1 Introduction

In [1, 7, 8], Aharonov and his coauthors introduced a new concept in quantum mechanics, namely the notion of weak measurement of a quantum observable (see also [19, 20, 22, 23]). This notion led to the discovery of an interesting family of functions, which had been observed as well in optical phenomena [12,13,14,15,16,17], and [24]. These functions, known as the Aharonov–Berry superoscillations, are band-limited functions with the apparently paradoxical property that they can oscillate faster than their fastest Fourier component.

An important question that was posed originally by both Aharonov and Berry is whether such superoscillatory behaviour can persist when we evolve a superoscillatory function according to some differential equations. In a series of papers [2,3,4,5, 9, 18] and the monograph [6], the authors have developed a powerful method to study the evolution of superoscillations propagated by the Schrödinger equation. This method employs essentially two steps: First one uses Fourier analysis, or the relevant Green function, to solve the Cauchy problem associated with the Schrödinger equation, and then, one translates the problem in the complex plane (essentially by complexifying both the functions and the operators acting on them) and demonstrates the permanence of the superoscillatory behaviour as a consequence of a continuity theorem for suitable convolution operators. The key ingredient for the continuity theorem is the understanding of the growth of the associated symbol (usually an entire function).

In this paper, we consider a very large, and so far not yet considered, class of potentials for the Schrödinger equation, and we prove how a new theorem on the growth of analytic functions (which we prove in Sect. 2) can be used to demonstrate superoscillatory longevity in time for those equations.

The novelty of the present paper consists in the fact that we will now study convolution operators with non-constant coefficient of the form:

$$\begin{aligned} \mathcal {U}\left( t,\frac{\partial }{\partial x}\right) :=\sum _{m=0}^\infty b_m(t,x)\frac{\partial ^{m}}{\partial x^{m}}. \end{aligned}$$

This situation will require a new continuity theorem, which we will prove in Sect. 2. This result will be applied in Sect. 3 to the study of the evolution of superoscillations by Schrödinger equations in which variable coefficients potential appear.

The first author would like to express his sincere thanks to Professor Susumu Yamasaki for valuable comments on the topology of the spaces of entire functions.

2 Theorem on continuity of a class of convolution operators

Let f be a non-constant entire function of a complex variable z. We define

$$\begin{aligned} M_f(r)=\max _{|z|=r}|f(z)|,\quad \text { for}\quad r\ge 0. \end{aligned}$$

The non-negative real number \(\rho \) defined by

$$\begin{aligned} \rho =\limsup _{r\rightarrow \infty }\frac{\ln \ln M_f(r)}{\ln r} \end{aligned}$$

is called the order of f. If \(\rho \) is finite, then f is said to be of finite order, and if \(\rho =\infty \), the function f is said to be of infinite order.

In the case f is of finite order \(\rho \), we define the non-negative real number

$$\begin{aligned} \sigma =\limsup _{r\rightarrow \infty }\frac{\ln M_f(r)}{ r^\rho }, \end{aligned}$$

which is called the type of f. If \(\sigma \in (0,\infty )\), we call f of normal type, while we say that f is of minimal type if \(\sigma =0\) and of maximal type if \(\sigma =\infty \). The constant functions are said to be of minimal type and order zero. In the sequel, we make use of the notions given in the next definitions. These notions are classical, see e.g. [11], and go back to Hörmander see [21]:

Definition 2.1

Let p be a positive number. We define the class \(A_p\) to be the set of entire functions such that there exists \(C>0\) and \(B>0\) for which

$$\begin{aligned} |f(z)|\le C \exp (B|z|^p), \qquad \forall z\in {{\mathbb {C}}}. \end{aligned}$$

The class \(A_{p,0}\) consists of those entire functions such that for all \(\varepsilon >0\) there exists \(C_\varepsilon >0\) such that

$$\begin{aligned} |f(z)|\le C_\varepsilon \exp (\varepsilon |z|^p), \quad \forall z\in {{\mathbb {C}}}. \end{aligned}$$

To define a topology in these spaces, we follow [11, Section 2.1]: For \(p>0\), \(c>0\) and for any entire function f, we set

$$\begin{aligned} \Vert f\Vert _c:=\sup _{z\in {{{\mathbb {C}}}}}\{|f(z)|\exp (-c|z|^p)\}. \end{aligned}$$

Let \(A_p^c\) denote the linear space of entire functions satisfying \(\Vert f\Vert _c<\infty \). Then, \(\Vert \cdot \Vert \) defines a norm in \(A_p^c\) which makes this space a Banach space. The natural inclusion mapping \(A_p^c\hookrightarrow A_p^{c'}\) is a compact operator for any \(0<c<c'\).

