Abstract
In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with \(\delta \)- and \(\delta '\)-potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converge in the topology of the entire function space \(A_1(\mathbb {C})\). Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave under \(\delta \)- and \(\delta '\)-potentials.
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1 Introduction
Superoscillating functions have an oscillatory behaviour which is locally faster than their fastest Fourier component. This paradoxical property was discovered by the first author and his collaborators in their work about weak measurements [1] and afterwards investigated from a mathematical and quantitative point of view by Berry [18]. In antenna theory, this phenomenon was discovered by Toraldo di Francia [47] as pointed out also in [19]. Several authors have contributed to this field and without claiming completeness we mention [10, 12, 22, 23] and also [34,35,36,37, 39, 45, 46]. More recently, a special emphasis was given to the mathematical aspect of superoscillations, see, e.g., [2,3,4,5,6,7,8, 27].
The special topic which we want to investigate in this paper is the Schrödinger time evolution of superoscillating functions F, that is, we consider
where V is the potential and F is the initial datum, which is assumed to be superoscillating. Already a number of different potentials were investigated, see the survey papers [9, 11, 14, 21] and references therein. Our aim is to add the one-dimensional \(\delta \)-potential and \(\delta '\)-potential to this list. We mention that the \(\delta \)-potential was already treated in [15]; however, a (technical) condition on the strength of the \(\delta \)-interaction was imposed there, which we are able to avoid in the present paper. A detailed discussion of Schrödinger operators with \(\delta \) and \(\delta '\)-point interactions can be found in the standard monograph [13]. For further reading on Schrödinger operators with singular potentials, we refer the reader to, e.g., [16, 17, 24,25,26, 29,30,31,32,33, 38, 40,41,44] and the references therein.
We briefly illustrate the concept of superoscillations. Consider for some fixed \(k\in {\mathbb {R}}\) with \(|k|>1\) the sequence of functions
with coefficients
The notion superoscillatory now comes from the fact that, although all the Fourier coefficients \(k_j(n)=1-\frac{2j}{n}\) are contained in the bounded interval \([-1,1]\), the whole sequence converges to
a plane wave with wave vector \(|k|>1\); cf. [28, Theorem 2.1] for more details. Besides the convergence (3), the important feature of the functions \(F_n\) is that also for finite n, they oscillate with frequencies close to k in certain intervals; the lengths of these intervals grow when n increases. Nevertheless, outside this interval, one obtains an exponential growth of the amplitude, which conversely means that the amplitude inside the superoscillatory region is exponentially small. Different types of functions, in the form of a square-integrable \(\mathrm{sinc}\) function, which are band-limited and, in some intervals, oscillate faster than its highest Fourier component, can be found in [20].
Inspired by (2), we define the notion of superoscillations as follows:
Definition 1.1
A sequence of functions of the form
with coefficients \(C_j(n)\in {\mathbb {C}}\) and \(k_j(n)\in {\mathbb {R}}\), is said to be superoscillating, if there exists some \(k\in {\mathbb {R}}\), such that
-
(i)
\(\sup _{n\in {\mathbb {N}}_0,j\in \{0,\ldots ,n\}}|k_j(n)|<k\), and
-
(ii)
for some \(B\ge 0\), one has \(\lim \nolimits _{n\rightarrow \infty }\big \Vert (F_n-\mathrm{e}^{ik\,\cdot })\mathrm{e}^{-B|\,\cdot \,|}\big \Vert _\infty =0\).
Note that (ii) is exactly the convergence in the space \(A_1({\mathbb {C}})\) introduced in Definition 2.1 below. Due to the above description of the exponential growth of the amplitude outside the superoscillatory region, it is reasonable to use the exponential weight \(\mathrm{e}^{-B|\,\cdot \,|}\) as a damping factor in the uniform convergence.
The purpose of this paper is now to consider a superoscillating sequence \((F_n)_n\) as the initial datum of the time-dependent Schrödinger equation (1) with either a \(\delta \)-potential or a \(\delta '\)-potential and investigate the corresponding sequence of solutions \((\Psi _n)_n\). The main result of this paper is the following:
Theorem 1.2
Let \((F_n)_n\) be a superoscillatory sequence with limit function
for some \(k\in {\mathbb {R}}\). Then, the solutions \(\Psi \) and \(\Psi _n\) of (1) with either \(V=2\alpha \delta \) or \(V=\frac{2}{\beta }\delta '\) for \(\alpha ,\beta \in {\mathbb {R}}{\setminus }\{0\}\), and initial data \(\mathrm {e}^{ik\,\cdot \,}\) and \(F_n\), respectively, satisfy
uniformly on every compact subset of \((0,\infty )\times {\mathbb {R}}\).
