Bethe-Sommerfeld conjecture for periodic Schrödinger operators in strip

J. Math. Anal. Appl. 479 (2019) 260–282

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Bethe-Sommerfeld conjecture for periodic Schrödinger operators in strip D.I. Borisov a,b,c,∗ a b c

Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russia Bashkir State University, Ufa, Russia University of Hradec Králové, Hradec Králové, Czech Republic

a r t i c l e

i n f o

Article history: Received 10 December 2018 Available online 12 June 2019 Submitted by P. Exner Keywords: Periodic operator Strip Band spectrum Gap Bethe-Sommerfeld conjecture

a b s t r a c t We consider the Dirichlet Laplacian in a straight planar strip perturbed by a bounded periodic symmetric operator. We prove the classical Bethe-Sommerfeld conjecture for this operator, namely, that this operator has finitely many gaps in its spectrum provided a certain special function written as a series satisfies some lower bound. We show that this is indeed the case if the ratio of the period and the width of strip is less than a certain explicit number, which is approximately equal to 0.10121. We also find explicitly the point in the spectrum, above which there is no internal gaps. We then study the case of a sufficiently small period and we prove that in such case the considered operator has no internal gaps in the spectrum. The conditions ensuring the absence are written as certain explicit inequalities. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The classical Bethe-Sommerfeld conjecture says that a multi-dimensional periodic differential operator has finitely many gaps in its spectrum. This conjecture was proved for a wide class of operators in multidimensional spaces. The case of Schrödinger operator with a periodic potential or, more generally, with a bounded periodic symmetric operator, was studied in [7], [19], [14], [9], [18], [17], [20], [10] and the BetheSommerfeld conjecture was proved under various conditions for the potential and the bounded periodic symmetric operator including various cases of unbounded potentials. In [11], [13], this conjecture was proved for the magnetic Schrödinger operator. Papers [15], [1], [16] were devoted to proving the Bethe-Sommerfeld conjecture for polyharmonic operators perturbed by a pseudodifferential operator of a lower order obeying certain conditions. Apart from operators in multi-dimensional spaces, the Bethe-Sommerfeld conjecture can be formulated also for differential operators in periodic domains. The examples of such domains are strips, cylinders or * Correspondence to: Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russia. E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2019.06.026 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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layers. Here the simplest model is the periodic Schrödinger operator in a two-dimensional planar strip. Such model was studied in PhD thesis [2]. Assuming that 1 T < , d 16

(1.1)

where d was the width of the strip and 2T was the period, it was proved in [2] that the considered operator has finitely many gaps in the spectrum. We also note that this result appeared only in the cited PhD thesis and was not published as a usual paper in a journal. One more example of differential operators on periodic domains are the operators on periodic graphs. Here the situation changes substantially and the operators on the graphs can have infinitely many gaps, or finitely many gaps or no gaps at all. More details on the Bethe-Sommerfeld conjecture for operators on periodic graphs can be found, for instance, in recent work [8] devoted to finding examples of quantum graphs with finitely many gaps in the spectra. The Bethe-Sommerfeld conjecture can be interpreted as the absence of the gaps above some point, that is, in the higher part of the spectrum. This suggests another problem on finding the periodic operators having no gaps at all; such problem can be called a strong Bethe-Sommerfeld conjecture. This issue was studied, for instance, in [7] and [17]. It was found that for the periodic Schrödinger operators in the multi-dimensional space this is true provided the potential is small enough, see Remark in [7] and Theorems 15.2 and 15.6 in [17, Ch. III, Sect. 15]. By a simple rescaling, this result can be also reformulated as follows: the periodic Schrödinger operator in entire space has no gaps if the period is small enough. The aforementioned results on the strong Bethe-Sommerfeld conjecture motivated very recent studies on periodic operators with a small period in [4], [5], [3], [6]. The considered operators were a periodic Schrödinger operator [4], a periodic magnetic Schrödinger operator [3], the Laplacian with frequently alternating boundary conditions [5] and the Laplacian with a periodic delta interaction [6]. The main result of the cited works was as follows: for a sufficiently small period, as T < T0 , the considered operators has no internal spectral gaps at least till certain point λT in the spectrum. The upper bound T0 for the period ensuring this result was found explicitly, as a particular number. The point λT was also found explicitly as a rather simple function of T . It was shown that λT behaved as O(T −6 ) as T goes to zero. We stress that this result does not state the absence of the gaps in the entire spectrum but only in its lower part. At the same time, we succeeded to consider more complicated operators and not only the periodic Schrödinger operator. Here it is important to stress that the approach used in [2] is rather limited and it can not be extended to the operators with stronger perturbations like in [5], [3], [6]. The technique in the latter works, namely, the key estimates, were based on different ideas in comparison with that in [2]. In the present work we study the same model as in [2], namely, the Dirichlet Laplacian in a strip perturbed by a bounded periodic symmetric operator. We again study the internal gaps in the spectrum but in greater details. Our first result is the proof of the classical Bethe-Sommerfeld conjecture under weaker conditions. Namely, we show that it is true provided T < ξ0 ≈ 0.10121 d

(1.2)

and this condition is better than (1.1). For other values of Td , the classical Bethe-Sommerfeld conjecture holds if a certain special function written explicitly as a series satisfies certain lower bound, see (2.2), (2.3). Here we also find explicitly a point in the spectrum, above which there is surely no gaps. We also study the case of a small period. Here we prove that if condition (1.2) is satisfied and the numerical range of the perturbation is not too wide, the considered operator has no internal gaps in the spectrum. The conditions for the perturbation are explicit and rather simple, see (2.6), (2.7). In particular, these conditions imply the following statement: varying the period of a potential and keeping its oscillation

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uniformly bounded, for sufficiently small periods the corresponding periodic Schrödinger operator in the strip has no internal gaps. This result fits very well what was said above about periodic Schrödinger operators in multi-dimensional space with small periods. The approach we use follows the same lines as in [2], namely, it is based on the ideas from [7]. But while proving the key estimates for the Fourier coefficients of the counting function, we succeeded to do this in a shorter and simpler way tracking at the same time all the constants explicitly. In the proof of the strong Bethe-Sommerfeld conjecture we also employ the approach developed in [3], [4], [5], [6]. 2. Problem and main results Let x = (x1 , x2 ) be Cartesian coordinates in R2 , Π := {x : 0 < x2 < d} be an infinite horizontal strip of a width d > 0, and  := {x : |x1 | < T, 0 < x2 < d} be a periodicity cell, where T > 0 is a constant. By L0 we denote a bounded symmetric operator in L2 () and S(n) stands for the translation operator in L2 (Π) acting as (S(n)u)(x) = u(x1 − 2T n, x2 ). By means of the operators L0 and S(n) we introduce one more operator in L2 (Π): Lu = S(−n)L0 S(n)u

on n ,

n ∈ Z,

where n := {x : (x1 − 2T n, x2 ) ∈ }. This definition of the operator L can be explained as follows: the restriction on n of a function u ∈ L2 (Π) belongs  to L2 (). Identifying then the spaces L2 () and L2 (n ), we apply the operator L0 to the restriction u and the result is translated to the cell n . This is the n action of the operator L on u on the cell n . The operator L is bounded, symmetric and periodic. The latter is understood in the sense of the identity S(m)L = LS(m)

for each

m ∈ Z.

