A unified approach to Schrödinger evolution of superoscillations and supershifts

Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper, we give a unified approach to determine the supershift property for the solution of the time-dependent one-dimensional Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green’s function, but not its explicit form. With this efficient general technique, we are able to treat various potentials.


Introduction
Superoscillations are band limited functions F that oscillate faster than their fastest Fourier component; they appear in connection with weak measurements in quantum mechanics [2,15,19,31] and as initial conditions in the time-dependent onedimensional Schrödinger equation (1.1) or in optics, signal processing, and other fields of physics and engineering as, e.g. antenna theory [25,43]. The theory of superoscillations and their applications has grown enormously in the last decades, and without claiming completeness, we mention the contributions [24], [26][27][28][29][30] and [35,37,38,41,42]. The standard example of a sequence of superoscillating functions is F n (x; k) = n l=0 C l (n; k)e ik l (n)x , (1.2) where k > 1 and the coefficients C l (n; k), k l (n), for n ∈ N 0 , l = 0, . . . , n, are given by C l (n; k) = n l 1 + k 2 n−l 1 − k 2 l and k l (n) = 1 − 2l n , using the binomial coefficient n l . If we fix x ∈ R and let n go to infinity, we obtain the limit lim n→∞ F n (x; k) = e ikx (1.3) uniformly on compact subsets of R. Observe that the frequencies k l (n) in (1.2) are in modulus bounded by 1, but the frequency k in the limit function in (1.3) can be arbitrary large; this (somewhat unexpected) behaviour gives rise to the notion superoscillations. Inspired by the above example, it has been shown that also other coefficients lead to the same phenomenon, and the theory of superoscillations was extended to a larger class of functions in [7,9,11] and to functions of several variables in [10]. More precisely, a sequence (F n ) n of the form (1.2) is said to be superoscillating if sup n∈N 0 , l∈{0,...,n} |k l (n)| < k for some k > 0 and there exists a compact subset K ⊂ R, called superoscillation set, such that lim n→∞ sup x∈K |F n (x) − e ikx | = 0.
The quantum mechanical evolution problem of superoscillations investigates the behaviour of the solution of the time-dependent Schrödinger equation (1.1) with superoscillatory initial data. It is of particular importance to understand if a frequency shift of the initial conditions at time t = 0, as in (1.3), survives the time evolution and leads to a similar shift for the solutions at later times t > 0. The first Schrödinger evolution problem of superoscillations that has been studied was the free particle in [8], where the solution gives a superoscillatory function in several variables. Later on, in the analysis of evolution problems with nonconstant potentials like the quantum harmonic oscillator [16,32,36], the electric field [12], or the uniform magnetic field [33], it turned out that the solution of the Schrödinger equation with superoscillatory initial datum does not formally belong to the class of superoscillatory functions, although a certain frequency shift still appears. These observations stimulated the notion of supershift, see, e.g. [4,13,[33][34][35]. Very roughly speaking, it is known that the supershift property of the initial datum is stable under the time evolution of the Schrödinger equation for the above mentioned explicit cases. The analysis was based on sophisticated tools involving spaces of holomorphic functions with growth conditions, infinite-order differential operators, but required for each of the potentials the explicit form of the Green's function. Although the Schrödinger evolution of superoscillations and supershifts was mainly studied in the one-dimensional situation (with the exceptions of the uniform magnetic field in three dimensions [33] or the constant electric field in n dimensions [12]), the above techniques in principle extend to the multidimensional setting, provided the Green's function is known. For more details on superoscillations and supershifts, we also refer the reader to [3,5,6,8,13,17,21,22] and the introductory papers [14,18,20,23,40].
A general approach to study the evolution of superoscillations and supershifts, which only relies on qualitative properties of the Green's function and avoids its explicit form, does not exist so far. It is the main objective of the present paper to fill this gap; in fact, we shall provide a unified method, where we just assume regularity and growth conditions on the Green's function. The starting point in this paper will be the (formal) representation for some α ∈ (0, π 2 ), and in certain situations under slightly stronger assumption, one even has cf. Remark 3.5. Moreover, we prove in Theorem 3.6 that the solution depends continuously on the initial datum F. This result will be one of the main reasons why the supershift property is stable for t > 0. In Sect. 4, we first recall the supershift property in Definition 4.1 (in a slightly more general form than in earlier publications), and discuss its connection to the concept of superoscillations. Afterwards, we prove in Theorem 4.5 the time persistence of the supershift property for potentials V , where the corresponding Green's function satisfies Assumption 3.1. In Sect. 5, we apply our main results to the one-dimensional Schrödinger equation with explicitly given potentials. Here, we consider the free particle as a warm up, and the time-dependent uniform electric field, the time-dependent harmonic oscillator, and the Pöschl-Teller potential as more sophisticated examples. In each case, we verify that the corresponding Green's functions satisfies Assumption 3.1, and hence fits into our general setting. Therefore, for each of these potentials, we conclude the time persistence of the supershift property of the initial condition.

