Abstract
This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, \(\mathcal {H}_{\hbar ,V}:=-\hbar ^{-2}\mathcal {L}_{\hbar }+V\) on the lattice \(\hbar \mathbb {Z}^{n},\) where V is a positive multiplication operator and \(\mathcal {L}_{\hbar }\) is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit \(\hbar \rightarrow 0\).
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1 Introduction
The aim of this manuscript is to investigate the semi-classical version of the general Caputo-type diffusion equation on the discrete lattice
with \(\hbar >0\) being a (small) discretization parameter. Furthermore, we investigate the behaviour of its solutions in the semi-classical limit \(\hbar \rightarrow 0\).
The fractional differential operators are nonlocal operators that can be thought of as a generalization of classical differential operators of arbitrary integer orders. These generalized differential equations can be used to model many fields of physics and processes, including fractals, optics, finance, traffic flow, fluid mechanics, mathematical biology, viscoelasticity, control, electrochemistry, signal processing, control theory, and material sciences; see, e.g., [1, 23, 26, 41] and the references therein.
Significant attention has been devoted by several researchers to studying the behavior of the solutions of time-fractional heat equations with constant, time-dependent, or space-dependent thermal coefficients in various frameworks due to their widespread application in developing laws of physics while studying real-world phenomena. The modeling of various classical and quantum mechanical issues heavily relies on the nonlocal character of fractional derivatives such as Riemann-Liouville, Caputo, Riesz fractional derivative, etc. In recent times, there has been a significant increase in interest towards initial value problems for fractional differential equations involving Caputo-type differential operators. We refer to recent papers [34, 39] for the theory of diffusion equations associated with the Caputo-type differential operators.
The general Caputo-type fractional derivative denoted by \(\mathbb {D}_{(g)}\) is defined as
where the kernel function \(g\in L^{1}_{loc}(0,\infty )\) satisfies the following conditions:
-
1.
the Laplace transform \(\widetilde{g}(p)=\mathscr {L}(g ; p)\) of g exists for all \(p>0\);
-
2.
\(\widetilde{g}(p)\) is a Stieltjes function;
-
3.
\(\widetilde{g}(p) \rightarrow 0\) and \(p \widetilde{g}(p) \rightarrow \infty \) as \(p \rightarrow \infty \); and
-
4.
\(\widetilde{g}(p) \rightarrow \infty \) and \(p \widetilde{g}(p) \rightarrow 0\) as \(p \rightarrow 0\).
We refer to Section 3 for an extensive discussion of the general Caputo-type fractional derivative. Recently, diffusion equations with the general Caputo-type differential operator have been considered by several authors; see [36,37,38]. Specifically, there is an abundance of literature on the Caputo-type time-fractional diffusion equation with constant, time-dependent, or space-dependent coefficient; see [13, 24, 34, 35, 39]. For the time-dependent coefficient settings that differ from the lattice \(\mathbb {Z}^n\), see [8,9,10, 16,17,18, 21, 22, 40, 42, 43]. However, a heat equation with a constant coefficient on the lattice framework \(\mathbb {Z}^n\) can be seen in the form of a parabolic Anderson model, see [3].
The primary objective of this paper is to investigate the semi-classical analog of the general Caputo-type diffusion equation in the Euclidean framework, which is given by the following Cauchy problem:
where the diffusion coefficient is \(a = a(t) > 0, f\) is the source term, the operator \(\mathbb {D}_{(g)}\) is given by (1.1), and the discrete Schrödinger operator \(\mathcal {H}_{\hbar ,V}\) on \(\hbar \mathbb {Z}^{n}\) is defined by
with the discrete Laplacian \(\mathcal {L}_{\hbar }\) given by
where \(v_{j}\) is the \(j^{th}\) basis vector in \(\mathbb {Z}^{n}\), having all the zeros except for 1 at the \(j^{th}\) component. Here, V is a positive multiplication operator by V(k) satisfying the conditions
We refer to [4, Section 3] for a detailed study on the spectral properties of the discrete Schrödinger operator \(\mathcal {H}_{\hbar , V}\) on \(\hbar \mathbb {Z}^{n}\).
To the best of our knowledge, the general Caputo-type diffusion equation in the lattice \(\hbar \mathbb {Z}^n\) framework has never been considered in the literature so far, even for the Caputo-type diffusion equation. Our primary objective of the present work is to establish the well-posedness results of the diffusion equation (1.2) in the semi-classical framework of \(\hbar \mathbb {Z}^n\). More precisely, we discuss the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation (1.2) when time-dependent coefficients and source terms are regular (irregular). Here, we want to emphasize that the discrete wave equation with regular propagation speed and discrete Klein-Gordon with regular potential have already been considered by the authors in [5] and [6], respectively, and prove that both are well-posed in \(\ell ^2\left( \hbar \mathbb {Z}^n\right) \). On the other hand, recently, the authors in [2] investigated the well-posedness of the discrete heat equation with regular (irregular) thermal conductivity and discrete tempered distributional data.
In our approach, a special feature is that we also allow the coefficient a to have distributional irregularities such as \(\delta ,~ \delta ^{2},~ \delta \delta ^{\prime }\) or Heaviside function, etc. Due to Schwartz’s impossibility result (see Schwartz [33]), the singularities of the coefficients would give rise to fundamental mathematical challenges, and in this scenario, the formulation of the Cauchy problem (1.2) may not be feasible from a distributional perspective. Garetto and the third author of this paper proposed a notion of a very weak solution to get around this problem in [15] and subsequently implemented in [14, 25, 30] for various physical models.