The main object of our study is the periodic operator H := −Δ + L

in L2 (Π)

˚ 2(Π) consisting of subject to the Dirichlet condition. The domain of this operator is the Sobolev space W 2 the functions in W22 (Π) with the zero trace on ∂Π. The operator H is self-adjoint. We denote ω− :=

(Lu, u)L2 () , 2 u∈L2 () uL () 2 inf

u=0

ω+ :=

(Lu, u)L2 () , 2 u∈L2 () uL2 () sup

ωL := ω+ − ω− ,

(2.1)

u=0

and ξ :=

T . d

Given ξ > 0, for  > 0, p ∈ N we introduce the function:    1 k2 2− π 2 sin 2π + p  2 ξ 4 1 ϕp () := .   34 πξ k2 2 k∈Z ξ2 + p

(2.2)

Theorem 2.1. Assume that for a given ξ there exists three constants c0 = c0 (ξ) > 0, 0 = 0 (ξ)  1 and γ = γ(ξ) < 14 such that for   0 the inequality holds:

D.I. Borisov / J. Math. Anal. Appl. 479 (2019) 260–282

sup |ϕp (ξ, )|  c0 −γ .

263

(2.3)

p∈N

Then, for this ξ, the spectrum of the operator H has finitely many internal gaps. Moreover, there are no internal gaps in the half-line [1 , +∞), where 2   √ 1−2γ  4

π2 4 2π + 6 1−4γ 1 25 1 := 2 max 0 , , 9+ , T 3πc0 8π 2 ξc0 1024π 2  4 T 1 3ξT 1−4γ ωL + + + ω− . 4πc0 ξ 2c0 4πc0

(2.4)

Our next main result states that condition (2.3) holds for sufficiently small ξ. Theorem 2.2. Let ξ < ξ0 ,

(2.5)

where

ξ0 :=

c1   2ζ 32

c1 := 

c2 c22 + 1

23 ≈ 0.10121, ,

√ √ √ 1 8 (78 3 + 54 11) 3 − 3 c2 := − √ √ 1 , 9 3(78 3 + 54 11) 3

where ζ(t) is the Riemann zeta function. Then condition (2.3) holds with 0 (ξ) = 1,

γ = 0,

  3 c1 − 2ζ 32 ξ 2 c0 = πξ

and the statement of Theorem 2.1 is true. In the next theorem we prove the strong Bethe-Sommerfeld conjecture. Theorem 2.3. Assume that condition (2.5) holds and  T2 ((A(ξ) − ξ)2 + 1)2 3 + 4ξ 2 + ξ 2 − A (ξ), A(ξ) := , 0  2 ωL < π 4 3  √  3 3π 2 3 (3 + 2 2)π + 3 ξ 25 T ωL 0 c1 − 2ζ ξ2 − ξ− ξ − . 9+ − 2 2 6 4 32π 1024π 4

(2.6) (2.7)

Then the spectrum of the operator H has no internal gaps. Let us discuss briefly the main results. The first theorem states the classical Bethe-Sommerfeld conjecture. Namely, provided estimate (2.4) holds, the considered operator has finitely many gaps in its spectrum and surely there are no gaps above the point 1 . The natural question is whether estimate (2.4) is true or not. Theorem 2.2 says that provided ξ is not too big, namely, if ξ obeys (2.5), estimate (2.4) is true and the classical Bethe-Sommerfeld conjecture holds. We stress that condition (2.4) is better than similar 1 condition (1.1) in [2] since 16 = 0.0625 is less than ξ0 . We failed to prove estimate (2.4) for other values of

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ξ but numerical tests show that this estimate is very likely true for all values of ξ. In Section 6 we discuss the functions ϕp () and condition (2.3) in more details. Theorem 2.3 is devoted to the case of a small period. Here we prove the absence of the internal gaps in the spectrum provided conditions (2.5), (2.6), (2.7) are satisfied. And as we see easily, these conditions hold for a sufficiently small period T assuming that the width d and the oscillation ωL are fixed. For instance, this implies that given a fixed bounded potential V (x1 , x2 ), which is 2π-periodic in x1 , the Schrödinger operator −Δ + V ( xε1 , x2 ) in the strip Π subject to the Dirichlet boundary condition has no internal gaps provided ε is small enough. In view of this result and the aforementioned results in [3], [4], [5], [6], [7], [17], we could formulate a strong Bethe-Sommerfeld conjecture: multi-dimensional periodic differential operators, for which the classical Bethe-Sommerfeld conjecture holds, have no internal gaps in their spectra if the period is small enough. In conclusion we stress that our technique and results can be also extended to the case of Neumann or Robin boundary condition (with a constant coefficient). Of course, for other boundary conditions all the constants in Theorems 2.1, 2.2, 2.3 are different. 3. Counting functions This section is devoted to the preliminary notations and statements used then the proofs of Theorems 2.1, 2.2, 2.3. Since the operator H is periodic, its spectrum has a band structure and it can be described in terms of the band functions. In order to do this, we first define the operator 2  πτ πτ ∂ πτ ∂2 H(τ ) := i + − + ei T x1 Le−i T x1 , 2 ∂x1 T ∂x2

  τ ∈ − 12 , 12 ,

in L2 () subject to the Dirichlet condition on ∂ ∩ ∂Π and to the periodic boundary conditions on the ˚ 2 () consisting of the functions in lateral boundaries of . The domain of this operator is the space W 2,per 2 W2 () satisfying the Dirichlet condition on ∂ ∩ ∂Π and the periodic conditions on the lateral boundaries of . The operator H(τ ) is self-adjoint and has a compact resolvent. The spectrum of the operator H(τ ) consists of countably many discrete eigenvalues. These eigenvalues are taken in the ascending order counting the multiplicities and are denoted by Ek (τ ), k ∈ N. By Ek0 (τ ) we denote the same eigenvalues in the case L = 0, that is, they are associated with the Dirichlet Laplacian in Π. The latter operator is denoted by H0 and the associated operator on the periodicity cell  is H0 (τ ). The well-known formulae for the spectra of the operators H and H0 are 

  Ek (τ ) : τ ∈ − 12 , 12 ,

σ(H) =

k∈Z

σ(H0 ) =



  Ek0 (τ ) : τ ∈ − 12 , 12 .

k∈Z

By N0 (, τ ) we denote the rescaled counting function of the operator H(τ ) in the case L = 0:   π2  0 0 N0 (, τ ) = # Ek (τ ) : Ek (τ )  2 . T

(3.1)

Since the function N0 (, τ ) is associated with the Dirichlet Laplacian in Π, we can calculate explicitly the eigenvalues of H0 (τ ): 

Ek0 (τ ),



k∈N =



 π 2 m2 π2 2 (τ + n) + , n ∈ Z, m ∈ N . T2 d2

(3.2)

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The eigenvalues in the right hand side correspond to the eigenfunctions e−i T x1 sin πm d x2 and they do not follow the ascending order. This is why we write (3.2) as the identity for two sets of the eigenvalues. The counting function N0 (, τ ) can be written as πn



N0 (, τ ) =



1=

n∈Z, m∈N (n+τ )2 +ξ 2 m2 



1+

n∈Z+ , m∈N (n+τ )2 +ξ 2 m2 

1.

