(x)} \, \mathrm {d}x\), and \(c_{2}=0\).
Remark 1.4
We want to go further in studying the operator \(H_{k,l}=-\frac{{{d}^{2k}}}{d{{x}^{2k}}}+{{x}^{2l}}\), where \(k,l\in {{\mathbb {N}}^{*}}\), then by giving k the value of “1”, we’ll reach important results that have a lot off applications in the field of physics, especially the quartic oscillator.
Our main tool is the averaging Method of Weinstein [8, 9], whose origins go back to the classical work on celestial mechanics [10]. Note that for \(m\in {{\mathbb {N}}^{*}}-\{ 1\}\), this method cannot be used directly in this case because H, viewed as a Pseudo-differential operator (OPD), doesn’t have a periodic flow, but the operator \({{H}^{\frac{1}{m}}}\) does have this property, so we start reducing ourselves to a perturbation (1.8) of the operator \({{H}^{\frac{1}{m}}}\)
where B is an operator to be determined. We apply the Averaging Method, firstly we replace B in perturbation (1.8) by the average
where T is the period of the flow of \(H^{\frac{1}{m}}\), T is given by (1.3) [11]. The main advantage of this method is that \(\overline{B}\) is a compact operator and \(L_{m}\), \({{\overline{L}}_{m}}=H^{\frac{1}{m}}+\overline{B}\) are almost unitary equivalent, that means it exists a unitary operator U such as \(U{{L}_{m}}{{U}^{-1}}-{{\overline{L}}_{m}}\) is compact. Note that \(L_{m}\) and \({{\overline{L}}_{m}}\) are also with compact resolvent. Using min-max theorem, the parts of their spectrum around \(\lambda _{k}^{\frac{1}{m}}\) are respectively of the form \(\lambda _{k}^{{\frac{1}{m}}}+{{\upsilon }_{k}}\) and \(\lambda _{k}^{\frac{1}{m}}+{{\overline{\upsilon }}_{k}}\). Then we study \({{\overline{\upsilon }}_{k}}\) by using a functional calculus of the operator H. We begin by establishing the link between \({{\upsilon }_{k}}\) and \({{\mu }_{k}}\).
Proposition 1.5
Using a functional calculus for H, we obtain
Proposition 1.6
The following proposition gives the relation between \({{\upsilon }_{k}}\) and \({{\overline{\upsilon }}_{k}}\)
Proposition 1.7
where \(\eta \in ] 0, \min (2,m-1+\tfrac{{\delta }}{2}) [\)
Remark 1.8
Note that for \(m=1\), H has a periodic flow of period \(\pi \), so we can directly apply the Averaging Method to H.
This paper is organized as follows. The next section contains auxiliary fact concerning the proprieties of Weyl pseudo-differential operators which are the main tool in this article. In Sect. 3, we study the operator \(L_{m}\) and show the relation between \(\mu _{k}\) and \(\upsilon _{k}\) by proving Proposition 1.5. The Sect. 4 is devoted to the functional calculus for the operator H, and we establish Proposition 1.6. In the last section, we study the relation between the spectrum of \({{L}_{m}}\) and \({\overline{L}_{m}}\) and we prove Proposition 1.7. Finally, we justify the asymptotic expansion given by Theorems 1.2 and 1.3
2 Weyl pseudo-differential operator
Let \(\rho \in [ 0,1]\),\(\;q\in \mathbb {R}\). \(\Gamma _{\rho }^{q}(\mathbb {R}\times \mathbb {R})\) denote the space symbols associated with the temperate weight function \({{\mathbb {R}}^{2}}: (x,\xi )\rightarrow {{(1+{{x}^{2}}+{{\xi }^{2} })}^{\frac{q}{2}}}\) [15]. Precisely the space of function \(a\in {{C}^{\infty }}({{\mathbb {R}}^{2}})\) satisfies \(\forall \alpha ,\beta \in \mathbb {N},\exists \, {{C}_{\alpha ,\beta }}>0\)
We will use the standard Weyl quantization of the symbols. To be precise, if \(a\in \Gamma _{\rho }^{q}\), then for \(u\in S(\mathbb {R})\) the operator associated is defined by :
Let us now introduce the notion of asymptotic expansion.