Fresnel integrals
In this section, we provide some preliminary material on the so-called Fresnel integral, which will be used in our main results in the next sections. Roughly speaking, the main purpose is to make sense of integrals of the form R e iy 2 f (y)dy, (2.1) in particular, in the case where the function f itself is not integrable.
(i) If f satisfies the estimate for some A, B ≥ 0, then for every where both integrands are absolutely integrable.
where the integrand on the right-hand side is absolutely integrable, and also the integrand on the left-hand side is absolutely integrable for every R > 0.
Proof. (i) Since the proof for arbitrary y 0 ∈ R follows the same steps, we will restrict ourselves to y 0 = 0. We use the abbreviation k = tan(α) > 0. For R > 0, we consider the integration path With the exponential bound (2.3), the integral along the curve γ 2 can be estimated as where the integrand on the left-hand side is absolutely integrable due to the factor e −εy 2 , but also the right-hand side is absolutely integrable due to the estimates which is (2.4).
(ii) In order to prove (2.6), we use the same integration path as in (i). We get the statement from (2.5) and from the fact that for every R > B 2 we have the estimate Hence, we also in this case we conclude Moroever, from (2.7) with ε = 0, we obtain where the integrand on the right-hand side is absolutely integrable due to the estimate The next corollary is a slight extension of Proposition 2.1 which includes some additional constants a > 0 and y 1 ∈ R. Corollary 2.2. Let a > 0, y 1 ∈ R and let f : → C be holomorphic on an open set ⊆ C, which contains the sector S + α from (2.2) for some α ∈ (0, π 2 ). Then, the following assertions hold. where both integrands are absolutely integrable.
(ii) If f satisfies the estimate where the integrand on the right-hand side is absolutely integrable, and also the integrand on the left-hand side is absolutely integrable for every R > 0. Proof.
Since, by (2.8), g satisfies the estimate |g(z)| = e 2 Im(z) √ we know by Proposition 2.1 (i) that (ii) In the same way as in (2.12) for ε = 0, we can write Since, by (2.10), g satisfies the estimate |g(z)| = e 2 Im(z) √ we know by Proposition 2.1 (ii) that The Fresnel integral technique in Proposition 2.1 and Corollary 2.2 can also be applied on the negative semi-axis. This leads to the following corollary.
for some A, B ≥ 0, then for every where both integrands are absolutely integrable.
where the integrand on the right-hand side is absolutely integrable, and also the integrand on the left-hand side is absolutely integrable for every R 1 , R 2 > 0.
The initial purpose of the Fresnel integral technique was to give meaning to the integral (2.1). For holomorphic functions f satisfying the growth condition (2.14), Corollary 2.3 (i) shows (with the choice a = 1 and y 1 = 0) that one can insert the Gaussian e −ε(y−y 0 ) 2 and view the integral (2.1) as the limit Under the stronger assumption (2.16) (which, in particular, implies that f is bounded on the real line) such a regularisation is not necessary and according to Corollary 2.3 (ii) one can regard the integral (2.1) as the limit However, under both assumptions (2.14) and (2.16), one has the absolutely convergent representation

