Quantum diffusion in the Kronig-Penney model
In this paper we consider the 1D Schrödinger operator with periodic point interactions. We show an bound for the time evolution operator restricted to each energy band with decay order as , which comes from some kind of resonant state. The order is optimal for our model. We also give an asymptotic bound for the coefficient in the high energy limit. For the proof, we give an asymptotic analysis for the band functions and the Bloch waves in the high energy limit. Especially we give the asymptotics for the inflection points in the graphs of band functions, which is crucial for the asymptotics of the coefficient in our estimate.
The one-electron models of solids are based on the study of Schrödinger operator with periodic potential. There are a lot of studies on the periodic potential, in particular, for periodic point interactions, we can show the spectral set explicitly (Albeverio et. al.  is the best guide to this field for readers). Most fundamental case is the one-dimensional Schrödinger operator with periodic point interactions, called the Kronig–Penney model (see Kronig–Penney ), given by
where is a non-zero real constant, and is the Dirac delta measure at . The positive sign of corresponds the repulsive interaction, while the negative one corresponds the attractive one. More precisely, is the negative Laplacian with boundary conditions on integer points:
Here and is the usual Sobolev space of order on the open set . From Sobolev’s embedding theorem , every elements of are continuous (classical sense) and uniformly bounded functions. It is well-known that is self-adjoint [4, 2] and is a model describing electrons on the quantum wire. The spectrum of this model is explicitly given by
is so-called discriminant and can be regarded as an entire function with respect to . The spectrum of consists of infinitely many closed intervals (spectral bands) and is purely absolutely continuous.
On the other hand, for the Schrödinger operator on with decaying potential , the dispersive estimate for the Schrödinger time evolution operator is stated as follows:
where denotes the spectral projection to the absolutely continuous subspace for . The dispersive estimate is a quantitative representation of the diffusion phenomena in quantum mechanics, and is extensively studied recently, because of its usefulness in the theory of the non-linear Schrödinger operator (see e.g. Journé–Soffer–Sogge , Weder , Yajima , Mizutani , and references therein). The estimate (5) is also obtained in the case of the one-dimensional point interaction. Adami–Sacchetti  obtain (5) when is one point potential, and so do Kovařík–Sacchetti  when is the sum of potentials at two points. The motivation of the present paper is to obtain a similar estimate for our periodic model (1). Though this problem is quite fundamental, we could not find such kind of results in the literature, probably because the deduction of the result requires a detailed analysis of the band functions, as we shall see below.
Since the spectrum of the Schrödinger operator with periodic potential is absolutely continuous, one may expect some dispersion type estimate holds also in this case. However, there seems to be few results about the dispersive type estimate for the time evolution operator of the differential equation with periodic coefficients.111Some authors study the pointwise asymptotics for the integral kernel of the time evolution operator in the large time limit; see e.g. Korotyaev  and references therein. But we could not find the dispersive type estimate for the periodic Schrödinger operator itself in the literature. An example is the paper by Cuccagna , in which the Klein–Gordon equation
is considered, where is a smooth real-valued periodic function with period . Cuccagna proves the solution satisfies
for , where is some bounded discrete set. The peculiar power comes from the following reason. The integral kernel for the time evolution operator is written as the sum of oscillatory integrals
where , , , , and . The function is called the band function for the -th band, which is a real-analytic function of with period . In the large time limit , it is well-known that the main contribution of the oscillatory integral (7) comes from the part nearby the stationary phase point (the solution of ), and the stationary phase method tells us the principal term in the asymptotic bound is a constant multiple of (see e.g. Stein [15, Chapter VIII] or Lemma 15 below). However, since is a periodic function, there exists a point so that . If the stationary phase point coincides with , then the previous bound no longer makes sense. Instead, the stationary phase method concludes the principal term is a constant multiple of . Since the integral kernel of our operator has the same form as (7) (see (125) below), we expect a result similar to (6) also holds in our case.
where the -th band is a closed interval of finite length (for the precise definition, see (22) below). Our main result is as follows.
Let be the spectral projection onto the -th energy band . Then, for sufficiently large , there exist positive constants and such that
for any and any , where . The coefficients obey the bound
The power in the first term of the coefficient in (8) is the same as in (5) with and , since it comes from the states corresponding to the energy near the band center, which behaves like a free particle. This fact can be understood from the graph of (Figure 1).222All the graphs are written by using Mathematica 9.0. The part of the graph corresponding to the band center is similar to the parabola or its translation, which is the band function for the free Hamiltonian . On the other hand, the power comes from part of the integral (7) given by
where is some open set including two solutions to the equation . Notice that is an inflection point in the graph of ; see Figure 1. The estimates for the coefficients are obtained from the lower bounds for the derivatives of . Actually, we can choose so that
where is a positive constant independent of . By (10) and the estimates for the amplitude function (Proposition 14), we can prove Theorem 1 by using a lemma for estimating oscillatory integrals, given in Stein’s book (see Stein [15, page 334] or Lemma 15 below). We can also prove the power is optimal, by considering the case (so, is a stationary phase point), and applying the asymptotic expansion formula in the stationary phase method (see e.g. Stein [15, Page 334]).