Definition 2.1
Let \({{a}_{j}}\in \Gamma _{\rho }^{{{q}_{j}}} (j\in {{\mathbb {N}}^{*}})\), we suppose that \(q_{j}\) is a decreasing sequence tending towards \(-\infty \). We say that \(a\in {{C}^{\infty }}(\mathbb {R}\times \mathbb {R})\) has an asymptotic expansion and we write
if
We use the notation \(G_{\rho }^{q}\) for the set of operators \(op^{w}(a)\) if \(a\in \Gamma _{\rho }^{q}\). In order to prove our main results, we shall recall some well-known results [12,13,14].
Theorem 2.2
(Calderon-Vailloncourt theorem) If \(a\in \Gamma _{0}^{0}\) then the operator \(op^{w}(a)\) is bounded.
Proposition 2.3
(Compactness) If \(a\in \Gamma _{\rho }^{q}\) and \(q<0,\,\rho \in [0,1 ]\), then the operator \(op^{w}(a)\) is compact.
We will need the following proposition for the composition of pseudo-differential operators.
Proposition 2.4
Let \(A\in G_{\rho }^{p}\), \(B\in G_{\rho }^{q}\), \(\rho \in ] 0,1 ]\), p and q\(\in \mathbb {R}\). Then the operator \(AB\in G_{\rho }^{p+q}\). Its Weyl symbol admits the following asymptotic behavior
In particular
where
a and b are respectively the Weyl symbol of A and B
In the next proposition we are giving an extension where the case “\(\rho =0\)”, we have the following result
Proposition 2.5
If \(A\in G_{1}^{m}\) and \({{({{B}_{i}})}_{i\in \{ 1,\ldots p \}}}\) is the set of operators such as \({{B}_{i}}\in G_{0}^{{{m}_{i}}}\) then:
- (i)
The operator \(A{{B}_{1}}\in G_{0}^{m+{{m}_{1}}}\), its Weyl symbol is giving by (2.3)
where \({{c}_{j}}\in \Gamma _{0}^{m+{{m}_{1}}-j}\)
- (ii)
The commutator \([ A,{{B}_{1}} ]\in G_{0}^{m+{{m}_{1}}-1}\)
- (iii)
If A is elliptic and \(m>0\), then \({{B}_{1}}\ldots {{B}_{P}}{{A}^{-\tfrac{{{m}_{1}}+\cdots +{{m}_{p}}}{m}}}\) is bounded.
Proposition 2.6
(Functional calculus) Let A be an elliptic operator included in \(G_{1}^{m}\), its Weyl symbol admits the development \(a\sim \sum \nolimits _{j\ge 0}{{{a}_{j}}}\). Then for any real number q we have \({{A}^{q}}\in G_{1}^{mq}\). Moreover, its weyl symbol admits the following asymptotic behavior
where \({{\sigma }_{{{A}^{q}},0}}={a_{0}}^{q}\), \({{\sigma }_{{{A}^{q}},1}}=qa_{1}.a_{0}^{q-1}\).
3 Reduction to a perturbation of \({H}^{{\frac{1}{m}}}\)
If we translate H by a strictly positive constant, we can always assume that L is positive and \(\left\| {{H}^{-1}}V \right\| <1\). We can reduce ourselves to a perturbation of \({{H}^{\frac{1}{m}}}\) by writing
consequently
where
We can write
Since \(\left\| {{H}^{-1}}V \right\| <1\), the operator \(\sum \nolimits _{k=0}^{+\infty }{{{\alpha }_{k+2}}}{{({{H}^{-1}}V)}^{k}}\) is bounded in \({{L}^{2}}(\mathbb {R})\). Using the fact that, \({{H}^{-1}}\in G_{1}^{-2m}\), \(V\in G_{0}^{0}\) and (iii) Proposition 2.5, we obtain that the operator \(({{L}^{\frac{1}{m}}}-{{L}_{m}}){{H}^{2-\frac{1}{m}}}\) is bounded. We deduce that there exists a constant \(c>0\) such that
According to the min-max theorem, we get
Using the fact that \(\{ {{\mu }_{k}} \}\) is bounded and the Taylor’s formula for the function \(t\rightarrow {{\left( 1+\frac{{{\mu }_{k}}}{t} \right) }^{\frac{1}{m}}}\), we obtain the estimate
This completes the proof of Proposition 1.5.