(3.3)

n∈Z+ , m∈N (n+1−τ )2 +ξ 2 m2 

  The definition of the function N0 implies immediately that this function is even in τ ∈ − 12 , 12 . By ap we denote the Fourier coefficients of this function: 1

1

2 a0 () :=

2 N0 (, τ ) dτ,

ap () :=

− 12

N0 (, τ ) cos(2πpτ ) dτ,

p ∈ N;

(3.4)

− 12

the Fourier series for N0 (, τ ) reads as N0 (, τ ) = a0 () + 2

∞ 

ap () cos 2πpτ.

p=1

The functions Ek0 (τ ) satisfy the estimate Ek0 (τ )  non-zero only for

π2 d2

and therefore, the counting function N0 (, τ ) is

  ξ2. In what follows we assume that this inequality is satisfied. 3.1. Coefficient a0 () In this subsection we calculate and estimate the coefficient a0 (). By · we denote the integer part of a number, while · stands for the fractional part. By straightforward calculations we get: 

a0 () =

1+

n∈Z+ , m∈N 1 (n+ 2 )2 +ξ 2 m2 

1

2

 n∈Z+ , m∈N (n+1)2 +ξ 2 m2 

1

2

n∈Z+ , m∈N 0 (n+τ )2 +ξ 2 m2    1 2 n2 +ξ 2 m2 < n+ 2 +ξ 2 m2



+2



1+2

dτ

n∈Z+ , m∈N 0 (n+1−τ )2 +ξ 2 m2  2  1 n+ 2 +ξ 2 m2 <(n+1)2 +ξ 2 m2 

=

1 2 ξ



  m=1

+2



 − ξ 2 m2 + 

1 2

 +

1 2 ξ



  m=1

n∈Z+ , m∈N   1 2 n2 +ξ 2 m2 < n+ 2 +ξ 2 m2

 − ξ 2 m2



   − ξ 2 m2 − n

dτ

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266





+2 

1 n+ 2

2

 − ξ 2 m2 − n −

1 2

 .

n∈Z+ , m∈N +ξ 2 m2 <(n+1)2 +ξ 2 m2

Hence, 



1 2 ξ





a0 () =

2

   − ξ 2 m2 +

 m=1  1 −ξ 2 m2 < 2 

1 2 ξ





 

2

   − ξ 2 m2

m=1  1 −ξ 2 m2 < 2





     − ξ 2 m2 + 1 + 2



+

1 2 ξ

m=1  1 −ξ 2 m2  2

1 2 ξ



     − ξ 2 m2 − 1 2

  m=1  1 −ξ 2 m2  2

and therefore,   12

a0 () = 2

ξ    − ξ 2 m2 .

(3.5)

m=1

Lemma 3.1. The function a0 () is monotonically increasing and for each ξ 2    ˜ the estimate holds: ˜ − a0 ()  a0 ()

 π ˜ ( − ) + ˜ − . 2ξ

Proof. By formula (3.5) we have: 

1 ˜ 2 ξ





1 2



ξ     2 2 ˜ ˜ a0 () − a0 () = 2  − ξ 2 m2 −ξ m −2

m=1



=2

1 2 ξ

m=1



 



˜ − ξ 2 m2 −



−

ξ 2 m2

 +2

 m=

=2

1 2 ξ



 m=1





m=1



1 ˜ 2 ξ



1 ˜ 2 ξ

1 2 ξ

 +1





˜ −   +2  ˜ − ξ 2 m2 +  − ξ 2 m2

 m=

1 2 ξ

 +1

 ˜ − ξ 2 m2



˜ − ξ 2 m2 .

Since the function t → 

1  2 2 ˜  − ξ t +  − ξ 2 t2

 1   is monotonically increasing as t ∈ 0, ξ2 , and the function t → ˜ − ξ 2 t2 is monotonically decreasing as  1 ˜1  t ∈ ξ2 , ξ2 , we can continue estimating as follows:

D.I. Borisov / J. Math. Anal. Appl. 479 (2019) 260–282 



1 2 ξ



2(˜ − ) dt   +  1 1 ˜ − ξ 2 t2 +  − ξ 2 t2 2 2 2 ˜  − ξ  ξ +  − ξ 2  ξ2 2

˜ − a0 ()  2(˜ − ) a0 () 0



1 ˜ 2 ξ







+2 



1 2 ξ

267

 π ˜ ( − ) + ˜ − . ˜ − ξ 2 t2 dt  2ξ

The proof is complete. 2 3.2. Coefficient ap In this subsection we calculate the coefficients ap and estimate them. As in the previous subsection, by (3.3) and the parity of N0 we have 1

2

1

2



ap () = 2

cos 2πpτ dτ + 2

n∈Z+ , m∈N 0 (n+τ )2 +ξ 2 m2  1

2

n∈Z+ , m∈N 0 (n+1−τ )2 +ξ 2 m2 



=2



cos 2πpτ dτ

n∈Z+ , m∈N 0   1 2 n2 +ξ 2 m2 < n+ 2 +ξ 2 m2 (n+τ )2 +ξ 2 m2  1

2



+2 0 

1 n+ 2



=2



2

+ξ 2 m2 <(n+1)2 +ξ 2 m2

(n+1−τ )2 +ξ 2 m2    −ξ 2 m2



1 2 ξ

cos 2πpτ dτ



0

m=1,...,   −ξ 2 m2 n=   1 0 −ξ 2 m2 < 2

+2

cos 2πpτ dτ

n∈Z+ , m∈N

1

2

 

1 2 ξ



cos 2πpτ dτ,   1− −ξ 2 m2

m=1,...,   −ξ 2 m2 n=   1 −ξ 2 m2 <1 2

and thus, 

ap () =

1 2 ξ



 1  sin 2πp  − ξ 2 m2 . πp m=1

cos 2πpτ dτ

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Our next step is to transform the above  identity to an integral form. We again employ the Euler-Maclaurin formula for the function t → sin 2πp  − ξ 2 t2 , see [12, Ch. 1, Sect. 1.1, Thm. 1.3]: 1 2

ap () :=

1 πp

ξ

sin 2πp 0

1  sin(2πp 2 )  − ξ 2 t2 dt − √ 2πp

1 2

ξ −2 0

 ξ 2 t φ(t)  cos 2πp  − ξ 2 t2 dt,  − ξ 2 t2

φ(t) := t − 12 .