The equation (1.2) is the semi-classical discretization of the following diffusion equation in the Euclidean framework \(\mathbb {R}^n\):
where \(\mathcal {H}_{V}\) is the usual Schrödinger operator on \(\mathbb {R}^{n}\) with potential V defined as
where \(\mathcal {L}\) is the usual Laplacian on \(\mathbb {R}^{n}\). In the case of a regular coefficient, using the technique of [34] with appropriate modifications (see Section 6), one can prove that the Cauchy problem (1.5) is well-posed in the Sobolev space \(\textrm{H}^{s}_{\mathcal {H}_{V}}\), defined as
with the norm \(\Vert f\Vert _{\textrm{H}_{\mathcal {H}_{ V}}^s}:=\left\| \left( I+\mathcal {H}_{ V}\right) ^{s / 2} f\right\| _{L^2\left( \mathbb {R}^n\right) }\), where \(\textrm{H}_{\mathcal {H}_{ V}}^{-\infty }\) is the space of \(\mathcal {H}_{ V}\)-distributions. The relation between the Sobolev space associated with the Schrödinger operator \(\textrm{H}_{\mathcal {H}_{ V}}^{s}\) and usual Sobolev spaces \(\textrm{H}^{s}(\mathbb {R}^{n})\), particularly for non-negative polynomial potentials, can be understood by using the inequalities obtained by Dziubanski and Glowacki [11] in the following theorem:
Theorem 1
[11] Let P(x) be a nonnegative homogeneous elliptic polynomial on \(\mathbb {R}^{n}\) and V is a nonnegative polynomial potential. For \(1<p<\infty \) and \(\alpha >0\) there exist constants \(C_1, C_2>0\) such that
and
for f in the Schwartz class \(\mathcal {S}(\mathbb {R}^{n})\).
Now applying the inequality (1.7) for Schrödinger operator \(\mathcal {H}_{V}\) with non-negative polynomial potential and using the fact that the Schwartz space \(\mathcal {S}(\mathbb {R}^{n})\) is dense in \(L^{2}(\mathbb {R}^{n})\), we deduce the following:
We refer to [4] for a detailed study on the Sobolev spaces for \(\mathcal {H}_{V},\textrm{H}_{\mathcal {H}_{ V}}^{s}\) and \(\textrm{H}^{-\infty }_{\mathcal {H}_{V}}\). We refer to [27,28,29] for Fourier analysis associated with positive self-adjoint operators with discrete spectrum.
To simplify the notation, we shall write \(A\lesssim B\) throughout the article if there is a constant C that is independent of the parameters that occur such that \(A \le CB\). The well-posedness result for the Cauchy problem (1.5) with regular coefficients is given by the following theorem:
Theorem 2
For \(s\in \mathbb {R}\), let \(f\in C([0,T];\textrm{H}^{2+s}_{\mathcal {H}_{V}})\) and \(u_{0}\in \textrm{H}^{2+s}_{\mathcal {H}_{V}}\). If we suppose that \(a\in C([0,T])\) is positive, then the Cauchy problem (1.5) has a unique solution \(u\in C([0, T] ; \textrm{H}_{\mathcal {H}_{V}}^{2+s})\) satisfying \(\mathbb {D}_{(g)}u\in C([0, T] ; \textrm{H}_{\mathcal {H}_{V}}^{s})\). Furthermore, the solution u also satisfies the estimate
for all \(t\in [0,T]\).
On the other hand, the Cauchy problem (1.5) is very weakly well-posed when the diffusion coefficient has a distributional singularity.
Theorem 3
For \(s \in \mathbb {R}\), let \(u_{0} \in \textrm{H}_{\mathcal {H}_{V}}^{2+s} \). Let a be the distribution with support included in [0, T] such that \(a \ge a_0>0\) for some positive constant \(a_{0}\), and also let the source term \(f(\cdot , x)\) be a distribution with support included in [0, T], for all \(x\in \mathbb {R}^{n}\). Then the Cauchy problem (1.5) has a very weak solution \(\left( u_{\varepsilon }\right) _{\varepsilon } \in C([0, T] ; \textrm{H}_{\mathcal {H}_{V}}^{s})^{(0,1]}\) of order s.
Concerning the classical as well as very weak solutions to the Cauchy problem (1.2) on \(\hbar \mathbb {Z}^n\), one may ask the following natural question:
-
Is it possible to recover the classical solution and very weak solution to the Cauchy problem (1.2) when the spatial variable is taken from the Euclidean space \(\mathbb {R}^n\) by taking \(\hbar \rightarrow 0\)?
In this paper, we provide affirmative answers to the abovementioned question with some additional Sobolev regularity.
We conclude the introduction with a brief outline of the organization of the paper.
-
In Section 2, we first recall the basic Fourier analysis associated with the Schrödinger operator \(\mathcal {H}_{V}\), developed by the authors in [4]. We then present our main results and their explanations. The concept of very weak solutions is introduced to address equations that might not have a meaningful solution in the conventional distributional sense.
-
In Section 3, we recall some essential tools and important results related to the general Caputo-type fractional derivative.
-
In Sect. 4, we provide proofs of our main results. In particular, we establish the well-posedness result to the Cauchy problem (1.2) in the case of regular coefficients and also provide the existence, uniqueness, and consistency of the very weak solution to (1.2) in the case of irregular coefficients.
-
In Section 5, we discuss the approximation of the classical solution in the Euclidean framework \(\mathbb {R}^n\) by the solutions in the discrete framework \(\hbar \mathbb {Z}^n\). More precisely, we recapture the classical solution and the very weak solution in certain Sobolev-type spaces in the semi-classical limit \(\hbar \rightarrow 0\).
-
Finally, in Sect. 6, we make a few comments about the classical solution and the very weak solution of the Cauchy problem (1.5).
2 Main results
In this section, we will summarize our key results for the Cauchy problem (1.2) with a regular (irregular) coefficient and a source term. In order to state our main results, some notations and basic preliminaries must be recalled. We refer to Sect. 3 for a thorough exposition of preliminaries.