For any sequence \(\{c_n\}_{n\ge 1}\) of positive numbers, strictly increasing to infinity, we can introduce an LF topology on \(A_p\) given by the inductive limit

$$\begin{aligned} A_p:=\lim _{\longrightarrow } A_p^{c_n} . \end{aligned}$$

Since this topology is stronger than the topology of the pointwise convergence, it is independent of the choice of the sequence \(\{c_n\}_{n\ge 1}\). In this inductive limit topology, a sequence \(\{f_k\}\) in \(A_p\) converges to f in \(A_p\) if and only if there exists n such that \(f_j\in A_p^{c_n}\) for all j, \(f\in A_p^{c_n}\) and \(\Vert f_j-f\Vert _{c_n}\rightarrow 0\) for \(j\rightarrow \infty \). The topology in \(A_{p,0}\) is given by the projective limit

$$\begin{aligned} A_{p,0}:=\lim _{\longleftarrow } A_p^{c_n}. \end{aligned}$$

It can be proved, see [11, Section 6.1], that \(A_p\) is a DFS space and \(A_{p,0}\) is an FS space, respectively.

To prove our main results, we need an important lemma that characterizes the coefficients of entire functions with growth conditions.

Lemma 2.2

The function

$$\begin{aligned} f(z)=\sum _{j=0}^\infty f_jz^j \end{aligned}$$

belongs to \(A_p\) if and only if there exists \(C_f>0\) and \(b>0\) such that

$$\begin{aligned} |f_j|\le C_f \frac{b^j}{\Gamma \left( \frac{j}{p}+1\right) }. \end{aligned}$$

Proof

First suppose that \(f(z)\in A_p\) and let us prove that the estimate on the coefficients \(f_j\) follows by the Cauchy formula. In fact, we have

$$\begin{aligned} f^{(j)}(z)=\frac{j!}{2\pi i}\int _\gamma \frac{f(w)}{(w-z)^{j+1}}\, \hbox {d}w , \end{aligned}$$

where the path of integration \(\gamma \) is the circle \(|w-z|=s|z|\), where s is a positive real number and \(z\not =0\). Then, we have

$$\begin{aligned} \begin{aligned} |f^{(j)}(z)|&\le \frac{j!}{(s|z|)^j} \max _{|w-z|=s|z|}|f(w)|\\&\le \frac{C_fj!}{(s|z|)^j}\exp (B(1+s)^p|z|^p) \end{aligned} \end{aligned}$$

for all \(s> 0\), where we have used the fact that \(f\in A_p\) and \(|w|\le (1+s)|z|\). The well-known estimate

$$\begin{aligned} (a+b)^p\le 2^p(a^p+b^p),\ \ a>0, \ \ b>0, \ \ p>0 \end{aligned}$$

gives

$$\begin{aligned} (1+s)^p\le 2^p(s^p+1) \end{aligned}$$

for all \(s>0\). Hence we have

$$\begin{aligned} |f^{(j)}(z)|\le C_f\frac{j!}{(s|z|)^j}\exp (B\cdot 2^ps^p|z|^p)\exp (B\cdot 2^p|z|^p) \end{aligned}$$

for all \(z\in {\mathbb {C}}\) and \(s>0\). We now take the minimum of the right-hand side of the above estimate with respect to s, i.e. the minimum of the function

$$\begin{aligned} g(s):=\frac{1}{(s|z|)^j}\exp (B\cdot 2^ps^p|z|^p) \end{aligned}$$

in \((0,\infty )\). The minimum is at the point

$$\begin{aligned} s_\mathrm{min}=\Big (\frac{j}{2^pBp}\Big )^{1/p}\frac{1}{|z|} \end{aligned}$$

so that we obtain

$$\begin{aligned} |f^{(j)}(z)|\le C_f j! \Big (\frac{2^pBp }{j}\Big )^{j/p} e^{j/p}\exp (A2^p|z|^p). \end{aligned}$$

