The Time Fractional Schrödinger Equation on Hilbert Space

We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative of order 0<α<1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha <1,$$\end{document} and a self-adjoint generator A. Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family {Uα(t)}t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ U_{\alpha }(t)\}_{t\ge 0}$$\end{document}. Moreover, we prove that the solution family Uα(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\alpha }(t)$$\end{document} converges strongly to the family of unitary operators e-itA,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-itA},$$\end{document} as α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} approaches to 1.


Introduction
The Schrödinger equation is the basic equation of quantum mechanics, it describes the evolution in time of a quantum system. More recently, N. Laskin has introduced the fractional Schrödinger equation, as a result of extending the Feynman path integral, the resulting equation is a fundamental equation in fractional quantum mechanics [11][12][13]. Furthermore, N. Laskin [11] states that "the fractional Schrödinger equation provides us with a general point of view on the relationship between the statistical properties of the quantum mechanical path and the structure of the fundamental equations of quantum mechanics". Naber [15] introduced and examined some properties of the timefractional Schrödinger equation, in which (−i) α = e −iα π 2 . It was shown in [15] that the above Eq. (1) is equivalent to the usual Schrödinger equations with a time dependent Hamiltonian. On the other hand, it was point out that the so-called quantum comb model [1,2,8,9], leads to a time-fractional Schrödinger equation with α = 1 2 . Equation (1) describes non-Markovian evolution in Quantum Mechanics. As a result this system has memory. Different aspects of the time fractional Schrödinger equation have already been studied. Particular solutions were sought in [2,4,15] and numerical analysis performed in [5]. Nevertheless, to the best of our knowledge there are no results in the literature which show in full generality the uniqueness and existence of solutions to the abstract Schrödinger equation on a Hilbert space.
The purpose of this paper is to consider the abstract fractional evolution Eq. (1) on a Hilbert space H, in which A is a positive self adjoint operator on H, and ∂ α u ∂t α is the Caputo fractional derivative of order α ∈ (0, 1). We show that A generates a family of bounded operators {U α (t))} t≥0 which are defined by the functional calculus of A via the Mittag-Leffler function when evaluated at A. Moreover if u 0 belongs to the domain of A then we show that u(t) = U α (t)u 0 is the unique strong solution of problem (1). We also study the problem of the continuous dependence on α for U α (t), and we show that where e −itA is the unitary group whose infinitesimal generator corresponds to the self adjoint operator A. Thus, we recover in the limit as α → 1 the classical Theorem of Stone [16].
The remainder of the paper is structured as follows. In Sect. 2, we introduce the notations and recall the notion of the Caputo derivative. We also give the definition of strong solution to the fractional Schrödinger equation. Moreover, we formulate and prove some technical but very crucial lemma. The main result about existence and uniqueness of solution is shown in Sect. 3. The properties of the solution operator are formulated and proven in Sect. 4.