Green's functions and solutions of the Schrödinger equation
The main goal of this section is to treat the Cauchy problem (1.1) for the timedependent one-dimensional Schrödinger equation and the integral representation (1.4) of the solution in a mathematical rigorous framework. In particular, we provide a class of Green's functions and initial conditions such that (1.4) can be interpreted as a Fresnel integral.
For our purposes, it is convenient to view solutions (and their derivatives) in the context of absolutely continuous functions. Recall that for an interval I ⊆ R, a function f : I → C is said to be absolutely continuous, if there exists some g ∈ L 1 loc (I ), such that The linear space of absolutely continuous functions on I will be denoted by AC(I ).
Observe that f ∈ AC(I ) is differentiable almost everywhere and its derivative f coincides with g in (3.1) almost everywhere. For T ∈ (0, ∞] we shall work with the space Let V : (0, T ) × R → C be some potential and let F : R → C be some initial condition. We call a function ∈ AC 1,2 The corresponding Green's function is a function G : (0, T ) × R × R → C (which depends on the potential V , but is independent of the initial condition F), such that admits the representation Next, we collect a set of assumptions on the Green's function G which ensure that the wave function (3.4) is well-defined and a solution of the Cauchy problem (3.3). The precise formulation of this statement, and also the set of allowed initial conditions, is given in Theorem 3.4.
Let ⊆ C be an open set, which contains the double sector S α defined in (2.13) for some α ∈ (0, π 2 ) and suppose that G admits a continuation to a function It will be assumed that the following properties (i)-(iv) hold.
(ii) There exists a ∈ AC((0, T )) with a(t) > 0 and lim t→0 + a(t) = ∞, such that the function G in the decomposition (iii) For all x ∈ R and z ∈ , one has lim t→0 + (3.10) (iv) There exist a nonnegative function A 1 ∈ L 1 loc ((0, T ) × R) and a nonnegative continuous function B 1 : (0, T ) × R → [0, ∞), such that for every fixed t ∈ (0, T ) the spatial derivatives of G and for every fixed x ∈ R the time derivative of G are exponentially bounded, that is, the bounds We briefly comment on some of the conditions in Assumption 3.1 and refer the reader to Sect. 5 for explicit examples of Green's functions that satisfy Assumption 3.1. In this context, we also point out that it is of particular interest to derive conditions on the potential V such that the corresponding Green's function satisfies Assumption 3.1.

Remark 3.2.
The assumption on the holomorphy of G on in the z-variable is needed to apply the Fresnel integral technique in Corollary 2.3. In explicit examples and applications, one typically starts with a Green's function as in (3.5) and verifies that it admits a holomorphic continuation to G : (0, T )×R× → C. The crucial assumption that allows to apply the Fresnel integral method is the decomposition (3.7), where the exponential e ia(t)(z−x) 2 (quadratic in z) is separated from the remainder G, which admits the (at most linear) exponential growth (3.8). The rotation of the integration path in (2.15) from the real line into the complex plane, turns the factor e ia(t)(z−x) 2 into a Gaussian which then dominates the linear exponential growth of G in the integral (3.4).

Remark 3.3.
We do not explicitly require the Green's function to converge to the delta distribution (3.12) In our situation, the counterpart of this standard assumption is the limit condition (3.10), which is the key ingredient to ensure the initial value (3.3b) of the wave function. For us, the limit condition (3.10) is convenient, since in examples it is often easier to check than (3.12).
The next theorem is the main result in the abstract part of this paper. It will be shown that under Assumption 3.1, the integral (3.4) is meaningful as a Fresnel integral and that the resulting function is a solution of the time-dependent Schrödinger equation

13)
for some A, B ≥ 0. Then, the wave function Remark 3.5. If we replace the growth condition (3.8) in Assumption 3.1 (ii) by the stronger condition (3.16) and also replace (3.13) by the stronger condition then it follows from Corollary 2.3 (ii) that the wave function (3.14) can be written in the equivalent form We point out that the stronger growth condition (3.16) on the Green's function is satisfied in all applications in Sect. 5. However, the growth condition (3.17) for the initial condition F is rather restrictive and it is desirable to allow also initial conditions that may be unbounded on the real line. See for example the type of initial conditions which arise naturally for the supershift property in Theorem 4.5.
Proof of Theorem 3.4. Step 1. In the first step, we apply Corollary 2.3, to show that the expression (3.14) for the wave function is meaningful. For this, we fix t ∈ (0, T ), x ∈ R and use the estimates (3.8) and (3.13) to get Hence, due to the decomposition (3.7), the assumptions of Corollary 2.3 are satisfied, which means that the wave function (3.14) exists and admits the absolutely convergent representation