1

In both integrals we make the change t →

2 ξ

sin t: 1

ap () := Sp(1) () + Sp(2) () −

sin(2πp 2 ) , 2πp

(3.6)

π

Sp(1) () :=

1 2

 πpξ

2

1

sin(2πp 2 cos t) cos t dt, 0



π

Sp(2) () := −2

1 2

2

φ

1 1 2 sin t cos(2πp 2 cos t) sin t dt. ξ

0

Here the first integral is the well-known representation for the Bessel function: 1

Sp(1) () =

1 2 J1 (2πp 2 ). 2pξ

(3.7)

The results of [21, Ch. VII, Sect. 7.3] imply the estimate    √       3  π  π  5  π  2 2  cos t + t − + t +  √ J1 (t) + cos  cos  5  πt 4  8 π t 32 4 4 16t 2 as t > 0. By (3.7) and the Cauchy-Schwarz inequality this leads us to the estimate    1    1 1 1 3  4 π  π   (1) 2 2 − 2πp cos 2πp + Sp () +  cos  3 5 1  4  32π 2 ξ p 2  4 4 2πp 2 ξ   1 5 π   2 + 2πp +  7 3 cos 4 32πp 2  4  1 25  9+ 5 1 2 p2  2 1024π 32π p 2 ξ 4 for all  > 0. Lemma 3.2. The identity holds:

2 1 1 ∞ 2  1 2 1 2 2πk sin t cos(2πp 2 cos t) sin t dt. Sp(2) () = sin π k ξ π

k=1

0

(3.8)

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269

Proof. We represent the function φ by its Fourier series φ(z) = −

∞ 1  sin 2πkz π k k=1

and we know that 2 R  N  sin 2πkz   lim φ(z) +  dz = 0 N →+∞  πk 

(3.9)

k=1

0

for each fixed R > 0. It was also stated in [12, Ch. 1, Sect. 1.1] that the estimate   N 1  sin 2πkz      C, π  k

z ∈ [0, R],

(3.10)

k=1

is true, where C is some constant independent of N and z. By the Hölder inequality we obtain 2  π  

2  1 N    1 1 2πk 2 1  (2)  sin t cos(2πp 2 cos t) sin t dt sin 1 Sp () −  πk ξ 2 2  k=1  0  π⎛ 2  ⎞ 1 2 

1 N sin 2πk 2 sin t    2 ξ 1   ⎜  ⎟ sin t + =  ⎝φ ⎠ cos(2πp 2 cos t) sin t dt   ξ πk k=1 0      2 π 1 2  1 N sin 2πk 2 sin t   2 ξ    sin t +  C φ  dt   ξ πk k=1   0 1

2 2 ξ  N  sin(2πkz)  dz  =C φ(z) +     πk  − ξ2z2 k=1

0

⎛

⎞ 13 ⎛ 1 ⎞ 23 2   6 ξ N ⎜  ⎟ ⎜ ⎟  sin(2πkz)  dz ⎜  ⎟ ⎜ ⎟  C ⎜ φ(z) + dz , ⎟ ⎜  3 ⎟  πk ⎝  ⎠ ⎝ ( − ξ 2 z 2 ) 4 ⎠ k=1 1 2 ξ

0

0

where the symbol C stands for some inessential constants independent of N . The function φ(z) is uniformly bounded and employing estimate (3.10), we continue estimating as follows: 2  π  

2 1 N    1 2πk 2 1  (2) 1  sin t cos(2πp 2 cos t) sin t dt sin 1 Sp () −   πk ξ 2 2  k=1  0 ⎛

⎞ 13   2 N ⎜  ⎟  sin(2πkz)  ⎜  ⎟  C ⎜ φ(z) +  dz ⎟ .  πk ⎝  ⎠ 1 2 ξ

0

k=1

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270

By (3.9), the right hand side of the obtained inequality tends to zero as N → +∞ and this proves the lemma. 2 By the formula

sin

    1 1 1 1 1 k k 2πk 2 1 sin t cos(2πp 2 cos t) = sin 2π 2 sin t + p cos t + sin 2π 2 sin t − p cos t ξ 2 ξ 2 ξ

we get: Sp(2) () =

1 ∞   2  1  (2,k) (2,k) Sp,+ () + Sp,− () , π k

(3.11)

k=1

2 :=





π

(2,k) Sp,± ()

sin 2π

1 2

k sin t ± p cos t ξ

sin t dt.

0

We denote ' αp,k

pξ = αp,k (T ) := arctan , k

ηp,k = ηp,k (, T ) := 2π

1 2

k2 + p2 . ξ2

(2,k)

Then the formulae for Sp,± can be rewritten as π 2 ±αp,k

π

(2,k)

2



sin(ηp,k sin(t ± αp,k )) sin t dt =

Sp,± () =

sin(ηp,k sin t) sin(t ∓ αp,k ) dt. ±αp,k

0

Hence, thanks to the parity properties of sin t and cos t, ⎛

π 2 −αp,k

π 2 +αp,k



⎜ (2,k) (2,k) Sp,+ + Sp,− = cos αp,k ⎝



sin(ηp,k sin t) sin t dt + −αp,k

⎛ ⎜ + sin αp,k ⎝

⎞ ⎟ sin(ηp,k sin t) sin t dt⎠

αp,k π 2 +αp,k

α p,k



sin(ηp,k sin t) cos t dt −

−αp,k

⎞ ⎟ sin(ηp,k sin t) cos t dt⎠

π 2 −αp,k

π

2

sin(ηp,k sin t) sin t dt.

= 2 cos αp,k 0

In the last integral we make the change of the variable t → sin

π 4

−

t 2



and we get:

1 √

(2,k)

(2,k)

2

Sp,+ + Sp,− = 4 cos αp,k 0

1 − 2t2 sin(ηp,k (1 − 2t2 )) √ dt. 1 − t2

We denote h(s) := √

1 2 √ − 1−s 1−s+ 1−s

(3.12)

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271

and we see that 1 − 2t2 √ = 1 − t2 h(t2 ). 1 − t2 We substitute this identity into the integral in (3.12):   (2,k) (2,k) Sp,+ + Sp,− = cos αp,k Sp(4,k) − Sp(3,k) , 1 √

1 √

2

2 sin(ηp,k (1 − 2t2 ))h(t2 )t2 dt,

Sp(3,k) := 4 (3,k)

sin(ηp,k (1 − 2t2 )) dt.

Sp(4,k) := 4

0

In the integral Sp

(3.13)

0

we integrate by parts as follows: 1 √

Sp(3,k) =

1

2 h(t2 )t d cos(ηp,k (1 − 2t2 ))

ηp,k 0

1

√ √ 2  2  1 2 − h(t ) + 2t2 h (t2 ) cos(ηp,k (1 − 2t2 )) dt. = ηp,k ηp,k

(3.14)

0

The function h(s) + 2sh (s) =

1 1 2 √ √ + + √ 3 2 (1 + 1 − s) (1 − s)(1 + 1 − s) (1 − s) 2 (1 + 1 − s)2

grows monotonically as s ∈ [0, 12 ] and hence, 0<

3 = h(0)  h(t2 ) + 2t2 h (t2 ). 2

Employing this estimate, we get   √1  √12 2    √   2    h(t2 ) + 2t2 h (t2 ) cos ηp,k (1 − 2t2 ) dt  h(t ) + 2t2 h (t2 ) dt = 2,     0 0 and by (3.14) we obtain |Sp(3,k) |

√ 2 2  . ηp,k

In view of the definition of αp,k we have 1 cos αp,k = 1 . k (k2 + p2 ξ 2 ) 2

(3.15)

  √  +∞ ∞  1 1 cos αp,k (3,k)  2ξ  1 coth(πpξ)  . Sp   − = √  1 1   k pξ πp2 ξ 2 π 2 k=1 k2 + p2 ξ 2 2 2 k=1

(3.16)

Hence,

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(4,k)

We proceed to estimating Sp

.