Preliminaries
We use the standard notation We recall the definition of the Riemmann Liouville integral by the convolution product, is the space of absolutely continuous functions on [0, ∞), then we can also realize the Caputo derivative as see [3,10] for further properties and definitions. Henceforth we shall denote the Caputo derivative either by D α u(t) or ∂ α u ∂t α (t), indistinctly.
Remark 2.1. We let E α (z) be the Mittag-Leffler function, that is, Let X be a Banach space, and suppose that u 0 ∈ X and ω ∈ C. If 0 < α < 1. Then the equation has a unique solution given by see [3,6,10]. Moreover, the uniqueness of the solution of (4) follows by the uniqueness theorem for the Laplace transform.
Let A be a densely defined self-adjoint operator on a Hilbert space H, and let 0 < α < 1. For a given u 0 ∈ H we study the following equation of fractional order α t>0, We first introduce the notion of strong solution for the abstract fractional Cauchy problem (5).  We will show that the strong solution of (5) is determined by the functional calculus for a self-adjoint operator when it is applied to the Mittag-Leffler function. Moreover, the following lemma will give us the necessary bounds we need in the proof of the qualitative properties of the solution operator.
In order to prove the next lemma we recall from [7, Theorem 2.3 Eq. 26] that the Mittag-Leffler function has the following representation for α ∈ (0, 1], |arg(z)| < π/α and z = 0, (6) in which Proof. First we show (8) that is α 0 ≤ α < 1/2. We notice that it suffices to prove assertion (8) for t = 1. Indeed, let us assume that (8) holds for t = 1, then for any t > 0 we have that, To begin we assume ω ≥ 1/α 0 . Next we recall that (−i) α = e −iαπ/2 . Then we proceed to estimate |K α (r, (−i) α ω)| for arbitrary ω ≥ 1/α 0 . Thus, where A = cos(πα), B = sin(πα), a = cos(απ/2), b = sin(απ/2) and these quantities are all positive since 0 < α < 1/2. Next we set u(r) and v(r) the real and imaginary parts respectively of the denominator on the right hand side of (10), that is Hence, On the other hand the quadratic u(r) = r 2 − 2raωA + ω 2 A is positive for all real r and its minimum equals to w 2 A(1 − a 2 A) > 0 since a 2 A < 1, and A > 0. But then, Vol. 87 (2017) The Time Fractional Schrödinger Equation 5 Hence the right side of (11) turns out to be less than or equals to Therefore, from (11) and (12) follows that Therefore, from (13) we obtain that the integral Now if ω ≤ 1/α 0 , then we have that Thus, from the very definition of the Mittag-Leffler function, we obtain that, Moreover, by the Stirling formula Hence, by the Lebesgue theorem, we obtain that the map Now, the proof of assertion (8) follows from (14) together with (15). Next we show (9). First we assume that ω ≥ 2 under the condition 1/2 ≤ α < 1 from the hypothesis. Again it suffices to prove assertion (9) 6 P . Ǵ orka et al. IEOT for t = 1. We notice that A ≤ 0, and B, a, and b are all positive. Thus Furthermore, Hence, reasoning as in the proof of (8) we obtain that there is a positive constant M which in this case does not depends on the value of α ∈ [1/2, 1), so that sup Now by an applications of the same argument as in (15) Thus the proof of (9) now follows from these last two inequalities.

Existence of the Dynamics
In this part of our paper we state and prove our principal assertion.
Moreover, there is a measure space (Ω, μ), a measurable function a on Ω and a unitary map W : L 2 (Ω) → H such that the unique solution of problem (16) has the following representation Proof. Let us recall that because of the spectral theorem for a self-adjoint for each f ∈ L 2 (Ω, μ) such that W f ∈ D (A). Moreover, if f ∈ L 2 (Ω, μ) is given, then W f ∈ D(A) if and only if M a f ∈ L 2 (Ω, μ); see e.g. [16,17], where M a f (x) = a(x)f (x). Thus, the spectral theorem ensure us that there exists a unitary map W from L 2 (Ω) onto H such that Now the proof of the theorem falls naturally into two parts.

Uniqueness
Let us assume that u is a strong solution to problem (16). We define v(t, ξ) = (W −1 u(t))(ξ). Then it follows from (17) that Let us observe that g 1−α * v ∈ C 1 ((0, ∞); L 2 (Ω)). Indeed, we shall show that Next we set Θ(t) = g 1−α * v. Then using the fact that W is an isometry, we get that the following expression , is equal to Furthermore, we have that (19) equals Now from the assumptions on the function u we have that (20) approaches to 0 when h → 0. Moreover, we can easily check that derivative Θ is a continuous function. Thus we obtain that g 1−α * v belongs to C 1 ((0, ∞); L 2 (Ω)). Furthermore, by the continuity of W, we obtain

Now the definition of the Caputo derivative implies that
where v 0 = W −1 u 0 and u 0 ∈ D(A). It follows from the Remark 2.1 that the unique solution of the above fractional differential Eq. (21) is given by )v 0 , we get that u is given by This finishes with the proof of the uniqueness property.