19)
Step 2. We show that the function in (3.14) is a solution of the Schrödinger equation (3.15a). Roughly speaking, since G is already a solution of (3.6) by Assumption 3.1 (i), one needs to check that wave function (3.19) belongs to the space AC 1,2 ((0, T ) × R) and that the derivatives can be carried inside the integral. Note first, that G( · , x, z) ∈ AC((0, T )) for every x ∈ R, z ∈ by Assumption 3.1 (i) and hence for any t 0 ∈ (0, T ), we have This leads to the following integral representation of the wave function (3.19) Using the decomposition (3.7), we can write the derivative as Then, we use (3.8), (3.11) and (3.13) to estimate the integrand in (3.20) by · e (B+2|x|a(τ ) sin(α))|y| e −a(τ ) sin(2α)y 2 .
Since the functions A 0 ( · , x), B 0 ( · , x), B 1 ( · , x), a( · ) are continuous and a ( · ), A 1 ( · , x) ∈ L 1 loc ((0, T )) by Assumption 3.1, it follows that the integrand in (3.20) is integrable on [t 0 , t]. Moreover, the factor e −a(τ ) sin(2α)y 2 also implies integrability with respect to y ∈ R, and hence the integrand is absolutely integrable on [t 0 , t] × R. Therefore, the order of integration in (3.20) can be interchanged and we obtain In particular, this shows ( · , x) ∈ AC((0, T )) and the derivative with respect to t exists almost everywhere and is given by Using the same argument, also (t, · ) and ∂ ∂ x (t, · ) are absolutely continuous, with spatial derivatives almost everywhere given by This means ∈ AC 1,2 ((0, T ) × R) and from (3.6), we conclude that the Schrödinger equation (3.15a) is satisfied for a.e. t ∈ (0, T ), x ∈ R.
Step 3. Now, we verify the initial condition (3.15b). For this, we fix x ∈ R and split up the integral (3.14) as where we have set and y 1 < 0 and y 2 > 0 are chosen such that x ∈ (y 1 , y 2 ). Starting with 2 , we use the fact that the shifted sector y 2 + S + α is contained in S + α . A similar estimate as in (3.18) shows that we may apply Corollary 2.2 (i) to G(t, x, y + y 2 )F(y + y 2 ) and find In this form, we can estimate 2 (t, x) as e −a(t)(y 2 sin(2α)+2(y 2 −x)y sin(α)) e (B+B 0 (t,x))|ye iα +y 2 | dy where, for a shorter notation, we used (ξ ) := e ξ 2 (1 − erf(ξ )) as a modification of the well-known error function; for the computation of the integral see [4, Lemma 2.1]. According to Assumption 3.1 (ii), we know that A 0 √ a and B 0 remain finite for t → 0 + , and also that a → ∞ for t → 0 + . Therefore, since lim ξ →∞ (ξ ) = 0, see [1,Formula 7.1.23], and x < y 2 we conclude lim t→0 + 2 (t, x) = 0. (3.22) In the same way, one verifies that (3.23) For the limit of 0 for t → 0 + , we first note that due to the dominated convergence theorem, we can write

G(t, x, y)F(y)dy.
Using the derivative d dξ erf(ξ ) = 2 √ π e −ξ 2 of the error function together with the decomposition (3.7), we rewrite the integral as