Note that the mathematical rigorous implementation of a singular \(\delta \)-potential or a \(\delta '\)-potential in the Schrödinger equation (1) is via jump conditions at the location \(x=0\) of the interaction; cf. (14) and (15).
The proof of Theorem 1.2 is postponed to the end of Sect. 4. Before, in Sect. 2, we introduce the space \(A_1({\mathbb {C}})\) which is used in the convergence (5). We also construct continuous operators acting in this space, which will play a crucial role in the proof of Theorem 1.2. Moreover, in Sect. 3, we explicitly calculate the solution with a plane wave \(F(x)=\mathrm {e}^{ikx}\) as initial condition.
2 Convolution operators in \(A_1({\mathbb {C}})\)
In this section, we recall the definition of the space \(A_1({\mathbb {C}})\) of entire functions with exponential growth, already mentioned below Definition 1.1. For the analysis of superoscillations, this space (or slight modifications of it) is a convenient choice; cf. [7, 11, 14]. Note also that \(A_1({\mathbb {C}})\) is one particular space in the theory of analytically uniform spaces, see, e.g., [7, Chapter 4].
Definition 2.1
Let \({\mathcal {H}}({\mathbb {C}})\) be the space of entire functions and define
A sequence \((F_n)_n\in A_1({\mathbb {C}})\) is said to be \(A_1\)-convergent to \(F\in A_1({\mathbb {C}})\) if
This type of convergence will be denoted by \(F_n\overset{A_1}{\longrightarrow }F\).
The following lemma shows that also the derivatives of functions in \(A_1({\mathbb {C}})\) are exponentially bounded.
Lemma 2.2
Let \(F\in A_1({\mathbb {C}})\) admit the estimate
for some \(A,B\ge 0\). Then, the derivatives \(F^{(n)}\), \(n\in {\mathbb {N}}\), of F admit the estimate
and, in particular, \(F^{(n)}\in A_1({\mathbb {C}})\) for all \(n\in {\mathbb {N}}\).
Proof
If \(B=0\) the statement holds, since in that case, the entire function F is bounded and hence constant, so that all its derivatives vanish identically. In the following, let \(B\ne 0\). By Cauchy’s integral formula, we have
for the nth derivative, where \(r>0\) is not yet specified. The exponential boundedness (7) of the integrand gives the estimate
It is easy to see that the right-hand side can be minimized choosing \(r=\frac{n}{B}\), which gives
Using \(\frac{n!}{n^n}\le 1\), this gives the estimate (8). \(\square \)
After this preparatory lemma, we consider an operator that can be used to reconstruct a given function from plane waves. Such operators will play the crucial role of time evolution operators in Sect. 4, acting on the initial datum of the Schrödinger equation and pointwise giving its solution.
Proposition 2.3
Let \(\Psi :{\mathbb {R}}\rightarrow {\mathbb {C}}\) be a function, which can be written as the absolute convergent power series
with coefficients \((c_m)_m\in {\mathbb {C}}\). Then, the operator \(U:A_1({\mathbb {C}})\rightarrow A_1({\mathbb {C}})\), defined by
is continuous in \(A_1({\mathbb {C}})\) and satisfies
Moreover, for \(F\in A_1({\mathbb {C}})\), such that \(|F(\xi )|\le A\mathrm{e}^{B|\xi |}\) and \(S_B:=\sum \nolimits _{m=0}^\infty |c_m|B^m\), the estimate
is valid.