So if we set

$$\begin{aligned} b:=(2^pBp e)^{1/p}, \end{aligned}$$

we obtain

$$\begin{aligned} |f^{(j)}(z)|\le C_f j!\frac{b^j}{j^{j/p}} \exp (B\cdot 2^p|z|^p) \end{aligned}$$

for all \(z\in {\mathbb {C}}\). Since \( f_j=\frac{f^{(j)}(0)}{j!} \), we have, by the maximum modules principle applied in a disc centred at the origin and with radius \(\epsilon >0\) sufficiently small,

$$\begin{aligned} \begin{aligned} |f_j|&\le C_f \ \frac{b^j}{j^{j/p}} \exp (B\cdot 2^p\epsilon ^p)\\&\le 2C_f \ \frac{b^j}{j^{j/p}}\\&= C'_f \ \frac{b^j}{(j!)^{1/p}}\\&\le C'_f \ \frac{b^j}{\Gamma (\frac{j}{p}+1)}. \end{aligned} \end{aligned}$$

The other direction follows form the properties of the Mittag–Leffler function. In fact,

$$\begin{aligned} E_{\alpha ,\beta }(z)=\sum _{n=1}^\infty \frac{z^n}{\Gamma (\alpha n+\beta )} \end{aligned}$$

is an entire function of order \(1/\alpha \) (and of type 1) for \(\alpha >0\) and \(Re(\beta )>0\), see [10]. So, in our case, f is entire of order p. \(\square \)

In order to prove our main results, we need some more notations and definitions:

Definition 2.3

Let \(p>0\). The set \(\mathcal {D}_{p,0}\) consists of operators of the form

$$\begin{aligned} P(z,\partial _z)=\sum _{n=0}^\infty a_n(z)\partial _z^n \end{aligned}$$

satisfying:

  1. (i)

    \(a_n(z)\) (\(n=0,1,2,\ldots \)) are entire functions.

  2. (ii)

    There exists a constant \(B>0\) such that for every \(\varepsilon >0\) one can take a constant \(C_\varepsilon >0\) for which

    $$\begin{aligned} |a_n(z)|\le C_\varepsilon \frac{\varepsilon ^n}{(n!)^{1/q}} \exp (B |z|^p), \end{aligned}$$

    holds, where

    $$\begin{aligned} \frac{1}{p}+\frac{1}{q}=1, \end{aligned}$$

    and \(1/q=0\) when \(p=1\).

Theorem 2.4

Let \(P(z,\partial _z)\in \mathcal {D}_{p,0}\) and let \(f\in A_p\). Then, \(P(z,\partial _z)f\in A_p\) and \(P(z,\partial _z)\) is continuous on \(A_p\), that is \(P(z,\partial _z)f\rightarrow 0\) as \(f\rightarrow 0\).

Proof

We apply the operator \(P(z,\partial _z)\) to the function \(f\in A_p\) and we get

$$\begin{aligned} \begin{aligned} P(z,\partial _z)f(z)&=\sum _{n=0}^\infty a_n(z)\partial _z^n\sum _{j=0}^\infty f_jz^j \\&=\sum _{n=0}^\infty a_n(z)\sum _{j=n}^\infty f_j \frac{j!}{(j-n)!} z^{j-n} \\&=\sum _{n=0}^\infty \sum _{k=0}^\infty a_n(z) f_{n+k} \frac{(k+n)!}{k!} z^{k}. \end{aligned} \end{aligned}$$

Now we observe that

$$\begin{aligned} \begin{aligned} |P(z,\partial _z)f(z)| \le \sum _{n=0}^\infty \sum _{k=0}^\infty |a_n(z)| \ |f_{n+k}| \frac{(k+n)!}{k!} |z|^{k}. \end{aligned} \end{aligned}$$

Using the hypothesis on P and f, which translate into conditions on the \(a_n\) and on \(f_j\), we get

$$\begin{aligned} \begin{aligned} |P(z,\partial _z)f(z)| \le C_f C_\varepsilon \sum _{n=0}^\infty \sum _{k=0}^\infty \frac{\varepsilon ^n}{(n!)^{1/q}} \exp (B |z|^p)\ \ \frac{b^{n+k}}{\Gamma \Big (\frac{n+k}{p}+1\Big )} \frac{(k+n)!}{k!} |z|^{k}. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} (n!)^{1/q}\ge \Gamma \Big (\frac{n}{q}+1\Big ) \quad \text {and} \quad (k+n)!\le 2^{k+n}k!n! \end{aligned}$$