Lemma 3.3. The estimate     1 ∞ ∞  1 1 cos αp,k (4,k) ξ 2  sin ηp,k − π4  coth(πpξ)   Sp − − 1 √   3 1  k pξ πp2 ξ 2  4 k=1 (k2 + p2 ξ 2 ) 4  2 2 k=1 holds true. Proof. We make the change of the variable t → √

Sp(4,k) =

2 2

1



sin(ηp,k − t2 ) dt =

1 2 ηp,k

√

2 ηp,k

√

2π

1 2 ηp,k

0

1

(4,k)

2 2ηp,k t in the integral Sp

:

+∞ √   π 2 2 − 1 sin ηp,k − sin(ηp,k − t2 ) dt. 4 2 ηp,k 1

(3.17)

2 ηp,k

For the latter integral we have: +∞ +∞ +∞    1 d cos(ηp,k − t2 ) cos(ηp,k − t2 ) 2 = − sin(ηp,k − t ) dt = − − dt, 1 2t 2t2 2 2ηp,k 1 1 1 2 ηp,k

2 ηp,k

2 ηp,k

       +∞ cos(η − t2 )  1   p,k dt  ,  1 2   2t 2  1  2ηp,k 2 ηp,k 

and therefore, +∞  1 0− sin(ηp,k − t2 ) dt  1 . 2 ηp,k 1 2 ηp,k

Hence, by (3.15) and (3.17) we get    √ 1 ∞ ∞ ∞  cos αp,k (4,k) ξ 2  sin ηp,k − π4  1 2ξ    Sp − 1   3 1 2 2 2 2  k  4 k=1 (k + p ξ ) 4  π 2 k=1 k + p2 ξ 2 k=1  1 1 coth(πpξ) − 2 2 . =√ 1 pξ πp ξ 2 2 The proof is complete. 2 Identities (3.11), (3.12), (3.13), estimate (3.16) and Lemma 3.3 yield:    √  1 1 ∞  1 ξ 2  4  sin ηp,k − π4  2 coth(πpξ)  (2) − 2 2 . Sp () − 3   1  π pξ πp ξ (k2 + p2 ξ 2 ) 4  2 k=1

The derivative 

coth z 1 − 2 z z

 =

1 z 2 sinh z

 1−

4 cosh z2  z z z z + sinh − cosh sinh z z 2 2 2

(3.18)

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273

is non-positive since 1−

z  0, sinh z

sinh z − z cosh z  0.

Two latter inequalities can be easily checked by calculating the values at zero and the derivatives of the functions in their left hand sides. Then √  √  √ 1 1 2 coth πz 2 coth πz 2 − 2  lim − 2 = . 0 z→0 π π z πz z πz 3 Hence, by (3.6), (3.8), (3.18) and definition (2.2) of the function ϕp we infer that the coefficient ap can be represented as 1

ap () =  4 ϕp (ξ, ) + Sp(5) (), √  1 1 25 2 (5) + + . 9+ |Sp ()|  3 2π 32π 2 ξ 14 p 52 1024π 2 p2 

(3.19) (3.20)

4. Finitely many gaps In this section we prove Theorems 2.1, 2.2. We follow the lines of work [7] with certain minor modifications. We begin with an auxiliary lemma. Lemma 4.1. The estimates sup

( ) 1 1 τ∈ − 2 , 2

N0 (, τ )  a0 () +

1 sup {|ap ()|}, 2 p∈N

 a0 () −

1 sup {|ap ()|}, 2 p∈N

(inf ) N0 (, τ ) 1 1 τ∈ − 2 , 2

hold true. Proof. We introduce the functions ˜ 0 (τ, ) := N0 (, τ ) − a0 (), N ˜ 0 (τ, ), 0}, ˜ 0 (τ, ) := max{N N + ˜ 0 (τ, ), 0}. ˜ 0 (τ, ) := min{N N − These functions obey the identities 1

2 ˜ 0 (τ, ) dτ = 0, N − 12 1

1

2 ˜ 0 (τ, )| dτ = |N − 12

1

2

2 ˜ 0 (τ, ) dτ − N +

− 12

˜ 0 (τ, ) dτ N − − 12

1

1

2 =2 − 12

2 ˜ 0 (τ, ) dτ = −2 N + − 12

˜ 0 (τ, ) dτ N −

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and therefore, ˜0 (inf ) N− (τ, ) 1 1 τ∈ − 2 , 2 0

=

˜ 0 (τ, )  0,

(inf ) N 1 1 τ∈ − 2 , 2

˜ 0 (τ, ) = N

sup

) ( 1 1 τ∈ − 2 , 2

sup

) ( 1 1 τ∈ − 2 , 2

˜ 0 (τ, ). N +

Hence, by definition (3.4) of ap () we obtain immediately 1

2 sup

) ( 1 1 τ∈ − 2 , 2

˜ 0 (τ, )  N

1

˜ 0 (τ, ) dτ = 1 N + 2

− 12

2 ˜ 0 (τ, )| dτ  |N − 12

1

2 ˜ 0 (τ, ) 

(inf ) N 1 1 τ∈ − 2 , 2

1 |ap ()|, 2

1

˜ 0 (τ, ) dτ = − 1 N − 2

− 12

2

˜ 0 (τ, )| dτ  − 1 |ap ()|. |N 2

− 12

˜ 0 (τ, ) imply the statement of the lemma. 2 These inequalities and the definition of the function N By identity (3.19) and inequality (3.20) we can estimate the supremum of |ap ()| from below: 1

1

sup{|ap ()|}   4 sup{|ϕp (ξ, )|} − sup{|Sp(5) ()|}   4 sup{|ϕp (ξ, )|} − S (6) (), p

√

S (6) () =

p

1 1 2 + + 3 2π 32π 2 ξ 14



p

9+

p

25 . 1024π 2 

(4.1) (4.2)

It is easy to confirm that c0 1 −γ 4 as   2 , 2 ⎫ ⎧ 2

1−2γ 4  ⎬ ⎨  4√2π + 6 1−4γ 1 25 , , 9 + 2 := max 0 , ⎭ ⎩ 3πc0 8π 2 ξc0 1024π 2 S (6) () 

where γ comes from condition (2.3). Hence, by condition (2.3) and Lemma 4.1, as   2 , the estimates hold: sup

N0 (, τ )  a0 () +

c0 1 −γ 4 , 2

) N0 (, τ )  a0 () −

c0 1 −γ 4 . 2

) ( 1 1 τ∈ − 2 , 2 (inf

1 1 τ∈ − 2 , 2

(4.3)

Let [ηk0 , θk0 ], k  1, be the kth band of the operator H in the case L = 0, that is, 0 (min ) Ek (τ ) 1 1 τ∈ − 2 , 2

= ηk0 ,

max (

1 1 τ∈ − 2 , 2

0 ) Ek (τ )

= θk0 .