4 Functional calculus for the operator H and the asymptotic behavior of \({{\overline{\upsilon }}_{k}}\)
Recall that \({{\overline{L}}_{m}}\) is obtained by replacing B in \({{L}_{m}}\) by \(\overline{B}\). Putting
and From (3.1)
We have the following proposition
Proposition 4.1
\(\overline{V}\in G_{0}^{-\delta }\), and its Weyl symbol checks
where
and \(\sigma _{H}\) is the Weyl symbol of H.
Proof
First we will study the Weyl symbol of the operator W(t). In order to apply the Egorov’s theorem (Theorem IV-10 [15]) checks his assumptions. Using Proposition 2.6, the operator \(H^{\frac{1}{m}}\in G_{1}^{2}\) and \({{\sigma }_{{{H}^{\frac{1}{m}}}}}\) admits the following development
where \({{\sigma }_{{{H}^{\frac{1}{m}}},0}}={{({{\sigma }_{H}})}^{\frac{1}{m}}}\), \({{\sigma }_{{{H}^{\frac{1}{m}}},1}}={0}\)
An elementary calculation shows that
and
uniformly with respect to t, where \(\varphi (t)\) denotes the Hamiltonian flow of \(\sigma _{H}^{\frac{1}{m}}\). We recall that \(\varphi (t)=(x(t),\xi (t))\) is a solution of the system
\((*)\) is the first integral of this system
Now we apply the Egorov’s theorem, this yields, for \(t\in \mathbb {R}\), W(t) is an (OPD), its Weyl symbol admits the following development \({{\sigma }_{W(t)}}\sim \sum \nolimits _{j\ge 0}{{{\sigma }_{W(t),j}}}\)
where
in particular
We start by determining the class of \({{\sigma }_{\overline{V},0}}=\frac{1}{T}\int _{0}^{T}{V(x(t))dt}\). We can suppose that \(x(0)>0\), \(\frac{dx(0)}{dt}>0\) as initial conditions, other cases are treated in the same way, for now we must study the properties of the function x(t) on [0, T]. From the system (S) we get
By combining the fact that x(t) is a smooth periodic function of period T with (4.6), we can ensure that the function x(t) reaches its maximum in \(t_{0}\), \( x(t_{0})={{E}^{\frac{1}{2m}}}\) and its minimum on \(t_{1}\), \( x(t_{1})=-{{E}^{\frac{1}{2m}}}\).
For now we have
We make the change of variable \(x(t)=u\), obviously x(t) is increasing on \([ 0,{{t}_{0}}]\), we get
After the same calculation on \([ {{t}_{0}},{{t}_{1}}]\) and \([ {{t}_{1}},T]\) we obtain
Now we apply the change of variable \(y=u{{E}^{\frac{-1}{2m}}}\), we have
Let’s now determine the class to which \(\displaystyle {{\sigma }_{\overline{V},0}}\) belongs. Using (1.4) we get
We have 2 cases
1st case\(0<s<1\)
2nd case\(1<s\)
We set \(\sigma _{H}^{\frac{1}{2m}}=a\) and we split the integral into two parts,
Since \(s>1\) we have
After applying change both of variables \(y=\frac{u}{1+u}\), and \(v=u\sqrt{1+a^2}\),
we get
On the other side we have
We obtain the following estimate
The same estimates hold for \(\partial _{x}^{\alpha }\partial _{\xi }^{\beta }\sigma _{\overline{V},0}(x,\xi )\), \(\alpha , \beta \in {\mathbb { N}} \).
From (4.12) and (4.14) we have
Combining (1.4), (4.4) and (4.5) we deduce that there exist \(c>0\) such that: for all \(t\in [0,T]\)
by integrating (4.15) along the interval [0, T] and following the same previous calculation we have
The same estimates hold for \(\partial _{x}^{\alpha }\partial _{\xi }^{\beta }\sigma _{\overline{V}}(x,\xi )\), \(\alpha , \beta \in {\mathbb { N}} \). Finally we conclude
\(\square \)
In the following we will use a functional calculus for the operator H, this allows us to give the asymptotic behavior of \({{\overline{\upsilon }}_{k}}\). The functional calculus on (OPD) was studied in the case where the functions are in the Hörmander class \(S_{1}^{r}\)\((r \in \mathbb {R})\) see [15, 16]. In our work we are dealing with the case of the operator H plus a function in the class \(S_{1-\frac{1}{2m}}^{r}\). More precisely, the set of functions \(f\in {{C}^{\infty }}(\mathbb {R})\) such that for all \(k\in \mathbb {N}\), there exist \({{C}_{k}}\ge 0\) such that
We recall that the main symbol of \(\overline{V}\) is written
where
a direct calculation shows that \(f\in S_{1-\frac{1}{2m}}^{-\frac{\delta }{2m}}\) for \(s\in \mathbb {R}_{+}^{*}-\{ 1\}\).