The Fourier analysis associated with the Schrödinger operator \(\mathcal {H}_{\hbar ,V}\) was developed by the authors in [4]. We are simply mentioning a handful of the most essential properties; for detailed information, see [4].
Lemma 1
Let \(V\ge V_{0}>0\) be a multiplication operator satisfying \(|V(k)|\rightarrow \infty \) as \(|k|\rightarrow \infty \). Then the discrete Schrödinger operator \(\mathcal {H}_{\hbar ,V}\) has a purely discrete spectrum.
Since \(-\mathcal {L}_{\hbar }\) is non-negative and V is a positive operator, it follows that \(\mathcal {H}_{\hbar ,V}\) is a positive operator. Let \(u_{\xi }\) be the eigenfunction of \(\mathcal {H}_{\hbar ,V}\) associated with the eigenvalue \(\lambda _{\xi }\) for each \(\xi \in \mathcal {I}_{\hbar }\), i.e.,
where \(\mathcal {I}_{\hbar }\) is a countable set. The collection of eigenvalues, i.e., \(\{\lambda _{\xi }> 0:\xi \in \mathcal {I}_{\hbar }\}\) will be the discrete spectrum of the Schrödinger operator \(\mathcal {H}_{\hbar ,V}\). Moreover, we have
Consequently, the set of eigenfunctions \(\{u_{\xi }\}_{\xi \in \mathcal {I}_{\hbar }}\) forms an orthonormal basis for \(\ell ^{2}(\hbar \mathbb {Z}^{n})\), i.e.,
where
is the usual inner product of the Hilbert space \(\ell ^{2}(\hbar \mathbb {Z}^{n})\).
We will now review the spaces of distribution generated by \(\mathcal {H}_{\hbar ,V}\), its adjoint, and the associated global Fourier analysis.
The space \(\textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}}:={\text {Dom}}\left( \mathcal {H}_{\hbar ,V}^{\infty }\right) \) is called the space of test functions for \(\mathcal {H}_{\hbar ,V}\), defined by
where \({\text {Dom}}\left( \mathcal {H}_{\hbar ,V}^{k}\right) \) is the domain of the operator \(\mathcal {H}_{\hbar ,V}^{k}\) defined as
The Fréchet topology of \(\textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}}\) is given by the family of norms
The space
of all continuous linear functionals on \(\textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}}\) is called the space of \(\mathcal {H}_{\hbar ,V}\)-distributions. For \(w \in \textrm{H}^{-\infty }_{\mathcal {H}_{\hbar ,V}}\) and \(u \in \textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}}\), we shall write
For any \(u \in \textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}}\), the functional
is a \(\mathcal {H}_{\hbar ,V}\)-distribution, which gives an embedding \(u \in \textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}} \hookrightarrow \textrm{H}^{-\infty }_{\mathcal {H}_{\hbar ,V}}.\)
Definition 1
(Schwartz Space \(\mathcal {S}(\mathcal {I}_{\hbar })\)) Let \(\mathcal {S}(\mathcal {I}_{\hbar })\) denote the space of rapidly decaying functions \(\varphi :\mathcal {I}_{\hbar }\rightarrow \mathbb {C}\), i.e., \(\varphi \in \mathcal {S}(\mathcal {I}_{\hbar })\) if for any \(M<\infty \), there exists a constant \(C_{\varphi , M}\) such that
where we denote
The topology on \(\mathcal {S}(\mathcal {I}_{\hbar })\) is given by the family of seminorms \(p_{k}\), where \(k \in \mathbb {N}_{0}\) and
We now define the \(\mathcal {H}_{\hbar ,V}\)-Fourier transform on \(\textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}}\).
Definition 2
The \(\mathcal {H}_{\hbar ,V}\)-Fourier transform \( \mathcal {F}_{\mathcal {H}_{\hbar ,V}}: \textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}} \rightarrow \mathcal {S}(\mathcal {I}_{\hbar })\) is given by the formula
where \(u_{\xi }\) satisfies (2.1). It can also be extended to \(\textrm{H}^{-\infty }_{\mathcal {H}_{\hbar ,V}}\) in the usual way.
Moreover, the \(\mathcal {H}_{\hbar ,V}\)-Fourier transform \(\mathcal {F}_{\mathcal {H}_{\hbar ,V}}:\textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}} \rightarrow \mathcal {S}(\mathcal {I}_{\hbar })\) is a homeomorphism and its inverse \( \mathcal {F}_{\mathcal {H}_{\hbar ,V}}^{-1}:\mathcal {S}\left( \mathcal {I}_{\hbar }\right) \rightarrow \textrm{H}^{\infty }_{\mathcal {H}_{\hbar ,V}},\) is given by
so that the \(\mathcal {H}_{\hbar ,V}\)-Fourier inversion formula becomes
The \(\mathcal {H}_{\hbar ,V}\)-Plancherel formula takes the form
The \(\mathcal {H}_{\hbar ,V}\)-Fourier transform of the discrete Schrödinger operator \(\mathcal {H}_{\hbar ,V}\) is
since \(\mathcal {H}_{\hbar ,V}\) is a self-adjoint operator. Furthermore, for \(s \in \mathbb {R}\), we introduce the Sobolev space \(\textrm{H}_{\mathcal {H}_{\hbar , V}}^s\) defined by
with the norm \(\Vert f\Vert _{\textrm{H}_{\mathcal {H}_{\hbar , V}}^s}:=\left\| \left( I+\mathcal {H}_{\hbar , V}\right) ^{s / 2} f\right\| _{\ell ^2\left( \hbar \mathbb {Z}^n\right) }\), where \(\textrm{H}_{\mathcal {H}_{\hbar , V}}^{-\infty }\) is the space of \(\mathcal {H}_{\hbar ,V}\)-distributions. Combining the Plancherel identity (2.4) with (2.5), we obtain
For more details about the discrete Schrödinger operator and the associated Sobolev spaces, one can refer to [4, Section 3].