Proof
To see that for \(F\in A_1({\mathbb {C}})\), the image UF is again an element in \(A_1({\mathbb {C}})\), we have to show that UF is entire and exponentially bounded. Since \(F\in A_1({\mathbb {C}})\) is entire, we can use the power series representation
to write (10) as the power series
In the last step, we interchanged the order of summation, which is allowed, since (8) ensures the estimate
and, hence, the absolute convergence of the double sum. This shows that UF can be represented as an everywhere convergent power series and, hence, is entire. Moreover, the estimate (13) gives the exponential boundedness
so that \(UF\in A_1({\mathbb {C}})\) and the estimate (12) is satisfied. Finally, (11) follows from
\(\square \)
3 Plane wave under a \(\delta \)- or \(\delta '\)-potential
In quantum mechanics, a particle or wave interacting with a \(\delta \)-potential or \(\delta '\)-potential is described by the Schrödinger equation (1), formally putting \(V=2\alpha \delta \) or \(V=\frac{2}{\beta }\delta '\), where \(\alpha ,\beta \in {\mathbb {R}}{\setminus }\{0\}\) model the strength of the respective peaks. The rigorous mathematical description of such singular potentials is via jump conditions at the location \(x=0\) of the potential. More precisely, for the \(\delta \)-potential, we have to consider the system
and for the \(\delta '\)-potential, we have to consider the system
where F is an appropriate initial condition. In the case that F is a plane wave, we find an explicit representation of the solutions \(\Psi _\delta \) and \(\Psi _{\delta '}\) in Theorem 3.2 below. To write down these solutions efficiently, we will use a certain modified error function \(\Lambda \) and its properties. For the convenience of the reader, we recall the following lemma from [15].
Lemma 3.1
For the function
the following statements hold:
-
(i)
\(\Lambda (-z)=2\mathrm{e}^{z^2}-\Lambda (z)\), \(z\in {\mathbb {C}}\);
-
(ii)
\(\Lambda '(z)=2z\Lambda (z)-\frac{2}{\sqrt{\pi }}\), \(z\in {\mathbb {C}}\);
-
(iii)
for \(|z|\rightarrow \infty \) one has
$$\begin{aligned} \Lambda (z)=\left\{ \begin{array}{ll} \frac{1}{\sqrt{\pi }z}+{\mathcal {O}}\left( \frac{1}{|z|^2}\right) , &{} \text {if }{\text {Re}}(z)\ge 0, \\ 2\mathrm{e}^{z^2}+\frac{1}{\sqrt{\pi }z}+{\mathcal {O}}\left( \frac{1}{|z|^2}\right) , &{} \text {if }{\text {Re}}(z)\le 0; \end{array}\right. \end{aligned}$$(16) -
(iv)
\(\Lambda \) admits the power series representation
$$\begin{aligned} \Lambda (z)=\sum \limits _{n=0}^\infty \frac{(-1)^n}{\Gamma (\frac{n}{2}+1)}z^n,\quad z\in {\mathbb {C}}. \end{aligned}$$(17)
The next theorem is the main result in this section.
Theorem 3.2
The Schrödinger equations (14) and (15) with initial datum \(F(x)=\mathrm {e}^{ikx}\), \(k\in {\mathbb {R}}\), admit the explicit solutions
where \(\Psi _{\mathrm{free}}(t,x;k)=\mathrm{e}^{ikx-ik^2t}\) is the solution of the unperturbed system and
Moreover, for \(t\rightarrow \infty \), the wave functions (18) satisfy
where \(1_{{\mathbb {R}}^-}\) denotes the characteristic function of the negative half line.
Remark 3.3
Note that the second terms on the right-hand sides in the asymptotics (20) only appear for the attractive cases \(\alpha ,\beta <0\). These terms are connected to the negative bound states which are only present in the attractive cases. More precisely, these terms can be seen as the time evolution of the eigenfunctions \(\mathrm {e}^{\alpha |x|}\) and \({\text {sgn}}(x)\mathrm {e}^{\beta |x|}\) corresponding to the eigenvalues \(-\alpha ^2\) and \(-\beta ^2\), respectively.
Proof of Theorem 3.2
Since the strategy of the proof is the same for \(\Psi _\delta \) and \(\Psi _{\delta '}\), and since the \(\delta \)-potential is already treated in [15], we restrict our considerations to the \(\delta '\)-case. The verification of (15) and (20b) is splitted into four steps.
Step 1: We check that (18b) satisfies the differential equation (15a). Since this is obvious for the free part \(\Psi _{\mathrm{free}}\), it suffices to check this for (19b). In fact, it suffices to verify that
is a solution of (15a) for every \(\omega \in {\mathbb {C}}\). With the help of Lemma 3.1 (ii), we first compute
Using this with \(\omega =\beta \) and \(\omega =-ik\) immediately shows that \(\varphi _{\delta '}\) and hence \(\Psi _{\delta '}\) solve (15a).
Step 2: We verify that the jump conditions (15b) and (15c) are satisfied. For this, we calculate the limits \(x\rightarrow 0^\pm \) of (21) and (22) and insert them into (19b), to get
Plugging this into (18b) gives
Finally, using property (i) in Lemma 3.1 of \(\Lambda (-z)\) leads to
from which it is clear that (15b) and (15c) are both satisfied.