By the definition of the counting function N0 (, τ ), for a fixed , the number of the band functions Ek0(τ ) 2 whose minima do not exceed πT 2 is equal to (sup ) N0 (, τ ), while (inf ) N0 (, τ ) is the number of the 1 1 τ∈ − 2 , 2

1 1 τ∈ − 2 , 2

band functions Ek0 (τ ) whose maxima do not exceed

π2  T2 .

Hence, for each k  1,

D.I. Borisov / J. Math. Anal. Appl. 479 (2019) 260–282

 sup

) ( 1 1 τ∈ − 2 , 2

N0

T 2 ηk0 ,τ π2



= k,

(inf ) N0 1 1 τ∈ − 2 , 2

T 2 θk0 ,τ π2

275

= k.

Assuming now ηk0 

π2 2 , T2

(4.4)

by (4.3) we obtain  k + 1  a0

T2 0 η π 2 k+1



c0 + 2



T2 0 η π 2 k+1

14 −γ

 k  a0

,

T2 0 θ π2 k



c0 − 2



T2 0 θ π2 k

14 −γ .

The operator H as L = 0 is the Dirichlet Laplacian and its spectrum has no internal spectral gaps. Therefore, 0 ηk+1  θk0 and by Lemma 3.1 and the inequality 4 2 α + 3β 2 , 3

α2 + 2αβ 

α, β  0,

this implies: T 2πξ



4 0 0 (θ − ηk+1 ) + 3ξ 2 3 k



T T 0 0 0 (θk0 − ηk+1 )+ θk − ηk+1 2πξ π  2  2 T 0 T 0 − a  a0 θ η 0 π2 k π 2 k+1  12 −2γ  0 1 −γ  1 T 0  c0 (θk ) 4 + (ηk+1 ) 4 −γ − 1 π  2 14 −γ T 0  2c0 η − 1. π2 k 

(4.5)

Hence, θk0

−

0 ηk+1

3πξc0  T



T2 0 η π2 k

14 −γ

−

9ξ 2 3πξ − . 2T 4

(4.6)

Since ηk0 → +∞ as k → +∞, the above estimate means that the length of the overlapping of the bands in the spectrum of H as L = 0 grows as k → +∞. Let [ηk , θk ], k  1, be the spectral bands of the operator H for a given operator L. In view of definition (2.1) of ω± and the minimax principle for each k we have ηk0 + ω−  ηk  ηk0 + ω+ ,

θk0 + ω−  θk  θk0 + ω+ .

Hence, by (4.6), the bands [ηk , θk ] overlap for sufficiently large k, namely, as 3πξc0 T



T2 0 η π2 k

14 −γ

−

9 3πξ − ξ 2  ωL , 2T 4

or, equivalently, as 

T2 0 η π2 k

14 −γ



T 1 3ξT ωL + + . 4c0 πξ 2c0 4πc0

(4.7)

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In addition, condition (4.4) should be satisfied. Both these conditions are true if ηk0  1 − ω− , where 1 was introduced in (2.4). And by (4.7) we conclude that the operator H surely has no spectral gaps in [1 , +∞). This completes the proof of Theorem 2.1. We proceed to proving Theorem 2.2. We have:       1  3 ∞   sin 2π 2 kξ22 + p2 − π4   3 ξ2 3   ξ2.  2 3 = 2ζ   34   2 2 k k2 2 k=1 k∈Z\{0}  ξ2 + p Therefore,

|ϕp ()| 

   1   sin 2π 2 p − π4  3

πξp 2

 3 3 ξ2 − 2ζ 2

and sup |ϕp ()|  p

1 πξ



 0  1 1 3 3 π  − 3   π 1  max sin 2π 2 − ξ2 .  , 3 2 sin 6π 2 −  − 2ζ 4 4 2

(4.8)

1

Denote z = 2π 2 − π4 , then 0 0  1 1 1 3 π  − 3   π 1  max sin 2π 2 −  , 3 2 sin 6π 2 −  = max | sin z|, 3− 2 | cos 3z| . 4 4 The function | sin z| is π-periodic and 3− 2 | cos 3z| is 3

| sin z| = | sin(π − z)|,

π 3 -periodic

(4.9)

and

| cos 3z| = | cos 3(π − z)|,

z ∈ [0, π].

The function | sin z| increases from 0 to 1 and the function | cos 3z| decreases from 1 to 0 as z ∈ [0, π6 ]. Then it is straightforward to confirm that 0 1 3 max | sin z|, 3− 2 | cos 3z| =



3− 2 | cos 3z|,

z ∈ [0, z0 ] ∪ [π − z0 , π],

| sin z|,

z0  z  π − z0 ,

3

where z0 ∈ (0, π6 ) is the root of the equation sin z − 3− 2 cos 3z = 0.

(4.10)

0 1 3 tan z0 min max | sin z|, 3− 2 | cos 3z| = sin z0 =  . z∈[0,π] tan2 z0 + 1

(4.11)

3

This implies immediately that

Equation (4.10) is reduced to the third order equation for tan z: 3

3

3 2 tan3 z + 3 tan2 z + 3 2 tan z − 1 = 0, which can be solved explicitly: tan z0 = c2 . Hence, by (4.11),

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277

0 1 3 c2 min max | sin z|, 3− 2 | cos 3z| = c1 :=  2 z∈[0,π] c2 + 1 and it follows from (4.8), (4.9) that   3 c1 − 2ζ 32 ξ 2 . sup |ϕp ()|  πξ p This completes the proof. 5. Absence of gaps In this section we prove Theorem 2.3. Replacing the operator L by L˜ := L −ω− , we just shift the spectrum ˜ The advantage of of the operator H and therefore, it is sufficient to prove the theorem for the operator L. using such operator instead of L is that the constant ω− defined by (2.1) is zero and we have ˜ u) ()  ωL u2 0  (Lu, 2 () 2

(5.1)

for all u ∈ 2 (). This is why from the very beginning we assume that for the operator L we have ω− = 0 and inequality (5.1) is satisfied. The proof consists of two parts. In the first part we prove the absence of the gaps in the lower part of 2 the spectrum, namely, below the point Tπ 2 . Here we employ the approach suggested in [4, Sect. 5.2]. In the second part of the proof we show the absence of the gaps in the higher part of the spectrum, that is, above 2 the point Tπ 2 . This will be done by the approach employed in the previous section. We begin with studying the lower part of the spectrum. Similar to (3.1), we introduce the counting function for the operator H with a given operator L:   π2  N (, τ ) = # Ek (τ ) : Ek0 (τ )  2 . T By the minimax principle and (5.1) we have the estimates Ek0 (τ )  Ek (τ )  Ek0 (τ ) + ωL

(5.2)

and therefore,  N0

T2  − 2 ωL , τ π

 N (, τ )  N0 (, τ ) .