we get

$$\begin{aligned} \begin{aligned} |P(z,\partial _z)f(z)|&\le C_f C_\varepsilon \sum _{n=0}^\infty \sum _{k=0}^\infty \frac{\varepsilon ^n}{\Gamma \Big (\frac{n}{q}+1\Big )} \ \ \frac{b^{n+k}}{\Gamma \Big (\frac{n+k}{p}+1\Big )} \frac{2^{k+n}k!n! }{k!} |z|^{k}\exp (B |z|^p) \\&\le C_f C_\varepsilon \sum _{n=0}^\infty \sum _{k=0}^\infty \frac{\varepsilon ^n}{\Gamma \Big (\frac{n}{q}+1\Big )} \ \ \frac{b^{n+k}}{\Gamma \Big (\frac{n+k}{p}+1\Big )} 2^{k+n}n! |z|^{k}\exp (B |z|^p)\\&\le C_f C_\varepsilon \sum _{n=0}^\infty \sum _{k=0}^\infty (2b)^k (2\varepsilon b)^n \frac{1}{\Gamma \Big (\frac{n}{q}+1\Big )} \ \ \frac{n!}{\Gamma \Big (\frac{n+k}{p}+1\Big )} |z|^{k}\exp (B |z|^p). \end{aligned} \end{aligned}$$
(1)

From the inequality

$$\begin{aligned} \frac{\Gamma (a+1)\Gamma (b+1)}{\Gamma (a+b+2)}\le 1, \end{aligned}$$

we deduce

$$\begin{aligned} \Gamma \Big (\frac{k+n}{p}+1\Big )\ge \Gamma \Big (\frac{k}{p}+\frac{1}{2}\Big )\Gamma \Big (\frac{n}{p}+\frac{1}{2}\Big ) \end{aligned}$$

and so we can write (1) as

$$\begin{aligned} \begin{aligned} |P(z,\partial _z)f(z)|&\le C_f C_\varepsilon \sum _{k=0}^\infty \frac{(2b)^k}{\Gamma \Big (\frac{k}{p}+\frac{1}{2}\Big )} |z|^{k}\exp (B |z|^p) \\&\quad \times \sum _{n=0}^\infty (2\varepsilon b)^n \frac{n!}{\Gamma \Big (\frac{n}{p}+\frac{1}{2}\Big )\Gamma \Big (\frac{n}{q}+1\Big )}. \end{aligned} \end{aligned}$$

Finally, consider the series

$$\begin{aligned} \sum _{n=0}^\infty (2\varepsilon b)^n \frac{n!}{\Gamma \Big (\frac{n}{p}+\frac{1}{2}\Big )\Gamma \Big (\frac{n}{q}+1\Big )} \end{aligned}$$

and observe that

$$\begin{aligned} (2\varepsilon b)^n \frac{n!}{\Gamma \Big (\frac{n}{p}+\frac{1}{2}\Big )\Gamma \Big (\frac{n}{q}+1\Big )} \sim \frac{n^n(2\varepsilon b)^n }{\Big (\frac{n}{p}\Big )^{n/p}\Big (\frac{n}{q}\Big )^{n/q} }, \end{aligned}$$

and

$$\begin{aligned} \frac{n^n(2\varepsilon b)^n }{\Big (\frac{n}{p}\Big )^{n/p}\Big (\frac{n}{q}\Big )^{n/q} }=\frac{n^n(2\varepsilon b)^n [p^{1/p}q^{1/q}]^n}{n^n} =(2\varepsilon b)^n [p^{1/p}q^{1/q}]^n. \end{aligned}$$

Since \(\varepsilon \) is arbitrary small, the series converges, i.e.

$$\begin{aligned} \sum _{n=0}^\infty (2\varepsilon b)^n \frac{n!}{\Gamma \Big (\frac{n}{p}+\frac{1}{2}\Big )\Gamma \Big (\frac{n}{q}+1\Big )} =C'. \end{aligned}$$

This finally gives

$$\begin{aligned} \begin{aligned} |P(z,\partial _z)f(z)|&\le C'C_f C_\varepsilon \sum _{k=0}^\infty \frac{(2b)^k}{\Gamma \Big (\frac{k}{p}+\frac{1}{2}\Big )} |z|^{k}\exp (B |z|^p) \end{aligned} \end{aligned}$$

and, by the properties of the Mittag–Leffler function, we have

$$\begin{aligned} \sum _{k=0}^\infty \frac{(2b)^k}{\Gamma \Big (\frac{k}{p}+\frac{1}{2}\Big )} |z|^{k}\le C'\exp (B' |z|^p). \end{aligned}$$