The operator f(H) is defined by a functional calculus of self-adjoint operators, then the spectrum of f(H) is the sequence \({{\{ f({{\lambda }_{k}})\}}_{k}}\). We have the following proposition
Proposition 4.2
f(H) is an OPD included in \(G_{0}^{-\delta }\) and its Weyl symbol admits the following development
where \({{d}_{j,k}}\in \Gamma _{1}^{2mk-4j}\) and \({{\sigma }_{f(H),2j}}\in \Gamma _{0}^{-\delta -j}\),
in particular
Proof
For studying f(H) We follow the same strategy in [16] , we will use the Mellin transformation, this later consist of
- (1)
To Study the operator \({{( H-\lambda )}^{-1}}\)
- (2)
To study the operator \({{H}^{-s}}\) using its Cauchy’s integral formula
$$\begin{aligned} \displaystyle {{H}^{-s}}=\frac{1}{2\pi i}\int _{\Delta }{{{\lambda }^{-s}}}{{(H-\lambda )}^{-1}}d\lambda \end{aligned}$$ - (3)
Studying f(H) using the representation formula
$$\begin{aligned} f(H)=\displaystyle \frac{1}{2\pi i}\int _{\rho -i\infty }^{\rho +i\infty }{M[f](s)}{{H}^{-s}}ds \end{aligned}$$
where \(r<0\) and \(\rho <-r\)
we only change the construction of the \((H-\lambda )^{-1}\)-parametrix. We prove by induction that the \((H-\lambda )^{-1}\) is an OPD and its Weyl symbol admits the development \({{b}_{\lambda }}\sim \sum {{{b}_{j,\lambda }}}\) where
\(\square \)
Proof of Proposition 1.6
using Proposition 4.2 and (4.19) formula we conclude
by combining (4.20) and Proposition 4.1 we get
From (4.21) and Proposition 2.5 (iii) we deduce that the operator \((\overline{V}-f(H)){{H}^{\frac{\delta +1}{2m}}}\) is bounded. We can write
Finally we get that the operator \([ {{{\bar{L}}}_{m}}-({{H}^{\frac{1}{m}}}+\tfrac{1}{m}{{H}^{\frac{1}{m}-1}}f(H)) ]{{H}^{1+\tfrac{\delta }{2m}-\tfrac{1}{2m}}}\) is also bounded. According to min-max theorem we have
Then
\(\square \)
Remark 4.3
We note that from (1.4) we have the following estimate
5 The relation between the spectrum of \({{L}_{m}}\) and \({\overline{L}_{m}}\)
Proof of Proposition 1.7
To establish Proposition 1.7, we need to prove the next result
Proposition 5.1
There exists a skew-symmetric operator \(Q\in G_{0}^{-(2m-2+\delta )}\) such as the operator \(({{e}^{Q}}{{L}_{m}}{{e}^{-Q}}-{{\overline{L}}_{m}}){{H}^{\frac{\delta -\eta }{m}+2-\frac{2}{m}}}\) is bounded, where \(\eta \in ] 0,2 [\).