(5.3)

  2  2  The operator H has no gaps in inf σ(H), Tπ 2 if for all  ∈ Tπ2 inf σ(H), 1 the estimate holds: sup

( ) 1 1 τ∈ − 2 , 2

N (, τ ) −

(inf ) N (, τ ) 1 1 τ∈ − 2 , 2

 1.

Since the function N (, τ ) is integer-valued, to ensure the above inequality, it is sufficient to find τmin , τmax ∈ [− 12 , 12 ] such that N (, τmax ) − N (, τmin ) > 0. Hence, in view of (5.3), it is sufficient to show that

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278

  2  T2 N0  − 2 ωL , τmax − N0 (, τmin ) > 0 as  ∈ Tπ2 inf σ(H), 1 . π

(5.4)

Exactly this inequality will be checked in the first part of the proof. It is straightforward to confirm that condition (2.6) implies the estimate 0

T2 1 ωL < + ξ 2 . 2 π 4

(5.5)

By (3.2), the band function E10 is given by the formula E10 (τ ) =

π2 2 (τ + ξ 2 m2 ) T2

and hence, due to (5.2) and (5.5), the first spectral band of H is at least 2  3 π2 1 2 inf σ(H), 2 . +ξ T 4 This is why, in what follows we need to prove the absence of gaps only for λ>



π2 T2

1 + ξ2 . 4

In terms of the parameter  used in (5.4), this means to study the case 1 + ξ2. 4

>

Apart from (3.3), the counting function N0 (, τ ) possesses one more representation: 5 4  − (n + τ )2 . ξ

1

 2 −τ 



N0 (, τ ) =

1 n=− 2

(5.6)

+τ 

Consider the equation

2

 −

T2 π 2 ωL

−

ξ

1 4

−1=

 −

T2 π 2 ωL

ξ

.

Its positive root is given by the formula 

∗ =

(

T2 3 + 4ξ 2 + ξ)2 + 2 ωL . 9 π

Conditions (5.5), (2.5) imply that 2 1 + ξ 2 < ∗ < . 4 3 1

As   ∗ , we let τmax := 0, τmin := 1 −  2 in (5.4) and by (5.6) we obtain 5 4 1 2 2 − 1 , N0 (, τmin ) = ξ

⎥ ⎢ ⎢  − T2 ω ⎥ ⎥ ⎢ 2 L π ⎦. N0 (, τmax ) = ⎣ ξ

(5.7)

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279

Hence, thanks to condition (2.6),  N0

2

T  − 2 ωL , τmax π

As ∗ <  < 1, we choose τmax =



1 2

 1 2 2 − 1 − N0 (, τmin )  − −1 ξ ξ   1 2 ∗ − Tπ2 ωL 2∗2 − 1  − − 1 > 0. ξ ξ T2 π 2 ωL

and by (5.6) we get

 N0

 −

T2  − 2 ωL , τmax π



⎛ = 2⎝

−

T2 π 2 ωL

−

ξ

1 4

⎞ ⎠.

Hence, by (5.7) and (2.6),  N0

2

T  − 2 ωL , τmax π



 2 −

 1 2 2 − 1 − N0 (, τmin )  − −2 ξ ξ   1 2 ∗ − Tπ2 ωL 2∗2 − 1 = − − 1 > 0. ξ ξ T2 π 2 ωL

−

1 4

In the remaining part of the proof we study the case   1 and here we shall employ the same approach as in the previous section. Thanks to Theorem 2.2 and condition (2.5), estimate (2.3) holds for   1. Then by (4.1) and Lemma 4.1 we can improve estimates (4.3):   3 c1 − 2ζ 32 ξ 2 1  4 − S (6) (), sup ) N0 (, τ )  a0 () + ( πξ 1 1

τ∈ − 2 , 2

  3 c1 − 2ζ 32 ξ 2 1  4 + S (6) (). (inf ) N0 (, τ )  a0 () − 1 1 πξ τ∈ − , 2 2

In the same way how inequalities (4.5), (4.6) were obtained, by Lemma 3.1 we get:   3  1  2 c1 − 2ζ 32 ξ 2 T 2 0 4 T T 0 T 0 0 0 (6) 0 (θ − ηk+1 )  2 θk − ηk+1 + η − 2S η − 1, π 2πξ k πξ π2 k π2 k     2 14 3 T 0 3πξ 9ξ 2 3 3 3πξ (6) T 2 0 0 0 θk − ηk+1  c1 − 2ζ ξ2 S − − . η − η T 2 π2 k 2T π2 k 2T 4 Therefore, by inequality (5.1) and the minimax principle, we have θk  ηk+1 once    2 14 3 T 0 3πξ 9πξ 2 3 3πξ (6) T 2 0 ξ2 S − −  T ωL 3 c1 − 2ζ η − η 2 π2 k 2 π2 k 2 4 

2

for all ηk0  Tπ 2 ; the latter condition corresponds to the assumed inequality   1. Denoting  := rewrite the above inequality as  

1 4

 3 3πξ 2 T ωL πξ − 1 (6) πξ 3 c1 − 2ζ ξ2 −  4 S () − − − 0 2 2 2 4 4

T2 0 π 2 ηk ,

we

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280

and this should hold for all   1. Explicit formula (4.2) for S (6) () implies immediately that this inequality is true for all   1 provided it holds as  = 1. As  = 1, up to obvious transformations, this inequality coincides with condition (2.7). This completes the proof of Theorem 2.3. 6. Discussion of condition (2.3) In this section we discuss the functions ϕp () and condition (2.3). Our main conjecture motivated by numerical tests is that condition (2.3) holds for all ξ with γ = 0. The first possible steps in proving this conjecture are as follows. We begin with a simple bound for ϕp (). We have ∞ 2 1 1  1 1 1 2 + ξ |ϕp ()|  3   34 =   34 . πξ π 2 2 2 πp ξ k∈Z k2 + p2 k=1 k2 + p2 ξ ξ

The function t → (t2 + p2 ξ 2 )− 4 decreases monotonically in t ∈ [0, +∞) and hence, 3

2 1 1 + ξ2 |ϕp ()|  3 π πp 2 ξ

+∞ 

0

2 1 dt + 3 = 3 1 2 2 2 (t + p ξ ) 4 πp 2 ξ πp 2

+∞ 

0

B( 14 , 12 ) 1 dt + , 3 = 3 1 (t2 + 1) 4 πp 2 ξ πp 2 1

where B(·, ·) is the Beta function. The obtained estimate yields that as p  C1  2 , C1 = const > 0, we have 

1

|ϕp ()| 

1

1

πC12  4

  B 14 , 12 +

1 1 C1 ξ 2

.

Comparing this inequality with condition (2.3), we immediately conclude that this condition can be reformulated as sup p∈N,

1 pC1  2

|ϕp (ξ, )|  c0 −γ .