The physical implication of the result is as follows. By definition, the parameter represents the propagation velocity of a quantum particle. The wave packet with energy near has the maximal group speed in the -th band, and the speed of the quantum diffusion is slowest in that band. Thus such state has a bit longer life-span (in the sense of -norm) than the ordinary state has; the state is in some sense a resonant state, caused by the meeting of two stationary phase points as tends to . It is well-known that the existence of resonant states makes the decay of the solution with respect to slower (see Jensen–Kato  or Mizutani ).
Since the estimate (8) is given bandwise, it is natural to ask we can obtain the dispersive type estimate for the whole Schrödinger time evolution operator , like Cuccagna’s result (6). However, it turns out to be difficult in the present case, from the following reason. A reasonable strategy to prove such estimate is as follows. First, we divide the integral into two parts and the rest, where is given in (9) with some open set including . Next, we show the sum of converges and gives , and the sum of the rests also converges and gives . However, for fixed and , we find that our upper bound for is not better than , and the sum of the upper bounds does not converge (see the last part of Section 4). One reason for this divergence is very strong singularity of our potential, the sum of -functions. Because of this singularity, the width of the band gap, say ( is the band number), does not decay at all in the high energy limit .333 Proposition 10 implies as . Then we cannot take the open set so small,444 We take in the proof of Theorem 1. and the sum of the lengths diverges; if this sum converges, we can use a simple bound to control the sum. Thus we do not succeed to obtain a bound for the sum of at present.
On the other hand, for the Schrödinger operator on with real-valued periodic potential , it is known that the decay rate of the width of the band gap reflects the smoothness of the potential . Hochstadt  says if is in , and Trubowitz  says ( is some positive constant) if is real analytic. So, if is sufficiently smooth, it is expected that we can control the sum of , and obtain the dispersive type estimate for the whole operator (i.e. (8) without the projection ). We hope to argue this matter elsewhere in the near future.
The paper is organized as follows. In Section 2, we review the Floquet–Bloch theory for our operator and give the explicit form of the integral kernel of . In Section 3, we give more concrete analysis for the band functions, especially give some estimates for the derivatives. In Section 4, we prove Theorem 1, and give some comment for the summability with respect to of the estimates (8).
2 Floquet–Bloch theory
In this section, we shall calculate the integral kernel of the operator by using the Floquet-Bloch theory. Most results in this section are already written in another literature (e.g. Reed-Simon [14, XIII.16] and Albeverio et. al. [2, III.2.3]), but we shall give it here again for the completeness.
First we shall calculate the generalized eigenfunctions for our model, i.e., the solutions to the equations
The proof is a simple calculation, so we shall omit it. Notice that is an entire function with respect to , since is an even function.
for some .
(i) It is easy to see a non-trivial solution to (11)-(13) and (17) exists if and only if has an eigenvalue , and the latter condition is equivalent to (18), since and . (ii) When (18) holds, the vector given in (19) is an eigenvector of with the eigenvalue . Thus the second equation in (19) follows from (15). ∎
as already stated in (4). For , we have
If , the right hand side of (21) is larger than and (20) does not hold for . Thus, there is no negative part in . If , then some negative value belongs to , and the corresponding is pure imaginary. However, we concentrate on the high energy limit in the present paper, and the existence of the negative spectrum does not affect our argument. So we sometimes assume in the sequel, in order to simplify the notation. In this case, the results for will be stated in the remark.
By an elementary inspection of the graph of , we find the following properties.
Assume . Then,
and for .
The equation has a unique solution in the open interval for .
The equation has a unique solution in the open interval for , and .
For convenience, we put . Then, is monotone decreasing on for even , and monotone increasing on for odd .
Remark. When , we denote the solution to in by , and the solution to in by , for .
When , the spectrum of is given as
by (20) and Proposition 4. The closed interval is called the -th band. By the expression (22), the band gap is non-empty for every . Proposition 4 also implies the function () has the unique inverse function . Then the band function is defined by
By definition, the band function is a real-analytic, periodic, and even function with respect to . The -th band is the range of the band function .
Let be the spectral projection for the self-adjoint operator corresponding to the -th band . The spectral theorem implies
where means the strong limit in . Let be the integral kernel of the operator , that is,
for . Let us calculate more explicitly.
Remark. When , the same result holds for , but the range of the function is .
Before the proof, we prepare a lemma about the Wronskian of the Bloch waves. The Wronskian of two functions and is defined as
for every . Thus is constant on .
In this proof, we denote the inner product in , that is, . Then we have from (29)
By differentiating both sides of (33) with respect to , we have
where . By differentiating both sides of (34) with respect to , we have
Proof of Proposition 5.
First, the Floquet-Bloch theory tells us
for any . Actually, is the normalized eigenfunction of , where is the Hilbert space defined by
Let and . Since , , and , we have from (38)
Moreover, we have by (27)
The Wronskian on the interval is calculated as follows.
This equality and (19) implies for
where is the boundary value of the resolvent . The integral kernel of the resolvent operator for (the resolvent set of ) is given as
3 Analysis of band functions
In this section, we analyze the band function , which is explicitly given by
especially its asymptotics as . Here is the inverse function of
is given in Proposition 4 and its remark, and we put . By the formula (47), we can draw the graphs of , 555 The inverse correspondence is useful for the numerical calculation. which are illustrated in Figure 1 in the introduction. From Figure 1, we notice that is similar to the parabola on the interval , except near the edge points. From this reason, we mainly consider the function on the interval , thereby we can simplify some formulas given below. Figure 1 also suggests us it is better to analyze when its value is near the band edge, and near the band center, separately.