Proof
The Q operator is built using the \(Q_{1}\) and \(Q_{2}\) operators as follows
where
and
Before starting the proof we could make sure that
where \(\overline{\overline{V}}=\frac{1}{2Ti}\int _{0}^{T}{\int _{0}^{t}{\left[ \frac{1}{m}{{H}^{\frac{1}{m}-1}}W(t),\frac{1}{m}{{H}^{\frac{1}{m}-1}}W(r) \right] }}drdt\)
We notice \(AdQ.{{L}_{m}}=\left[ Q,{{L}_{m}} \right] \). The differential equation
has a unique solution
To complete the proof of proposition we need the following lemma
Lemma 5.2
where \(\eta \in ]0,2 [\)
Proof
Using Proposition 2.5 (i) we can prove in analog way of Proposition 4.1 that \({{Q}_{1}}\in G_{0}^{-(2m+2-\delta )}\). Now let’s determine the class of \(\overline{\overline{V}}\), we can write
where
let’s start by clarifying the class of the operator \(\int _{0}^{T}{S(t)B(t)dt}\). For now we are interested in the operator S(t)B(t), its Weyl symbol \(c_{t}\) is given in [15] by
We split the oscillator integral \(c_{t}\) into two parts \(c_{t}^{(1)}\) and \(c_{t}^{(2)}\), then we use the cutoff functions
where \(\chi \in C_{0}^{\infty }(\mathbb {R})\), \(\chi \equiv 1\) in \(\left[ -1,1 \right] \), \(\chi \equiv 0\) in \(\mathbb {R}\backslash ] -2,2 [\), \(R={{\omega }^{2}}+{{\rho }^{2}}+{{r}^{2}}+{{\tau }^{2}}\), \(\varepsilon >0\) and \(\eta >0\). Let’s consider
\(c_{t}^{(1)}\) (resp \(c_{t}^{(2)}\)) the integral obtained in (5.4) by replacing the amplitude by \(d_{1}\) (resp \(d_{_{2}}\))
Study of \(c_{t}^{(2)}\)
On the support of \(d_{2}\) we have \(R\ge \varepsilon {{(1+{{x}^{2}}+{{\xi }^{2}})}^{\tfrac{\eta }{2}}}\). We make an integration by parts using the operator
We have for all \(k\in \mathbb {N}\)
then we obtain for all \(k>0\)
Uniformly with respect to \(t\in [ 0,T]\)
Study of \(c_{t}^{(1)}\)
On the support of \(d_{1}\) we have
By using (4.4) and (1.4) we can deduce for all \(\alpha ,\beta \in \mathbb {N}\)
by integrating along the interval [0, T] and following the same reasoning in Proposition 4.1 we get
On the support of \(d_{1}\), for \(\varepsilon \) small enough and since \(\eta \in ] 0,2 [\), there are positive constants \(c, c ', C, C'\) such that
It follows that
Finally
At the end by denoting \(\sigma \) the Weyl symbol of the operator \(\displaystyle \int _{0}^{T}{S(t)B(t)dt}\),
we have
We obtain the same estimates for \(\partial _{x}^{\alpha }\partial _{\xi }^{\beta }\sigma (x,\xi )\), we prove by the same way that \(Q_{2}\)\(\in G_{0}^{-(4m-4+\delta -2\eta )}\)\(\square \)
We return to the proof of Proposition 5.1, since \(V\in G_{0}^{0},\overline{V}\in G_{0}^{-\delta }\), \({{Q}_{1}},Q\in G_{0}^{-(2m-2+\delta )}\) and \(\overline{\overline{V}}\), \(Q_{2}\)\(\in G_{0}^{-(4m-4+2\delta -2\eta )}\), by using Proposition 2.5 we have
From what precedes we deduce that \(({{e}^{Q}}{{L}_{m}}{{e}^{-Q}}-{{\overline{L}}_{m}}){{H}^{\frac{\delta -\eta }{m}+2-\frac{2}{m}}}\) is bounded. \(\square \)
Come back to the proof of Proposition 1.7. We deduce from Proposition 5.1 that there exists a constant \(c> 0\) such that
According to the min-max theorem
To have a good estimate, let us specify the best choice of \(\eta \). Combining Remark 4.3, (5.11) and Proposition 5.1, we choose
\(\square \)
Now we prove Theorems 1.2 and 1.3. It is enough to combine Propositions 1.5, 1.6 and 1.7 we deduce
and for \(m=1\) (harmonic oscillator case)
where \(\eta \in ] 0,\frac{\delta }{2} [\).
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Aarab, I., Tagmouti, M.A. Harmonic oscillator perturbed by a decreasing scalar potential. J. Pseudo-Differ. Oper. Appl. 11, 141–157 (2020). https://doi.org/10.1007/s11868-019-00284-4
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DOI: https://doi.org/10.1007/s11868-019-00284-4