x, y)F(y)dy.
Applying integration by parts then leads to the four terms Consider now the fourth term in (3.24). Due to the estimate (3.8), we have which is uniformly bounded for t → 0 + by (3.9). Since also the error function ξ → erf(i √ i ξ) is bounded on R, there exists a t-uniform majorant, and hence, the limit can be carried inside the integral. This gives For the third term in (3.24), we note that the compact interval [y 1 , y 2 ] is contained in the open set . Hence, there exists some r > 0 such that for every y ∈ [y 1 , y 2 ] the closed ball B r (y) is contained in . The Cauchy integral formula then gives  (3.26) and from (3.9), we see that it admits an t-independent upper bound near t = 0 + . Hence, we are allowed to carry the limit inside the integral and get Moreover, by the representation (3.25) and the estimate (3.26), we get Again, by (3.9), we obtain an integrable and t-independent upper bound of the third term in (3.24). Hence, we are allowed to carry the limit t → 0 + inside the integral, which vanishes according to (3.27). Altogether this shows that (3.24) converges to Together with (3.22) and (3.23), this finally proves the initial value (3.15b).
In the next theorem, we show that the solution of the Schrödinger equation depends continuously on the initial condition. Note that the assumed convergence (3.28) of the initial condition is stronger than the resulting uniform convergence on compact sets (3.29) at times t > 0. However, the stronger convergence (3.28) is justified, since this is the standard type of convergence in which superoscillations are normally treated, see for example (4.6). For convenience, we use the notation (t, x; F) to emphasize the initial condition. Proof. Since the double sector S α is contained in , there exists for every x ∈ R some β(x) ∈ (0, α], continuously depending on x, such that the shifted double sector x + S β(x) is contained in . In the same way as in (3.19), we get the representation  0 (t,x))|x| R e −a(t) sin(2β(x))y 2 +(C+B 0 (t,x))|y| dy where the analytic value of the last integral is similar to the one in (3.21). Since L n n→∞ −→ 0 by (3.28) and the right-hand side is continuous in t and x by (3.9) and the continuity of β, it follows that the convergence (3.29) is uniform on compact subsets of [0, T ) × R.

Supershifts and superoscillations
The aim of this section is to investigate the time evolution of superoscillations and the supershift property of the solution of the one-dimensional Schrödinger equation (3.3). As already mentioned in the introduction, the main novelty of our unified approach, with respect to the existing literature, is, that we are able to consider potentials, where the explicit form of the Green's function is not known. Instead, our results are only based on the regularity and growth conditions on the Green's functions, see Assumption 3.1.
We start with the abstract definition of a supershift and explain its usefulness and meaning afterwards.   for some C ≥ 0, which agrees with the assumption (3.28) in Theorem 3.6.
In the next remark, we explain the connection between the notion of supershift in Definition 4.1 and the concept of superoscillations, which has attracted a lot of attention in the physical and mathematical literature, see the references mentioned in the introduction. Below, we use the definition of superoscillations from [4,11].