We conclude that there exists \(B''>0\) such that

$$\begin{aligned} \begin{aligned} |P(z,\partial _z)f(z)|&\le C'C_f C_\varepsilon \exp (B'' |z|^p) \end{aligned} \end{aligned}$$

which means that \(P(z,\partial _z)f(z)\in A_p\) and for \(C_f\rightarrow 0\) the same estimate proves the continuity, i.e. \(|P(z,\partial _z)f(z)|\rightarrow 0\) when \(f\rightarrow 0\). \(\square \)

We conclude this section with a result which is the integral counterpart of the previous theorem and is of independent interest.

We define the action of the operator denoted by \(\partial _z^{-n}\) (\(n=1,2,3,\dots \)) on the space of entire functions by the Riemann–Liouville integral

$$\begin{aligned} \partial _z^{-n}f(z)=\frac{1}{(n-1)!}\int _0^z(z-t)^{n-1}f(t)\hbox {d}t. \end{aligned}$$

Definition 2.5

Let \(\mathcal {E}_p\) denote the set of all formal power series

$$\begin{aligned} P(z,\partial _z^{-1})=\sum _{n=0}^\infty a_n(z) \partial _z^{-n} \end{aligned}$$

of \(\partial _z^{-1}\) satisfying

  1. (i)

    \(a_n(z)\) (\(n=0,1,2,\ldots \)) are entire functions.

  2. (ii)

    There exist constants \(B>0\) and \(C>0\) for which

    $$\begin{aligned} |a_{n}(z)|\le C^{n+1}n!^\frac{1}{q}\exp (B|z|^p) \end{aligned}$$

    holds for \(n=0,1,2,\ldots \) and where

    $$\begin{aligned} \frac{1}{p}+\frac{1}{q}=1, \end{aligned}$$

    with \(1/q=0\) when \(p=1\).

Theorem 2.6

Let \(\displaystyle P(z,\partial _z^{-1})=\sum _{n=0}^\infty a_n(z) \partial _z^{-n}\in \mathcal {E}_p\), and define the action of \(P(z,\partial _z^{-1})\) on \(A_p\) by

$$\begin{aligned} P(z,\partial _z^{-1})f(z)=a_0(z)f(z)+\sum _{n=1}^\infty a_{n}(z) \partial _z^{-n}f(z). \end{aligned}$$

This action is well defined, that is, if \(P(z,\partial _z^{-1})\in E_p\) and \(f\in A_p\), then \(P(z,\partial _z^{-1})f\in A_p\) and continuous.

Proof

Suppose that

$$\begin{aligned} |f(z)|\le C_1\exp (B_1|z|^p) \end{aligned}$$

holds for \(C_1>0,\ B_1>0\). We rewrite the Riemann–Liouville integral in the form

$$\begin{aligned} \partial _z^{-n}f(z)=\frac{z^n}{(n-1)!}\int _0^1(1-s)^{n-1}f(zs)\hbox {d}s. \end{aligned}$$

Using this expression, we have

$$\begin{aligned} |\partial _z^{-n}f(z)|&\le \frac{|z|^n}{(n-1)!}\int _0^1(1-s)^{n-1}C_1\exp (B_1|zs|^p)\hbox {d}s\\&\le C_1\frac{|z|^n}{n!}\exp (B_1|z|^p). \end{aligned}$$

By the condition (b), we have

$$\begin{aligned} |a_{n}(z)\partial ^{-n}_zf(z)|&\le C_1C^{n+1}|z|^n\frac{n!^\frac{1}{q}}{n!}\exp ((B+B_1)|z|^p)\\&\le C_1C^{n+1}\frac{|z|^n}{n!^\frac{1}{p}}\exp (B_2|z|^p) \end{aligned}$$

for \(n\in {\mathbb {N}}\). Here we set \(B_2=B+B_1\). Thus, there exist \(C_2>0\) and \(B_3>0\) for which

$$\begin{aligned} \sum _{n=1}^\infty |a_{n}(z)\partial ^{-n}_zf(z)|\le C_2\exp (B_3|z|^p) \end{aligned}$$

hold for all \(z\in {{\mathbb {C}}}\). This implies \(Pf\in A_p\). The continuity of P can be proved as in the proof of the previous theorem. \(\square \)