Let us begin by outlining the well-posedness result for the Cauchy problem (1.2) with regular coefficient a and the source term f:
Theorem 4
For \(s\in \mathbb {R}\), let \(f\in C([0,T];\textrm{H}^{2+s}_{\mathcal {H}_{\hbar ,V}})\) and \(u_{0}\in \textrm{H}^{2+s}_{\mathcal {H}_{\hbar ,V}}\). If we suppose that \(a\in C([0,T])\) is positive, then the Cauchy problem (1.2) has a unique solution \(u\in C([0,T];\textrm{H}^{2+s}_{\mathcal {H}_{\hbar ,V}})\) satisfying \(\mathbb {D}_{(g)}u\in C([0,T];\textrm{H}^{s}_{\mathcal {H}_{\hbar ,V}})\). Furthermore, the solution u can be represented by
where \(u_{\xi }\) are the eigenfunctions of \(\mathcal {H}_{\hbar ,V}\) given in (2.1), and also satisfies the estimate
for all \(t\in [0,T]\), where the constant \(C_{a}\) is given by
with \(\varLambda :=\inf \limits _{\hbar>0}\{\lambda _{\xi }:\xi \in \mathcal {I}_{\hbar }\}>0\).
Next, we will permit singularities for the coefficient a and the source term f in the time variable. Following the technique from [15], by regularizing the distributional coefficient a in the manner described below, we can form a net of smooth functions \((a_{\varepsilon })_{\varepsilon }\) given by:
where \(\omega (\varepsilon )\rightarrow 0\) as \(\omega \rightarrow 0\), and \(\psi \) is a Friedrichs-mollifier, i.e., \(\psi \in C_{0}^{\infty }(\mathbb {R}),\psi \ge 0\), and \(\int _{\mathbb {R}}\psi =1\).
For a net of functions/distributions, the notions of moderateness and negligibility are as follows:
Definition 3
(Moderateness)
-
(i)
A net \(\left( a_{\varepsilon }\right) _{\varepsilon } \in C(\mathbb {R})^{(0,1]}\) is said to be C-moderate if, for all \(K \Subset \mathbb {R}\), there exists \(N \in \mathbb {N}_{0}\) such that
$$\begin{aligned} \left\| a_{\varepsilon }\right\| _{C(K)} \lesssim \varepsilon ^{-N}, \end{aligned}$$for all \(\varepsilon \in (0,1]\).
-
(ii)
A net \(\left( u_{\varepsilon }\right) _{\varepsilon } \in C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{s})^{(0,1]}\) is said to be \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{s})\)-moderate if there exists \(N \in \mathbb {N}_{0}\) such that
$$\begin{aligned} \Vert u_{\varepsilon }\Vert _{C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{s})} \lesssim \varepsilon ^{-N}, \end{aligned}$$for all \(\varepsilon \in (0,1]\).
Definition 4
(Negligibility)
-
(i)
A net \(\left( a_{\varepsilon }\right) _{\varepsilon } \in C(\mathbb {R})^{(0,1]}\) is said to be C-negligible if, for all \(K \Subset \mathbb {R}\) and \(q\in \mathbb {N}_{0}\), the following condition is satisfied
$$\begin{aligned} \left\| a_{\varepsilon }\right\| _{C(K)} \lesssim \varepsilon ^{q}, \end{aligned}$$for all \(\varepsilon \in (0,1]\).
-
(ii)
A net \(\left( u_{\varepsilon }\right) _{\varepsilon } \in C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{s})^{(0,1]}\) is said to be \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{s})\)-negligible if for all \(q\in \mathbb {N}_{0}\), the following condition is satisfied
$$\begin{aligned} \Vert u_{\varepsilon }\Vert _{C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{s})} \lesssim \varepsilon ^{q}, \end{aligned}$$for all \(\varepsilon \in (0,1]\).
The constant appearing for \(``\lesssim "\) in the definition of negligibility may depend on q but not on \(\varepsilon \). Furthermore, using the structure theorems for the distributions, we have
Therefore, even though a solution to a Cauchy problem might not exist in the space on the left side of the above embedding, it might exist in the space on the right side.
We may now introduce the concept of a very weak solution in our settings in the following manner:
Definition 5
Let \(s \in \mathbb {R}\) and \(u_{0} \in \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s}.\) The net \(\left( u_{\varepsilon }\right) _{\varepsilon } \in \) \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})^{(0,1]}\) is said to be a very weak solution of order s of the Cauchy problem (1.2), if there exist
-
1.
C-moderate regularization \(a_{\varepsilon }\) of the coefficient a; and
-
2.
\(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-moderate regularization \(f_{\varepsilon }\) of the source term f,
such that \(\left( u_{\varepsilon }\right) _{\varepsilon }\) is \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-moderate and solves the regularized problem
for all \(\varepsilon \in (0,1]\).
Noting that the distributions a and b, are distributions with domain [0, T], it suffices to take into account that \({\text {supp}}(\psi ) \subseteq K,\) where \(K = [0, T]\), throughout the article. A distribution a is said to be positive if
where \(\langle a, \psi \rangle \) denotes the action of the distribution a on the test function \(\psi \). In a similar way, a distribution a is said to be strictly positive if there exists a constant \(\alpha >0\) such that \(a\ge \alpha >0\), where the meaning of the expression \(a\ge \alpha \) is that the distribution \(a-\alpha \) is positive, i.e.,
Furthermore, the following definition may be employed to understand the uniqueness of a very weak solution to the Cauchy problem (1.2):
Definition 6
We say that the Cauchy problem (1.2) has a \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-unique very weak solution, if
-
1.
for all C-moderate nets \(a_{\varepsilon },\tilde{a}_{\varepsilon }\) such that \((a_{\varepsilon }-\tilde{a}_{\varepsilon })_{\varepsilon }\) is C-negligible; and
-
2.
for all \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-moderate nets \(f_{\varepsilon },\tilde{f}_{\varepsilon }\) such that \((f_{\varepsilon }-\tilde{f}_{\varepsilon })_{\varepsilon }\) is \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-negligible,
the net \((u_{\varepsilon }-\tilde{u}_{\varepsilon })_{\varepsilon }\) is \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-negligible, where \((u_{\varepsilon })_{\varepsilon }\) and \((\tilde{u}_{\varepsilon })_{\varepsilon }\) are the families of solutions corresponding to the \(\varepsilon \)-parametrised problems
and
respectively.