(6.1)

In a similar way we can simplify the functions ϕp () by replacing them with truncated series. Namely, given N ∈ N, we have      ∞ 1 +∞ k2 π  ∞ 2 1 1  1 2  ξ2 + p − 4  2ξ 2 2ξ 2 4ξ 2 2   sin 2π dt dt   =   3 3 1 .   34  πξ  π π (t2 + p2 ξ 2 ) 4 t2 πN 2 k2 2 k=N +1  N k=N ξ2 + p

(6.2)

We fix a constant C2 > 0 and we truncate the series in (2.2):    1 k2 2− π 2 sin 2π + p 2 ξ 4 1 Φp () := .   34 πξ 2 k 1 2 k=−[C2  2 ] ξ2 + p 

1

k=[C2  2 ]

Then by (6.2) we get: 1

|ϕp () − Φp ()| 

4ξ 2 1

1

πC22  4

.

(6.3)

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Hence, we can replace ϕp by Φp in (6.1) and this leads us to an equivalent condition: |Φp (ξ, )|  c0 −γ .

sup p∈N,

1 pC1  2

(6.4)

Despite the functions Φp are given explicitly by formula (6.3), the structure of these functions is quite complicated. As  varies, the functions Φp () oscillate in a non-periodic way having infinitely many zeroes. This non-periodic oscillation is the main obstacle in calculating the supremum in (6.4). A possible way to find such supremum could be to understand the behavior of Φp() or of φp () for large , that is, the asymptotics as  → +∞. A naive attempt is to replace the series in (2.2) by the integral     sin 2π 12 t22 + p2 − π dt ξ 4 ,   34 t2 2 + p R 2 ξ

(6.5)

to calculate then the asymptotics of such integral and to try to estimate the error made while passing from the series in (2.2) to integral (6.5). The asymptotics of the latter integral can be found by the stationary phase method; the leading term is     sin 2π 12 t22 + p2 − π dt 1 1 ξ 4 1 p 2 sin(2πp 2 ) = + O(− 2 ). 1   34 π 4  t2 2 R ξ2 + p This leading term decays as − 4 . The oscillating part, the function  → sin(2πp 2 ), is periodic in  2 . But calculating the functions Φp () numerically, we see that they do not show such behavior for large , namely, 1 these functions do not decay and oscillate non-periodically in  2 . This means that trying to replace the series in (2.2) or in (6.3) by an integral like (6.5) is likely not a proper way in studying the functions ϕp and Φp . One more property of the functions ϕp () is that they solve certain differential equation. We define the function 1

1

u = u(l, μ) =

1

   sin l k2 + μ − π4 3

(k2 + μ) 4

k∈Z

and we see immediately that 1

ξ2 u ϕp () = π



1

2π 2 2 2 ,p ξ ξ

.

By straightforward calculations we check that the function u solves the equation ∂ ∂l



∂2u l 1 + u − u = 0, ∂l∂μ 2 4

μ > 0,

l ∈ R.

We can also write various initial conditions for the function u like  u

μ=0

=

   sin lk − π 4 k∈Z

k

3 2

,

 1  1 ul=0 = − √ 3 . 2 2 k∈Z (k + μ) 4

(6.6)

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The issue how to sum these series is open. We can only say that the right hand in the first condition is a 2π-periodic function and the right hand side in the second condition is a positive monotone function decaying as μ → +∞. But here the main question is how to solve equation (6.6) or, at least, how to study the behavior of the solutions for large l. Acknowledgments The author thanks Yu.A. Kordyukov for very stimulating discussions while working on this paper and an anonymous referee for useful remarks allowed to improve the initial version of the paper. The reported study was funded by RFBR according to the research project no. 18-01-00046. References [1] G. Barbatis, L. Parnovski, Bethe-Sommerfeld conjecture for pseudo-differential perturbation, Comm. Partial Differential Equations 34 (4) (2009) 383–418. [2] C.B.E. Beeken, Periodic Schrödinger Operators in Dimension Two: Constant Magnetic Fields and Boundary Value Problems, PhD thesis, University of Sussex, Brighton, 2002. [3] D.I. Borisov, On lacunas in the lower part of the spectrum of the periodic magnetic operator in a strip, Contemp. Math. Fundam. Dir. 63 (3) (2017) 373–391 (in Russian). [4] D.I. Borisov, On spectral gaps of a Laplacian in a strip with a bounded periodic perturbation, Ufa Math. J. 10 (2) (2018) 14–30. [5] D.I. Borisov, On absence of gaps in a lower part of spectrum of Laplacian with frequent alternation of boundary conditions in strip, Theoret. Math. Phys. 195 (2) (2018) 690–703. [6] D.I. Borisov, Gaps in the spectrum of the Laplacian in a band with periodic delta interaction, Proc. Inst. Math. Mech. Ural Branch RAS 24 (2) (2018) 46–53 (in Russian). [7] B.E.J. Dahlberg, E. Trubowitz, A remark on two dimensional periodic potentials, Comment. Math. Helv. 57 (1) (1982) 130–134. [8] P. Exner, O. Turek, Periodic quantum graphs from the Bethe-Sommerfeld perspective, J. Phys. A 50 (45) (2017) 455201. [9] B. Helffer, A. Mohamed, Asymptotics of the density of states for the Schrödinger operator with periodic electric potential, Duke Math. J. 92 (1) (1998) 1–60. [10] Y. Karpeshina, Perturbation Theory for the Schrödinger Operator with a Periodic Potential, Lect. Notes in Math., vol. 1663, Springer, Berlin, 1997. [11] Y. Karpeshina, Spectral properties of the periodic magnetic Schrödinger operator in the high-energy region. Twodimensional case, Comm. Math. Phys. 251 (3) (2004) 473–514. [12] E. Krätzel, Lattice Points, Kluwer Academic Publishers, Dordrecht, 1988. [13] A. Mohamed, Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential, J. Math. Phys. 38 (8) (1997) 4023–4051. [14] L. Parnovski, Bethe-Sommerfeld conjecture, Ann. Henri Poincaré 9 (3) (2008) 457–508. [15] L. Parnovski, A. Sobolev, On the Bethe-Sommerfeld conjecture for the polyharmonic operator, Duke Math. J. 107 (2) (2001) 209–238. [16] L. Parnovski, A.V. Sobolev, Bethe-Sommerfeld conjecture for periodic operators with strong perturbations, Invent. Math. 181 (3) (2010) 467–540. [17] M.M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Proc. Steklov Inst. Math. 171 (1987) 1–121. [18] M.M. Skriganov, A.V. Sobolev, Variation of the number of lattice points in large balls, Acta Arith. 120 (3) (2005) 245–267. [19] M.M. Skriganov, A.V. Sobolev, Asymptotic bounds for spectral bands of periodic Schrödinger operators, St. Petersburg Math. J. 17 (1) (2006) 207–216. [20] O.A. Veliev, Asymptotic formulas for the eigenvalues of a periodic Schrödinger operator and the Bethe-Sommerfeld conjecture, Funct. Anal. Appl. 21 (2) (1987) 87–100. [21] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1945.