3 Evolution of superoscillations for a class of potentials

The prototypical superoscillating sequence is

$$\begin{aligned} F_n(x,a):= \left( \cos \Big (\frac{x}{n}\Big )+ia \sin \Big (\frac{x}{n}\Big )\right) ^n, \end{aligned}$$
(2)

where x is a real variable and \(a>1\) is a parameter. By using Euler’s identity for the exponential, and the Newton binomial formula, it is immediate to show that

$$\begin{aligned} F_n(x,a)=\sum _{j=0}^n C_j(n,a)e^{ix(1-\frac{2j}{n})}, \end{aligned}$$
(3)

where

$$\begin{aligned} C_j(n,a)={n\atopwithdelims ()j}\left( \frac{1+a}{2}\right) ^{n-j}\left( \frac{1-a}{2}\right) ^j. \end{aligned}$$
(4)

The reason for the term superoscillations is easily understood if one considers that all the frequencies that appear in (3) are in modulus less than one, but that the sequence \(F_n(x,a)\) itself converges uniformly (on compact subsets of \(\mathbb {R}\)) to \(e^{iax}\). In our works, see e.g. [3, 4, 6], we considered some Cauchy problems in which the datum when the time t equals 0 is \(F_n(x,a)\), and we asked whether the solution \(\psi _n(x,t)\) to the problem maintained superoscillatory characteristics. In order to show that superoscillations perpetually persist in time, i.e. when the time tends to infinity, we need to explicitly compute the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\psi _n(x,t). \end{aligned}$$

We now consider a potential V(t, x) and the Cauchy problem for the Schrödinger equation

$$\begin{aligned} i\frac{\partial \psi (t,x)}{\partial t}=\Big ( -\frac{1}{2}\frac{\partial ^2 }{\partial x^2} + V(t,x) \Big )\psi (t,x),\quad \psi (0,x)= F_n(x,a). \end{aligned}$$
(5)

According to the type of potential, in some cases it is possible to determine explicitly the Green function \(G_V(t,x,y)\), but in most of the cases this is not possible. The solution of Cauchy problem (5) is given by

$$\begin{aligned} \psi (t,x)=\int _{{\mathbb {R}}} {G}_V(t,x,y)\psi (0,y)\hbox {d}y. \end{aligned}$$
(6)

Since the superoscillatory functions such as \(F_n(x,a)\) are linear combinations of exponential functions, we determine the solution of the Cauchy problem

$$\begin{aligned} i\frac{\partial \varphi (t,x)}{\partial t}=\Big ( -\frac{1}{2}\frac{\partial ^2 }{\partial x^2} + V(t,x) \Big )\varphi (t,x),\quad \varphi (0,x)= e^{iax}, \end{aligned}$$
(7)

that is

$$\begin{aligned} \varphi _a(t,x)=\int _{{\mathbb {R}}} {G}_V(t,x,y)e^{iay}\hbox {d}y. \end{aligned}$$
(8)

The solution of the Cauchy problem is obtained by linearity

$$\begin{aligned} \psi _n(t,x)=\sum _{k=0}^nC_k(n,a)\varphi _{(1-2k/n)}(t,x). \end{aligned}$$
(9)

We will consider the following classes of potentials such that a very general structure of the Green function is of the form

$$\begin{aligned} G_V(t,x,y)=e^{ig_1(t)x^2+2ig_2(t)xy+ig_3(t)y^2} f(x,y,t) \end{aligned}$$
(10)

where \(g_j, g: {\mathbb {R}}\rightarrow {\mathbb {R}}\) are given functions of the time and f is an analytic function; all those functions depend on the potential V.

Definition 3.1

Let \(V=V(t,x)\) be the potential and let \(G_V\) be the Green function of the Schrödinger equation associated with V and let \(a\in {\mathbb {R}}\).