The existence and uniqueness of a very weak solution for the Cauchy problem (1.2) with the distributional coefficient as well as the source term can be stated as follows:
Theorem 5
(Existence and uniqueness) For \(s\in \mathbb {R}\), let \(u_{0} \in \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s}\). Let a be a distribution with support contained in [0, T] such that \(a\ge a_{0}>0\) for some positive constant \(a_{0}\), and also let the source term \(f(\cdot ,k)\) be a distribution with support contained in [0, T], for all \(k\in \hbar \mathbb {Z}^{n}\). Then the Cauchy problem (1.2) has a \(C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-unique very weak solution of order s.
Our next theorem is devoted to the consistency of the very weak solution with the classical solution, i.e., whenever the classical solution u of the Cauchy problem (1.2) exists, the very weak solution \((u_{\varepsilon })_{\varepsilon }\rightarrow u\) in \(C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\) as \(\varepsilon \rightarrow 0\).
Theorem 6
(Consistency) For \(s\in \mathbb {R}\), let \(f \in C([0, T], \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\) and \(u_{0} \in \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s}\). Let \((u_{\varepsilon })_{\varepsilon }\) be the very weak solution and u be the classical solution of the Cauchy problem (1.2) given by Theorem 4. If we suppose that \(a \in C([0, T])\) satisfies \(\inf \limits _{t \in [0, T]}a(t)= a_{0}>0\), then for any regularization \((a_{\varepsilon })_{\varepsilon }\) and \( (f_{\varepsilon })_{\varepsilon }\), we have
In the following theorem, we approximate the classical solution in Euclidean settings by the solutions in discrete settings, namely, we prove that under the assumption that the solutions in \(\mathbb {R}^{n}\) exist, they can be recovered in the limit as \(\hbar \rightarrow 0\).
Theorem 7
Let \(V\ge V_{0}>0\) be a positive polynomial potential in (1.3). Let u and v be the solutions for the Cauchy problems (1.2) on \(\hbar \mathbb {Z}^{n}\) and (1.5) on \(\mathbb {R}^{n}\), respectively, with the same Cauchy data \(u_{0}\) and the source term f. Assume that the initial Cauchy data \(u_{0} \in \textrm{H}_{\mathcal {H}_{V}}^{2+s} \) with \(s>2+\frac{n}{2}\) satisfies \(u_{0}^{(4v_{j})}\in \textrm{H}_{\mathcal {H}_{V}}^{2+s} \) for all \(j=1,\dots ,n\), where \(v_{j}\)’s are the \(j^{th}\) standard basis vectors in \(\mathbb {Z}^{n}\). Then for every \(t \in [0, T]\), we have
and the convergence is uniform on [0, T].
Remark 1
The evaluation of initial data and source term from (1.5) on the lattice \(\hbar \mathbb {Z}^{n}\) is the initial Cauchy data \(u_{0}\) and the source term f of the Cauchy problem (1.2).
The analogous of the above semi-classical theorem for a very weak solution can be stated as follows:
Theorem 8
Let \(V\ge V_{0}>0\) be a positive polynomial potential in (1.3). Let \((u_{\varepsilon })_{\varepsilon }\) and \((v_{\varepsilon })_{\varepsilon }\) be the very weak solutions of the Cauchy problems (1.2) on \(\hbar \mathbb {Z}^{n}\) and (1.5) on \(\mathbb {R}^{n}\), respectively, with the same Cauchy data \(u_{0}\) and the source term f. Assume the initial Cauchy data \(u_{0}\in \textrm{H}_{\mathcal {H}_{V}}^{2+s}\) with \(s>2+\frac{n}{2}\) satisfies \(u_{0}^{(4v_{j})}\in \textrm{H}_{\mathcal {H}_{V}}^{2+s}\) for all \(j=1,\dots ,n\), where \(v_{j}\)’s are the \(j^{th}\) standard basis vectors in \(\mathbb {Z}^{n}\). Then for every \(\varepsilon \in (0,1]\) and \(t\in [0,T]\), we have
where the convergence is uniform on [0, T] but pointwise for \(\varepsilon \in (0,1]\).
3 Preliminaries
In this section, we will recall some necessary tools and results related to the general Caputo-type fractional derivative and discrete Schrödinger operator that will be necessary for the analysis that we will follow.
Definition 7
[19] The generalized Mittag-Leffler function denoted by \(E_{\alpha ,\beta }(z)\) is defined by
Definition 8
[32] A real-valued function f on \((0,\infty )\) is said to be completely monotone if f is infinitely differentiable and satisfies
Here are some examples of completely monotone functions on \((0,\infty )\):
Definition 9
[32] The Laplace transform of a non-negative function f on \([0,\infty )\) or a measure \(\mu \) on the line \([0,\infty )\) is defined as
respectively, wherever these integrals converge.
3.1 Generalized Caputo-type fractional derivative
The general Caputo-type fractional derivative was introduced by Kochubei in [20] and subsequently considered by many authors for relaxation equations, diffusion equations, evolution equations, etc. (see [36,37,38]). We must recall the Stieltjes function in order to define the general Caputo-type fractional derivative.