Step 3: We check the initial condition (15d). In fact, due to the asymptotic behaviour (16), we have \(\varphi _{\delta '}(t,x;k)\rightarrow 0\) for \(t\rightarrow 0^+\), and hence
Step 4: In this step, we show the large time asymptotics (20b) of the solution. Note first that for large t, we have
Hence, from (16), we get for every fixed \(x\in {\mathbb {R}}{\setminus }\{0\}\) the asymptotic behaviour
Similarly, we get the asymptotics
for the second term in (19b). Substituting this into (19b) gives
Since the third summand in this expression is of order \(t^{-3/2}\) and hence, in particular, \({\mathcal {O}}(\frac{1}{t})\), the expansion reduces to
Using this in (18b), immediately gives the asymptotics (20b) of \(\Psi _{\delta '}\). \(\square \)
4 Proof of Theorem 1.2
For the proof of Theorem 1.2, some preparatory statements are needed. The following lemma provides a simple algebraic reformulation of the power series of the difference quotient of an entire function.
Lemma 4.1
Let \(F\in {\mathcal {H}}({\mathbb {C}})\) be an entire function, that is, F admits the power series representation
with the coefficients \(f_n=\frac{F^{(n)}(0)}{n!}\). Then, for every \(a\in {\mathbb {C}}{\setminus }\{0\}\), the difference quotient admits the series representation
Proof
Inserting the power series (23) into the difference quotient gives
We can now use the binomic formula
to rewrite the series (24) as
\(\square \)
The following Lemma 4.2 for the \(\delta \)-potential and its counterpart Lemma 4.3 for the \(\delta '\)-potential are further useful ingredients in the proof of Theorem 1.2. Here, Lemma 4.1 is used to construct time evolution operators, which take the initial value of the Schrödinger equation and pointwise give the solutions (18). In the case of a \(\delta \)-potential, a similar operator was already provided in [15]; however, there the (technical) restriction \(|k|<|\alpha |\) (wave vector smaller than the potential strength) appeared, which is avoided here.
Lemma 4.2
For fixed \(x\in {\mathbb {R}}{\setminus }\{0\}\) and \(t>0\), there exists a continuous linear operator \(U_\delta (t,x):A_1({\mathbb {C}})\rightarrow A_1({\mathbb {C}})\), such that (18a) can be represented as
Furthermore, for every \(B\ge 0\), there exists \({\widetilde{S}}_{B,\delta }(t,x)\), continuous in t and x, such that the estimate
holds for every \(F\in A_1({\mathbb {C}})\) satisfying \(|F(\xi )|\le A\mathrm{e}^{B|\xi |}\).
Proof
Since \(\Psi _\delta \) decomposes into (18a), it is sufficient to ensure (25) and (26) for its components \(\Psi _\text {free}\) and \(\varphi _\delta \). Starting with \(\Psi _\text {free}\), we can write it as the power series
which coincides with (9). Hence, by Proposition 2.3, there exists a continuous operator \(U_\text {free}(t,x):A_1({\mathbb {C}})\rightarrow A_1({\mathbb {C}})\), such that
Arguing in the same way as in (27) the respective constant \(S_{B,\text {free}}(t,x)\) in the bound (12) can be estimated by
Next, we rearrange the terms of the function \(\varphi _\delta \) in (19a) in the form
It follows that this is a difference quotient of the form (23) with \(z=\frac{|x|}{2\sqrt{it}}-ik\sqrt{it}\) and \(a=(\alpha +ik)\sqrt{it}\). Hence, we obtain the series expansion
where we used the coefficients in the power series representation (17) of \(\Lambda \). In a similar way as in (27) and (29) for the free part, we can use the binomic formula to further expand this series into the form (9). Hence, we get an operator \(U_{\delta ,+}(t,x):A_1({\mathbb {C}})\rightarrow A_1({\mathbb {C}})\) satisfying
with a constant \(S_{B,\delta ,+}(t,x)\) in the corresponding bound (12). This constant can be estimated by
The same reasoning for \(\varphi _\delta (t,-x;-k)\) leads to an operator \(U_{\delta ,-}(t,x):A_1({\mathbb {C}})\rightarrow A_1({\mathbb {C}})\) and a constant \({\widetilde{S}}_{B,\delta ,-}(t,x)\). Finally, we add all three terms together to get
which satisfy (25) and (26). \(\square \)
Although the proof is slightly different, an analogous statement as in Lemma 4.2 holds for the \(\delta '\)-potential.