Remark 4.4 (Superoscillations). Sequences of the form
with coefficients C j (n) ∈ C, k j (n) ∈ R, are often called generalized Fourier sequences. Note that with ϕ k (x) = e ikx , x ∈ R, as in cf. [4,Definition 5.1]. Note that the sequence ( n ) n converges to a plane wave e ikx with frequency |k| > k , that is, the shift in the k-variable is manifested in a shift of the frequencies, which leads to the terminology superoscillations. Observe that superoscillations can be viewed as a special case of the supershift in Definition 4.1 by choosing In fact, the supershift property is heavily inspired by the concept of superoscillations and Definition 4.1 is designed in such a way that it applies to the time evolution of solutions of the Schrödinger equation subject to superoscillatory initial data.
The first main result of this section is Theorem 4.5 on the supershift property of the solution of the Schrödinger equation, which can be viewed as a corollary of the continuous dependence result from Theorem 3.6. Roughly speaking, we consider a family of initial conditions that admits a supershift (with respect to a slightly stronger form of convergence as in Definition 4.1) and conclude that the corresponding solutions of the Schrödinger equation admit a similar type of supershift; see also Remark 4.6.
for some A(κ), B(κ) ≥ 0 continuously depending on κ. If a sequence of initial conditions (F n ) n of the form F n (z) = n l=0 C l (n)ϕ κ l (n) (z), z ∈ , (4.10) with coefficients C l (n) ∈ C, κ l (n) ∈ U, converges as for some C ≥ 0, to some ϕ κ with κ ∈ O\U, then the sequence of solutions of the Schrödinger equation converges as uniformly on compact subsets of [0, T ) × R.
Proof of Theorem 4.5. The fact that the convergence (4.11) leads to the convergence (4.12), was already proven in Theorem 3.6. Moreover, splitting the solutions (t, x; F n ) into the given linear combination is allowed due to the linearity of the Schrödinger equation with respect to the initial condition, i.e. In the next result, we continue the theme of Theorem 4.5 and return to the analyticity issue mentioned in Remark 4.2. In fact, the following Theorem 4.7 shows that analyticity in the κ-variable in the initial condition implies analyticity in the κ-variable in the wave function.
is holomorphic, then for every fixed t ∈ (0, T ), x ∈ R, the mapping is holomorphic as well.
Proof. Fix t ∈ (0, T ), x ∈ R. Then for any triangle ⊆ O, we have the path integral (4.14) due to the representation (3.19) of the wave function. Here, α ∈ (0, π 2 ) is the angle of the double sector S α in Assumption 3.1. In order to interchange the order of integration, we have to prove absolute integrability of the double integral. Firstly, the estimate x)+2|x|a(t) sin(α))|y| (4.15) follows from (3.7), (3.8) and (4.13) and shows that the y-integral is absolutely convergent. Moreover, the coefficients A(κ), B(κ) are assumed to be continuous and hence this upper bound can be uniformly estimated on the compact triangle . This means that the right hand side of (4.15) can be estimated by some κ-independent and yintegrable upper bound. Hence, the double integral (4.14) is absolutely convergent, and we are allowed to interchange the order of integration and get Since the mapping O κ → ϕ κ (ye iα ) is holomorphic the path integral along vanishes, and we get (t, x; ϕ κ )dκ = 0.
Due to the Theorem of Morera this implies analyticity of O κ → (t, x; ϕ κ ).
In order to appreciate our main results, the following corollary shows how the above  uniformly on compact subsets of [0, T )×R, that is, ( (t, x; F n )) n admits a supershift. Moreover, for every fixed t ∈ (0, T ), x ∈ R, the mapping is analytic.
Proof. According to (4.6), the initial conditions (F n ) n converge as lim n→∞ sup z∈C |F n (z) − e iκz |e −C|z| = 0, for some C ≥ 0. Hence, it follows from Theorem 4.5 that the sequence of solutions ( (t, x; F n )) n converges as (4.17) uniformly on compact subsets of [0, T )×R. Moreover, since the mapping C κ → e iκz is analytic for every fixed z ∈ C, it follows from Theorem 4.7 that also the mapping is analytic for every fixed t ∈ (0, T ), x ∈ R.