Existence
Next, we shall show that u(t) given by formula (22) is indeed a strong solution to the initial value problem (16). First of all, we prove that u ∈ C(R + ; D(A)).
We need to show that u(t) ∈ D(A), for all t ≥ 0. For this purpose let us recall that

Thus, by the spectral theorem we know that h ∈ D(A) if and only if a(·)
(W −1 h)(· ) ∈ L 2 (Ω); see [16,17]. Hence, u 0 ∈ D(A) if and only if a(ξ)(W −1 u 0 ) (ξ) belongs to L 2 (Ω). But then, from the fact that ξ → E α ((−it) α a(ξ)) is bounded by Lemma 2.2, it follows that the function is in L 2 (Ω) for all t ≥ 0 and effectively we get that u(t) ∈ D(A). Moreover, since the mapping t → E α ((−it) α a(ξ)) is continuous, the map u is continuous. Indeed, let us take t 0 , t ∈ R + , then we have that .
Thus, by an application of the Lebesgue dominated convergence theorem the proof of the continuity of the function u defined in (22) is finished. Next, we prove that the map Vol. 87 (2017) The Time Fractional Schrödinger Equation 9 belongs to C 1 ((0, ∞); H). For this purpose we consider the following mapping Once more by the definition of Caputo derivative we get Let us notice that Moreover, by the Mean Value Theorem and Lemma 2.2 we have that Since W is unitary it follows that Therefore from (24) and (25) we obtain that .
Hence by Lebesgue dominated convergence we have Thus, the proof of (23) is complete and hence we have the differentiability of the function Φ. Furthermore, arguing as above, we get that Φ ∈ C ((0, ∞); H). IEOT It remains to prove that the function u defined in (22) satisfies Eq. (16). In order to show this last claim we compute the Caputo derivative of the function u. Consequently, and the whole proof of Theorem 3.1 is now finished.
Remark 3.2. Let A be a self-adjoint operator. Then we shall denote by U α (t) the corresponding solution operator family given by Theorem 3.1. To be more explicit

Properties of the Solution Operator U α
In this section we study the properties of the solution operator U α . Next, the commutation property [U α (t), A] = 0 on D(A) follows from the fact that Next, we state some further properties of the solution operator U α .

Proposition 4.2.
Let α ∈ (0, 1). Then the solution operator enjoys the following properties Proof. (i) Let us take φ, ψ ∈ H. Using the fact that W is a unitary operator we get Hence, we obtain that, The proof of (ii) follows from the very definition of U α (t). Next, we show (iii). We will prove that Since, W is an isometry and we have that (27) According to Lemma 2.2 part (b), for each t > 0 the function |E α ((−it) α a(ξ))| is bounded independently of ξ ∈ M and α ∈ [1/2, 1). But then, there is M such that for all α ∈ [1/2, 1), and ξ ∈ Ω and W −1 φ ∈ L 2 (Ω). Then the dominated convergence theorem applies to (27) when α → 1 − , and thus the proof of (iii) is finished. Remark 4.3. In the paper of Dong and Xu [4] it has been pointed out that the quantity U α (t)u 0 is not conserved during the evolution. IEOT

An Example
We consider A = −Δ, the Laplace operator on L 2 (R n ). Then by the Spectral Theorem we have that Next we find the strong solution of the following fractional evolution equation. Suppose that 0 < α < 1 and consider the initial value problem We will show that the strong solution of (28) is defined by a convolution kernel which is given by the Fourier transform in the distributional sense of the Mittag-Leffler function. To prove this claim we first recall some basic facts. We denote by S(R n ) and by S (R n ) the Schwartz space and the space of tempered distributions respectively. Let ϕ be a function of S(R n ). Then we recall that the action of the dilation operator on ϕ is defined as ϕ λ (x) = ϕ(λx), λ ∈ R, x ∈ R n . Furthermore the action on the Fourier transform F is (Fϕ) λ = 1 λ n Fϕ 1/λ and Fϕ λ = 1 λ n (Fϕ) 1/λ n λ > 0.
We see that the hypothesis of Theorem 3.1 are satisfied. Hence the strong solution of (28) is given by u(t, x) = F −1 (e t α/2 (Fg))(x).
Vol. 87 (2017) The Time Fractional Schrödinger Equation 13 Since e t α/2 ∈ C ∞ ∩ L ∞ , we have that e t α/2 (Fg) ∈ S. Thus, from formula (30), we get that the function x → u(t, x) belongs to the Schwartz space for each t ≥ 0. Using Proposition 4.2 and the above considerations we close the paper with the following observation.