Lemma 4.3
For fixed \(x\in {\mathbb {R}}{\setminus }\{0\}\) and \(t>0\), there exists a continuous linear operator \(U_{\delta '}(t,x):A_1({\mathbb {C}})\rightarrow A_1({\mathbb {C}})\), such that (18b) can be represented as
Furthermore, for every \(B\ge 0\), there exists \({\widetilde{S}}_{B,\delta '}(t,x)\), continuous in t and x, such that the estimate
holds for every \(F\in A_1({\mathbb {C}})\) satisfying \(|F(\xi )|\le A\mathrm{e}^{B|\xi |}\).
Proof
Since \(\Psi _{\delta '}\) decomposes into (18b), it is sufficient to ensure (31) and (32) for its components \(\Psi _\text {free}\) and \(\varphi _{\delta '}\). Since \(\Psi _\text {free}\) is already treated in (28) and (29), we only consider \(\varphi _{\delta '}\). For this, we first split (19b) into \(\varphi _{\delta '}=\varphi _{\delta '}^{(0)}+\varphi _{\delta '}^{(1)}\), where
By comparing (33a) with (30), we see that, besides the presence of the prefactor \({\text {sgn}}(x)\), both coincide if \(\alpha \) is replaced by \(\beta \). Hence, the same computations as in then proof of Lemma 4.2 lead to operators \(U_{\delta ',\pm }^{(0)}(t,x)\) and corresponding constants \({\widetilde{S}}_{B,\delta ',\pm }^{(0)}(t,x)\) for \(\varphi _{\delta '}^{(0)}(t,\pm x;\pm k)\).
For the treatment of (33b), we first use (17) to rewrite \(\varphi _{\delta '}^{(1)}(t,x;k)\) as the series
Proposition 2.3 then gives a continuous operator: \(U_{\delta ',+}^{(1)}(t,x):A_1({\mathbb {C}})\rightarrow A_1({\mathbb {C}})\), such that
as well as a corresponding constant \(S_{B,\delta ',+}^{(1)}(t,x)\) in the bound (12). The constant can be estimated by
In the same way, we get: \(U_{\delta ',-}^{(1)}(t,x):A_1({\mathbb {C}})\rightarrow A_1({\mathbb {C}})\) and \({\widetilde{S}}_{B,\delta ',-}^{(1)}\) for \(\varphi _{\delta '}(t,-x;-k)\). Now, we sum up all terms to end up with
and it follows that (31) and (32) are satisfied. \(\square \)
Now, we are ready to prove Theorem 1.2. Since the argument is the same for the \(\delta \)-potential and the \(\delta '\)-potential, we will only consider the \(\delta '\)-case. From Lemma 4.3, we know that the solution \(\Psi _{\delta '}\) of the Schrödinger equation (15) with the initial datum \(F(x)=\mathrm{e}^{ikx}\) can be represented in the form (31). Since the operator \(U_{\delta '}(t,x)\) and the Schrödinger equation are both linear, we also get
for every solution \(\Psi _n\) of (15) with the initial datum \(F_n\) of the form (4).
Let, now, \((F_n)_n\) be the superoscillating sequence from (5). Then, from the definition of the \(A_1\)-convergence (6), we get the estimate
where \(B\ge 0\) is as in (6) and \(A_n=\sup _{\xi \in {\mathbb {C}}}\vert (F_n(\xi )-\mathrm {e}^{ik\xi })\mathrm {e}^{-B\vert \xi \vert }\vert \rightarrow 0\) for \(n\rightarrow \infty \). Consequently, from the estimate (32), we get for any compact \(K\subseteq (0,\infty )\times {\mathbb {R}}\) the uniform convergence
note that \(\sup _{(t,x)\in K}{\widetilde{S}}_{\mathrm {e}B,\delta '}(t,x)\) is finite, which follows from the continuity of \({\widetilde{S}}_{\mathrm {e}B,\delta '}\); cf. Lemma 4.3. This completes the proof of Theorem 1.2.
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Aharonov, Y., Behrndt, J., Colombo, F. et al. Schrödinger evolution of superoscillations with \(\delta \)- and \(\delta '\)-potentials. Quantum Stud.: Math. Found. 7, 293–305 (2020). https://doi.org/10.1007/s40509-019-00215-4
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DOI: https://doi.org/10.1007/s40509-019-00215-4
Keywords
- Superoscillating functions
- Convolution operators
- Schrödinger equation
- Singular potential
- Entire functions with growth conditions