Definition 10
A non-negative function f on \((0,\infty )\) is said to be a Stieltjes function if f admits the integral representation
where \(a, b \ge 0\) are non-negative constants and \(\mu \) is a positive Borel measure on \((0, \infty )\) such that
Here are some examples of Stieltjes functions on \((0,\infty )\):
The class of Stieltjes functions forms a subclass of completely monotone functions. Furthermore, in view of [32, Proposition 1.2], the constants a, b and the measure \(\mu \) in the Definition 10 are uniquely determined by f. For more information about the Stieltjes function, one can refer to [32]. Now we are in a position to define the general Caputo-type fractional derivative.
Definition 11
[20] The general Caputo-type fractional derivative denoted by \(\mathbb {D}_{(g)}\) is defined as
where the kernel function \(g\in L^{1}_{loc}(0,\infty )\) satisfies the conditions (1)-(4) in the introduction.
Depending on the choice of a kernel function g, there are several variants of the Caputo-type fractional derivative; we list a few of them below. For \(0<\alpha <1\), the general Caputo-type fractional derivative is called:
-
1.
Caputo-Dzhrbashyan derivative when \(g(t)=\dfrac{t^{-\alpha }}{\varGamma (1-\alpha )}\);
-
2.
Caputo-Fabrizio derivative when \(g(t)=\dfrac{M(\alpha )}{1-\alpha }\exp \left( -\dfrac{\alpha }{1-\alpha }t\right) \); and
-
3.
Atangana-Baleanu derivative when \(g(t)=\dfrac{B(\alpha )}{1-\alpha }E_{\alpha }\left( -\dfrac{\alpha }{1-\alpha }t\right) \),
where \(M(\alpha )\) and \(B(\alpha )\) are normalization factors satisfying \(M(0)=M(1)=B(0)=B(1)=1\), see [7]. For more information about the general Caputo-type fractional derivative, one can refer to [20].
Next, we will review some results for the Cauchy problem associated with the general Caputo-type fractional derivative \(\mathbb {D}_{(g)}\):
and make some appropriate remarks. The following lemma is dedicated to the corresponding homogeneous fractional differential equation.
Lemma 2
[20, Theorem 2] The fractional differential equation (3.1) with initial Cauchy data \(w_{0}=1\) and the source term \(f\equiv 0\), has a unique solution \(w_{\lambda }(t)\) satisfying the following:
-
1.
\(w_{\lambda }\) is continuous on \([0,\infty )\);
-
2.
\(w_{\lambda }\) is infinitely differentiable on \((0,\infty )\); and
-
3.
\(w_{\lambda }\) is completely monotone on \((0,\infty )\).
In the case of the Caputo-Dzhrbashyan derivative, the generalized Mittag-Leffler function can be utilized to express the solution \(w_{\lambda }\) as follows:
Remark 2
In continuation of the above theorem, using the monotonicity of \(w_{\lambda }\) and the Karamata-Feller Tauberian theorem in [12], the author also proved that
Remark 3
In light of Lemma 2 and the above remark, it is easy to note that the solution \(w_{\lambda }(t)\) satisfies
Lemma 3
[37, Lemma 3.1] Let \(\lambda>0, T>0\), and \(f \in C([0, T])\). Then the fractional differential equation (3.1) with the initial Cauchy data \(w_{0}=0\), has a unique solution in C([0, T]). In particular, the solution has the form
where \(w_{\lambda }(t)\) is the solution of the homogeneous Cauchy problem obtained in Lemma 2.
Similar to Lemma 2, the solution w(t) obtained in the above lemma in the case of the Caputo-Dzhrbashyan derivative can be expressed as
Remark 4
Combining Lemmas 2 and 3, the fractional differential equation (3.1) has a unique solution in C([0, T]) expressed as
Remark 5
Using the fact that \(w_{\lambda }^{\prime }(t)\le 0\) for all \(t\in [0,T]\), we can conclude that the solution \(w(t)\ge (\le )0\) for all \(t\in [0,T]\), if the initial Cauchy data \(w_{0}\ge (\le )0\) and the source term \(f\ge (\le )0\) for all \(t\in [0,T]\).
4 Proof of the main results
In this section, we first prove the existence and uniqueness of the classical solution u of the Cauchy problem (1.2), and then we will conclude the proof of Theorem 4 by establishing the regularity estimates. Throughout this section, we will specify the coefficients as u(t, k; a) and \(u(t,k;a_{0})\) to represent the solution corresponding to the Cauchy problems with coefficients a(t) and \(a_{0}\), respectively.
4.1 Existence and uniqueness
Taking the \(\mathcal {H}_{\hbar ,V}\)-Fourier transform of the Cauchy problem (1.2) with respect to the variable \(k\in \hbar \mathbb {Z}^{n}\) and using (2.5), we obtain the collection of Cauchy problems with Fourier coefficients:
that is,
where \(a_{1}\) is a positive constant given by
Since \(\widehat{f}(t,\xi )+\lambda _{\xi }\left( a_{1}-a(t)\right) \widehat{u}(t, \xi ;a)\in C([0,T])\), by using Remark 4 for the Cauchy problem (4.2), we obtain a unique generalized solution \(\widehat{u}(t, \xi ;a)\in C([0,T])\) which can be represented as
where the function \(\widehat{u}_{\lambda _{\xi }a_{1}}(t,\xi )\) is the solution of the corresponding homogeneous equation of the Cauchy problem (4.2) with initial Cauchy data \(\widehat{u}(0, \xi ;a)\equiv 1\). Furthermore, the solution \(u(t,k)=u(t,k;a)\) can be expressed in the form (2.8), since the set of eigenfunctions \(\{u_{\xi }\}_{\xi \in \mathcal {I}_{\hbar }}\) in (2.1) forms an orthonormal basis of \(\ell ^{2}(\hbar \mathbb {Z}^{n})\). The uniqueness of solution u(t, k) follows from the uniqueness of solution \(\widehat{u}(t,\xi ;a)\) in (4.3). This completes the existence and uniqueness of solution u(t, k) in Theorem 4.