Some applications of the main results
We are now in the position to apply the results of the previous sections to the timedependent one-dimensional Schrödinger equation with specific potentials. Here, we consider the case of the free particle, the time-dependent uniform electric field, the time-dependent harmonic oscillator, and the Pöschl-Teller potential. In particular, we formulate a variant of Theorem 4.5. The proof of Theorem 5.1 is direct; in fact, for each potential, we derive the Green's function and verify Assumption 3.1. From our discussion below, it is also immediate that for each of the following potentials (I)-(IV), the representation of the wave function in Theorem 3.4, the continuous dependency on the initial value in Theorem 3.6, and the analyticity property in Theorem 4.7 hold.
We also refer the reader to [8] for the free particle, [12] for the constant electric field and [16,32,36] for the harmonic oscillator that are included in our general setting, while the Pöschl-Teller potential is treated here for the first time. The difference with respect to the previous literature is that the initial condition considered here is not necessarily a superoscillatory function but a supershift.
assume that the potential V in (5.1a) is one of the following: cosh 2 (x) for some l ∈ N. Moreover, with O, U ⊆ C such that U O, and ⊆ C some open set containing the double sector S α from (2.13) for some α ∈ (0, π 2 ), we consider a family of analytic functions ϕ κ : → C, κ ∈ O, and initial conditions of the form where C l (n) ∈ C, κ l (n) ∈ U. If we assume that We point out that Theorem 5.1 shows (similarly as Theorem 4.5 and Remark 4.6) that for the potentials (I)-(IV), the supershift property of the initial datum (F n ) n , with the stronger convergence (5.3), carries over to a supershift property of the solutions ( (t, x; F n )) n . Indeed, the convergence (5.3) implies the uniform convergence on all compact subsets of . Hence, (F n ) n admits a supershift according to Definition 4.1 in the metric space X = with respect to the functions ϕ κ . Furthermore, the convergence (5.4) shows the supershift property of the wave functions ( (t, x; F n )) n in the metric space X = [0, T ) × R with respect to the functions φ κ (t, x) := (t, x; ϕ κ ).
For the proof of Theorem 5.1, the Green's functions for the respective potentials (I)-(IV) are investigated in the following paragraphs. The first potential is the free particle, where it is almost obvious that Assumption 3.1 is satisfied.
We show that in this case the Green's function satisfies Assumption 3.1 with T = ∞. First of all it is clear that for every t ∈ (0, ∞) and x ∈ R, we can extend G(t, x, · ) to an entire function by simply replacing y → z.
A direct computation shows that G satisfies the differential equation (3.6). Moreover, the Green's function admits the decomposition (3.7) with The bound (3.8) and the extension properties (3.9) are satisfied with the choices It is also clear that the limit condition (3.10) holds and the estimates for the derivatives in (3.11) follow immediately from the explicit form of G.
It will be assumed that λ : [0, ∞) → R is continuous. This type of potential was already investigated with respect to the time persistence of superoscillations; cf. [12,Theorem 3.6]. In the present setting, the Green's function is of the form where the coefficients α, β : [0, ∞) → R are the solutions of the ordinary differential equations with initial conditions α(0) = β(0) = lim t→0 + tα (t) = 0. It will be shown that G satisfies Assumption 3.1 with T = ∞. As above, it is clear that for t ∈ (0, ∞) and x ∈ R, one can extend G(t, x, · ) to an entire function by simply replacing y → z. Differentiation and the use of the differential equations (5.7) show that G is indeed a solution of (3.6). Moreover, the Green's function admits the decomposition (3.7) with a(t) = 1 4t and G(t, x, z) = 1 2 √ iπ t e iβ(t)+i xtα (t)+izα(t) .
Using the initial conditions of the coefficients α and β, one easily verifies that the limit condition (3.10) holds. Finally, when computing the derivatives ∂ ∂ x G, ∂ 2 ∂ x 2 G, ∂ ∂t G, one obtains functions of the form

P(t, x, z)e iβ(t)+i xtα (t)+izα(t) ,
where P(t, x, z) is continuous in t and x, and a polynomial in the z-variable. In this form it is not difficult to see that the derivatives can be estimated as in (3.11).
(III) Time-dependent harmonic oscillator V (t, x) = λ(t)x 2 It will be assumed that λ : [0, ∞) → R is continuous. In contrast to the potentials (I) and (II), it turns out that the expression for the Green's function for the harmonic oscillator may only be valid on a finite time interval (0, T ). It is of the form , t ∈ (0, T ), x, y ∈ R, (5.8) where the coefficients α, β are the solutions of the ordinary differential equations α (t) = −4λ(t)α(t) and β (t) = −4λ(t)β(t), (5.9) with initial conditions α(0) = β (0) = 0 and β(0) = α (0) = 1. It will be shown that G satisfies Assumption 3.1 with T > 0 chosen as the smallest positive zero of either α or β, T = ∞ if α and β have no positive zeros. With this choice of T it follows from the initial conditions that α(t) > 0 and β(t) > 0 for t ∈ (0, T ). Again it is clear that for every t ∈ (0, T ) and x ∈ R, one can extend G(t, x, · ) to an entire function by simply replacing y → z. Note that α and β are linearly independent solutions of the differential equation (5.9) and hence the Wronskian has the constant value α (t)β(t) − α(t)β (t) = 1, t > 0.
Using β(0) = 1 and it follows that the extension properties (3.9) hold. Using β(0) = 1, the limit (5.10), as well as one verifies that the limit condition (3.10) is satisfied. Finally, when computing the derivatives ∂ ∂ x G, ∂ 2 ∂ x 2 G, ∂ ∂t G, one obtains functions of the form where P(t, x, z) is continuous in t and x, and a polynomial in the z-variable. In this form, it is not difficult to see that the derivatives can be estimated as in (3.11).
(IV) Pöschl-Teller potential V (t, x) = − l(l+1) cosh 2 (x) , l ∈ N For the Pöschl-Teller potential, it turns out that the Green's function cannot be extended to the whole complex plane, due to singularities on the imaginary axis. For x, y ∈ R and t > 0, the Green's function is given by