  1. (T1)

    We say that \(G_V\) is of type (I) if

    $$\begin{aligned} \int _{{\mathbb {R}}} {G}_V(t,x,y)e^{iay}\hbox {d}y=h_1(t,x)\sum _{\ell =0}^\infty b_\ell (t,x)a^{\ell }e^{ia t} \end{aligned}$$
    (11)

    where \(h_1\) is a given function and the coefficients \(b_\ell (t,x)\) extend analytically to an entire function \(b_\ell (z,x)\) that satisfy the growth condition: There exists \(A>0\), and for every \(\varepsilon >0\), there exists \(C_\varepsilon >0\), for all \(x\in {\mathbb {R}}\), such that

    $$\begin{aligned} |b_\ell (z,x)|\le C_\varepsilon \frac{A^\ell }{(\ell !)^{1/q}} \exp (\varepsilon |z|^p). \end{aligned}$$
    (12)
  2. (T2)

    We say that \(G_V\) is of type (II) if

    $$\begin{aligned} \int _{{\mathbb {R}}} {G}_V(t,x,y)e^{iay}\hbox {d}y=h_2(t,x)\sum _{\ell =0}^\infty c_\ell (t,x)a^{ \ell }e^{ia x} \end{aligned}$$
    (13)

    where \(h_2\) is a given function and the coefficients \(c_\ell (t,x)\) extend analytically to an entire function that satisfy the growth condition: There exists \(A>0\), and for every \(\varepsilon >0\), there exists \(C_\varepsilon >0\), for all \(t\in {\mathbb {R}}\), such that

    $$\begin{aligned} |c_\ell (t,z)|\le C_\varepsilon \frac{A^\ell }{(\ell !)^{1/q}} \exp (\varepsilon |z|^p). \end{aligned}$$
    (14)

The following result is now immediate to prove.

Lemma 3.2

Let \(a\in {\mathbb {R}}\).

  1. (I)

    Let \(G_V\) be a Green function of type (I). Then, the solution of Cauchy problem (7) can be represented by

    $$\begin{aligned} \varphi _a(t,x)=h_1(t,x)\sum _{\ell =0}^\infty b_\ell (t,x)i^{-\ell } \partial _t^{\ell }e^{ia t}. \end{aligned}$$
    (15)
  2. (II)

    Let \(G_V\) be a Green function of type (II). Then, the solution of Cauchy problem (7) can be represented by

    $$\begin{aligned} \varphi _a(t,x)=h_2(t,x)\sum _{\ell =0}^\infty c_\ell (t,x)i^{-\ell } \partial _x^{\ell }e^{ia x}. \end{aligned}$$
    (16)

Inspired by the above lemma, we define the operators

$$\begin{aligned} \mathcal {U}_1(t,x,\partial _t)=h_1(t,x)\sum _{\ell =0}^\infty b_\ell (t,x)i^{-\ell } \partial _t^{\ell }, \end{aligned}$$
(17)

and

$$\begin{aligned} \mathcal {U}_2(t,x,\partial _x)=h_2(t,x)\sum _{\ell =0}^\infty c_\ell (t,x)i^{-\ell } \partial _x^{\ell }. \end{aligned}$$
(18)

So we can prove the main result.

Theorem 3.3

Under the hypothesis of the previous lemma, the solution can be written as

$$\begin{aligned} \psi _n(t,x)=\mathcal {U}_1(t,x,\partial _x)F_n(x,a), \end{aligned}$$
(19)

or

$$\begin{aligned} \phi _n(t,x)=\mathcal {U}_2(t,x,\partial _t)F_n(t,a). \end{aligned}$$
(20)

Moreover, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\psi _n(t,x)=\varphi _a(t,x), \end{aligned}$$
(21)

or

$$\begin{aligned} \lim _{n\rightarrow \infty }\phi _n(t,x)=\varphi _a(t,x). \end{aligned}$$
(22)

Proof

We consider just one case, since the second is analogous

$$\begin{aligned}&\lim _{n\rightarrow \infty }\psi _n(t,x)= \lim _{n\rightarrow \infty }\mathcal {U}_1(t,x,\partial _t)F_n(x,a)\\&\quad =\mathcal {U}_1(t,x,\partial _t)\lim _{n\rightarrow \infty }F_n(x,a)=\varphi _a(t,x). \end{aligned}$$

\(\square \)

We conclude this article by showing how non-constant coefficients differential operator may naturally appear in the study of these problems. Consider the Cauchy problem

$$\begin{aligned} \left\{ \begin{aligned}&i\frac{\partial \phi (t,x)}{\partial t}=-a(t)\frac{\partial ^2 \phi (t,x)}{\partial x^2}+ b(t)x^2\phi (t,x)-i\Big (c(t)x\frac{\partial \phi (t,x)}{\partial x} +d(t)\phi (t,x)\Big ),\\&\phi (0,x)=e^{iax}. \end{aligned}\right. \end{aligned}$$
(23)