4.2 Regularity estimates
In order to establish the regularity estimate (2.9), we split the Cauchy problem (1.2) into homogeneous and the non-homogeneous Cauchy problems. Then, using the technique followed by the authors in [34] and [35], we will establish the regularity estimates for these Cauchy problems. Consequently, we will obtain the required estimates (2.9).
Before beginning the proof, the following introductory lemma is required, which establishes the regularity of the solution \(\widehat{u}(t,\xi ; a)\).
Lemma 4
Let \(\xi \in \mathcal {I}_{\hbar }\) and \(\widehat{u}(t,\xi ;a)\) be the unique solution of the Cauchy problem (4.1). Then the following statements hold:
-
1.
if \(\widehat{f}(t,\xi ) \le 0\) on [0, T] and \(\widehat{u}_{0}(\xi ) \le 0\), then we have \(\widehat{u}(t,\xi ; a) \le 0\) on [0, T],
-
2.
if \(\widehat{f}(t,\xi ) \ge 0\) on [0, T] and \(\widehat{u}_{0}(\xi ) \ge 0\), then we have \(\widehat{u}(t,\xi ; a) \ge 0\) on [0, T].
Proof
The proof follows from Remark 5. \(\square \)
Let \(u^{u_{0}}(t,k;a)\) and \(u^{f}(t,k;a)\) be the solutions of the Cauchy problems
and
respectively. In the light of Theorem 4, the solution u(t, k; a) of the Cauchy problem (1.2) can be written as
In the following Lemma 5 and 6, we establish the regularity estimate for the homogeneous Cauchy problem (4.4) and non-homogeneous Cauchy problem (4.5), respectively.
Lemma 5
For \(s\in \mathbb {R}\), let \(u_{0}\in \textrm{H}^{2+s}_{\mathcal {H}_{\hbar ,V}}\). If we assume that \(a\in C([0,T])\) is positive, then the solution \(u^{u_{0}}(t,k;a)\) of the Cauchy problem (4.4) satisfies the estimate
for all \(t\in [0,T]\).
Proof
Consider the Cauchy problems
and
Applying Lemma 4 to the Cauchy problem (4.7), we deduce that \(\widehat{u}^{u_{0}}(t, \xi ;a)\) holds the same sign as \(\widehat{u}_{0}(\xi )\) and applying the Lemma 2 to the Cauchy problem (4.8), we get
where \(\widehat{u}^{u_{0}}_{\lambda _{\xi }a_{0}}(t,\xi )\) is the solution of the Cauchy problem (4.8) with initial Cauchy data \(\widehat{u}^{u_{0}}(0,\xi ;a_{0})\equiv 1\). From the equations (4.7) and (4.8), denoting \(w^{u_{0}}(t,\xi ;a):=\widehat{u}^{u_{0}}(t, \xi ;a_{0})-\widehat{u}^{u_{0}}(t, \xi ;a)\), we get
for all \(\xi \in \mathcal {I}_{\hbar }\). Now using the regularity of \(\widehat{u}^{u_{0}}(t, \xi ;a)\) and Lemma 4 for the Cauchy problem (4.10), we deduce that the solution \(w^{u_{0}}(t, \xi ;a)\) also holds the same sign as \(\widehat{u}_{0}(\xi )\), since \(\lambda _{\xi }\left( a(t)-a_{0}\right) \ge 0\) for all \(t\in [0,T]\). Now combining the regularity of solutions \(w^{u_{0}}(t, \xi ;a)\) and \(\widehat{u}^{u_{0}}(t, \xi ;a_{0})\) with equation (4.9), we get
Recalling Remark 3, i.e., \(0\le \widehat{u}^{u_{0}}_{\lambda _{\xi }a_{0}}(t,\xi )\le 1\), for all \((t,\xi )\in [0,T]\times \mathcal {I}_{\hbar }\), we obtain
Additionally, using the above estimate and the Plancherel formula (2.7), we get
Similarily, using the Plancherel formula (2.7) and equation (4.4) with estimate (4.11), we get
for all \(t\in [0,T].\) Putting together (4.12) and (4.13), we get
for all \(t\in [0,T]\). This completes the proof. \(\square \)
Lemma 6
For \(s\in \mathbb {R}\), let \(f\in C([0,T];\textrm{H}^{2+s}_{\mathcal {H}_{\hbar ,V}})\). If we assume that \(a\in C([0,T])\) is positive, then the solution \(u^{f}(t,k;a)\) of the Cauchy problem (4.5) satisfies the estimate
for all \(t\in [0,T]\), where \(\varLambda \) is given by (2.2).
Proof
Consider the Cauchy problems
and
If we write
then we can denote
and
In view of the fact that \(\textrm{Re}\widehat{f}=\widehat{f}^{\max }_{\textrm{Re}}+\widehat{f}^{\min }_{\textrm{Re}}\) and \(\textrm{Im}\widehat{f}=\widehat{f}^{\max }_{\textrm{Im}}+\widehat{f}^{\min }_{\textrm{Im}}\), the Cauchy problems (4.14) and (4.15) can be split into the following Cauchy problems:
and
From the above equations, denoting
and
we get
and
Now applying Lemma 4 to the Cauchy problems (4.16), (4.18), (4.20), and (4.22), we deduce that
for all \(t\in [0,T]\). Consequently, using the regularity of solutions in (4.28) with Lemma 4 to the Cauchy problems (4.24)-(4.27), we conclude that
and
for all \(t\in [0,T]\), since \(\lambda _{\xi }(a(t)-a_{0})\ge 0.\) This gives
for all \(\xi \in \mathcal {I}_{\hbar }\).
Now applying Lemma 3 to the Cauchy problems (4.15), (4.17), (4.19), (4.21), and (4.23), we get
and
respectively.
