Stark-Wannier Ladders and Cubic Exponential Sums
On , we consider the Schrödinger operator
where is a real analytic -periodic function and is a positive constant. This operator is a model to study a Bloch electron in a constant electric field (). The parameter is proportional to the electric field. The operator (1.1) was studied both by physicists (see, e.g., the review ) and by mathematicians (see, e.g., ). Its spectrum is absolutely continuous and fills the real axis. One of main features of is the existence of Stark-Wannier ladders. These are -periodic sequences of resonances, which are poles of the analytic continuation of the resolvent kernel in the lower half plane through the spectrum (see, e.g., ). Most of the mathematical work studied the case of small (see, e.g., [9, 3] and references therein). When is small, there are ladders exponentially close to the real axis. Actually, only the case of finite gap potentials was relatively well understood. For these potentials, there is only a finite number of ladders exponentially close to the real axis. It was further noticed that the ladders non-trivially “interact” as changes, and conjectured that the behavior of the resonances strongly depends on number theoretical properties of (see, e.g., ).
In the present note, we only consider the periodic potential and study the reflection coefficient of the Stark-Wannier operator (1.1) in the lower half of the complex plane of the spectral parameter . The resonances are the poles of the reflection coefficient. We show that, as , the function can be asymptotically described in terms of a regularized cubic exponential sum that is a close relative of the cubic exponential sums often encountered in analytic number theory. This explains the dependence of the reflection coefficient on the arithmetic nature of . For , we describe the asymptotics of the Stark-Wannier ladders situated far from the real axis.
For the sake of simplicity, assume that the potential is entire. Assume also . For any , there are unique solutions to (1.2) that admit the asymptotic representations
where the determinations of and are analytic in and positive along . Consider also the solution . The solutions and being linearly independent, one has
where the coefficient is independent of and the function is entire. The ratio is the reflection coefficient. It is an -periodic meromorphic function of . The reflection coefficient is analytic in , and, for , one has . The poles of are the resonances of .
Let us now state the first of our results. Represent by its Fourier series for . Let . One has
Let . Then, as ,
where, for real, denotes the fractional part of , and . This estimate is locally uniform in .
Clearly, the asymptotic behavior of as is determined by the Fourier series terms with large positive , and so, roughly,
It is worth to compare the function with the cubic exponential sums . Such sums were extensively studied in analytic number theory, see, e.g., . They were proved to depend strongly on the arithmetic nature of . This appears to be true in our case too. We have
Let . Assume that and represent it in the form , where are co-prime integers. If , we take . For , we set . As , one has
where , , , and
The error estimates are locally uniform in .
Let us discuss this result. First, assume that . By Theorem 2,
where the determination of is analytic in and positive along . Recall that is -periodic. Let . Representation (1.8) implies
Assume . The resonances located in have the following properties :
for sufficiently large , the resonances with are located in the domain , where is a constant;
let be the number of resonances in the rectangle ; then, one has
The first statement immediately follows from Theorem 2;
to prove the second one has to use Jensen formula and Levin lower bounds
for the absolute values of entire functions, see, e.g., .
When , it is difficult to obtain the asymptotics of the resonances as, in a neighborhood of the line , they are determined by the first Fourier coefficients of , i.e., by with . Hence, the problem is not asymptotic in nature.
If , then the description of the resonances is determined by the values of for (the map is -periodic). The are cubic complete rational exponential sums, see, e.g., . One easily checks
For any , .
This implies that, for , there is at least one integer such that .
If is non zero for only one (this happens, for example, for ), then one can characterize the resonances as when . Now, they live near the lines , .
For large , there are actually many non-zero values :
There exists a constant such that, for any co-prime , one has .
In general, the behavior of is nontrivial; it is known to depend strongly on the prime factorization of . Computer calculations lead to the following conjecture: if is prime, , and , then .
If is non zero for at least two values of such that , then, using (1.5), one can describe asymptotically all the resonances with sufficiently negative imaginary part. One has
Assume that, for some integers such that , one has , , and for all . Then, for sufficiently large , in the vertical half-strip
there are resonances, and they are described by the asymptotic formulas:
This statement easily follows from Theorem 2.
Finally, let us describe very briefly the ideas leading to Theorems 1 and 2. Buslaev’s solutions used to define the reflection coefficient (see (1.3)) are entire functions of and ; they satisfy the relations . It appears that the analytic properties of such solutions can naturally be described in terms of a system of two first order difference equations on the complex plane (see, for example, ). To get the asymptotics of the Fourier coefficients of the reflection coefficient, we study the solutions of this system far from the origin. The idea leading from Theorem 1 to Theorem 2 is analogous to one used to study the behavior of the exponential sums with for large , see . However, to use it successfully, one has to carry out a non trivial analysis of properties of the error term in (1.5).
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