The solution of this problem is given by

$$\begin{aligned} \phi _a(t,x)=\frac{1}{\sqrt{2\pi i\mu (t)}}\,\int _{{\mathbb {R}}} e^{i(\alpha (t)x^2+\beta (t)xy+\gamma (t)y^2)}e^{iay}\hbox {d}y \end{aligned}$$

and also

$$\begin{aligned} \phi _a(t,x)=\frac{1}{\sqrt{2\pi i\mu (t)}}\,e^{i\alpha (t)x^2}\int _{{\mathbb {R}}} e^{i(\gamma (t)y^2 +(a+\beta (t)x)y)}\hbox {d}y . \end{aligned}$$

Using the integral

$$\begin{aligned} \int _{{\mathbb {R}}}e^{i(\delta y^2+2\xi y)}\, dy=\left( \frac{i\pi }{\delta }\right) ^{1/2}e^{-i\xi ^2 /\delta },\ \ \ Im(\delta )\ge 0, \end{aligned}$$

we obtain

$$\begin{aligned} \phi _a(t,x)= & {} \frac{1}{\sqrt{2\pi i\mu (t)}}\,e^{i\alpha (t)x^2}\,\left( \frac{i\pi }{\gamma (t)}\right) ^{1/2}e^{-i(a+\beta (t)x)^2 /(4\gamma (t))}\\ \phi _a(t,x)= & {} \frac{1}{\sqrt{2\mu (t)\gamma (t)}}\,e^{i[\alpha (t)+\beta ^2(t)/(4\gamma (t))]x^2}\, e^{-ia^2/(4\gamma (t))}\, e^{-ia x \beta (t) /(4\gamma (t))}. \end{aligned}$$

Consider the term

$$\begin{aligned} {\mathcal {G}}(t,x):=e^{-ia^2/(4\gamma (t))}\, e^{-ia x \beta (t) /(4\gamma (t))}; \end{aligned}$$

with a change of variables, we have

$$\begin{aligned} \beta (t) /(4\gamma (t)):=z. \end{aligned}$$

If we suppose that this change of variable is invertible, we can write

$$\begin{aligned} t=K(z). \end{aligned}$$

Replacing this expression in \(\frac{1}{4\gamma (t)}\), we can make the substitution

$$\begin{aligned} \eta (z)=\frac{1}{4\gamma (K(z))}, \end{aligned}$$

obtaining

$$\begin{aligned} {G}(z,x):=e^{-ia^2\eta (z)}\, e^{-ia x z}=\sum _{m\ge 0}\frac{(-i\eta (z))^m}{m!}\, a^{2m}\,e^{-ia x z}. \end{aligned}$$

It is clear that a non-constant coefficient operator appears, in fact

$$\begin{aligned} a^{2m}\,e^{-ia x z}=(-ix)^{-2m} \partial _z^{2m}e^{-ia x z} \end{aligned}$$

and so we have

$$\begin{aligned} G(z,x):=e^{-ia^2\eta (z)}\, e^{-ia x z}=\sum _{m\ge 0}\frac{(-i\eta (z))^m}{m!}\, (-ix)^{-2m} \partial _z^{2m}\,e^{-ia x z} \end{aligned}$$

and the operator

$$\begin{aligned} {\mathcal {G}}(z,x,\partial _z):=\sum _{m\ge 0} (-ix)^{-2m}\frac{(-i\eta (z))^m}{m!}\, \partial _z^{2m}. \end{aligned}$$

To summarize, the solution of the Cauchy problem

$$\begin{aligned} i\frac{\partial \psi (t,x)}{\partial t}= & {} -a(t)\frac{\partial ^2 \psi (t,x)}{\partial x^2}+ b(t)x^2\psi (t,x)\nonumber \\&-i\Big (c(t)x\frac{\partial \psi (t,x)}{\partial x} +d(t)\psi (t,x)\Big ),\qquad \psi (0,x)=F_n(x,a). \end{aligned}$$
(24)

can be written as

$$\begin{aligned} \psi _n(t,x)={\mathcal {G}}(z,x,\partial _z)F_n(z,a), \end{aligned}$$

where \({\mathcal {G}}(z,x,\partial _z)\) is a non-constant coefficients differential operator.