Combining the facts that
with the above equations and inequality (4.30), we get
for all \(t\in [0,T]\) and \(\xi \in \mathcal {I}_{\hbar }\), where the last inequality follows from the fact that \(0\le w_{\lambda _{\xi }a_{0}}(t,\xi )\le 1\), for all \((t,\xi )\in [0,T]\times \mathcal {I}_{\hbar }\), see Remark 3.
Further, using the Plancherel formula (2.7) with the estimate (4.31), we get
where \(\varLambda \) is given by (2.2). Similarly, combining the Plancherel formula (2.7) and equation (4.5) with the estimate (4.31), we get
for all \(t\in [0,T]\). Putting together estimates (4.32) and (4.33),
for all \(t\in [0,T]\). This completes the proof. \(\square \)
Now, we are in a position to prove the well-posedness theorem for the Cauchy problem (1.2) with the regular coefficient and the source term.
Proof of Theorem 4
Combining the equality (4.6) with Lemmas 5 and 6 we obtain the following regularity estimate
This completes the proof. \(\square \)
In the above result, we proved the well-posedness to (1.2) with regular coefficient in the Sobolev-type spaces. Now, we prove the existence and uniqueness of very weak solutions in the case of distributional coefficients.
Proof of Theorem 5
Let \(a_{\varepsilon }\) be the C-moderate and \(f_{\varepsilon }\) be the \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-moderate regularization of the coefficient a and the source term f, respectively. Now, fix \(\varepsilon \in (0,1]\), and consider the regularized problem
Since \(a\in \mathcal {D}^{\prime }(\mathbb {R})\), by the structure theorem for compactly supported distributions, there exists \(L_{1}\in \mathbb {N}\) such that
where \(\omega (\varepsilon )\rightarrow 0\) as \(\varepsilon \rightarrow 0\). Since \(a\ge a_{0}>0\), it follows that
where \(\tilde{\psi }(x)=\psi (-x), x \in \mathbb {R}\) and \(\tau _t\) is a translation operator. Using Theorem 4 to the Cauchy problem (4.34), we deduce that
for all \(t\in [0,T]\). Using the estimate (4.35) and the inequality \(1+x\le e^x,x\in \mathbb {R}\), we get
for all \(t\in [0,T]\). Putting \(\omega (\varepsilon )^{-L_{1}}=\log (\varepsilon ^{-1})\), we obtain
for all \(t\in [0,T]\). Since \(f_{\varepsilon }\) is \(C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-moderate, there exists \(L_{2}\in \mathbb {N}\) such that
Combining the estimates (4.38) and (4.39), we get
Thus, we conclude that \(u_{\varepsilon }\) is \(C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-moderate. This proves the existence of a very weak solution for the Cauchy problem (1.2).
Now we will prove the uniqueness of a very weak solution. Let \((u_{\varepsilon })_{\varepsilon }\) and \((\tilde{u}_{\varepsilon })_{\varepsilon }\) solve the Cauchy problems (2.13) and (2.14), respectively. Denoting \(w_{\varepsilon }:=u_{\varepsilon }-\tilde{u}_{\varepsilon }\), we obtain
where the source term \(g_{\varepsilon }\) is given by
Using the fact that \((\tilde{a}_{\varepsilon }-a_{\varepsilon })_{\varepsilon }\) and \((f_{\varepsilon }-\tilde{f}_{\varepsilon })_{\varepsilon }\) are C-negligible and \(C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-negligible, respectively, it is evident that \(g_{\varepsilon }\) is \(C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\)-negligible. Since \(w_{\varepsilon }(0,k)\equiv 0\), it follows from Theorem 4, estimate (4.35) and the inequality \(1+x\le e^{x}\) that
Putting \(\omega (\varepsilon )^{-L_{1}}=\log (\varepsilon ^{-1})\) and using the fact that \(g_{\varepsilon }\) is \(C([0,T];\textrm{H}^{2+s}_{\mathcal {H}_{\hbar ,V}})\)-negligible, it follows that
Thus \(w_{\varepsilon }\) is \(C([0,T];\textrm{H}^{2+s}_{\mathcal {H}_{\hbar ,V}})\)-negligible. This proves the uniqueness of a very weak solution and, hence, completes the proof. \(\square \)
In the above result, we discussed the uniqueness of the very weak solution in the sense of Definition 6. Now, we prove the consistency of the very weak solution with the classical solution.
Proof of Theorem 6
Let u be the classical solution of the Cauchy problem:
and \((u_{\varepsilon })_{\varepsilon }\) be the very weak solution, i.e., \((u_{\varepsilon })_{\varepsilon }\) satisfies the following regularized Cauchy problem
In the light of [31, Theorem 1.5.13], we note that
and
for \(a\in C([0,T])\) and \(f\in C([0,T];\textrm{H}^{2+s}_{\mathcal {H}_{\hbar ,V}})\), respectively. Further, we can rewrite the Cauchy problem (4.42) as follows:
where \(g_{\varepsilon }\) is given by
\(g_{\varepsilon } \in C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\) and \(g_{\varepsilon }\rightarrow 0\) in \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\) as \(\varepsilon \rightarrow 0\). Using the Cauchy problems, (4.43) and (4.46), denoting \(w_{\varepsilon }:=u-u_{\varepsilon }\), we obtain
Since the coefficients and source term are regular enough, it follows from Theorem 4 that
for all \(t\in [0,T]\). Taking sup-norm over [0, T] in the above estimate, we obtain
with the constant independent of \(\hbar \) and \(t\in [0,T]\). Since \(g_{\varepsilon } \rightarrow 0\) in \(C([0, T] ; \textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})\), therefore, we have
i.e.,
This concludes the proof. \(\square \)
5 Semi-classical limit \(\hbar \rightarrow 0\)
In this section, we mainly compare the classical (resp. very weak) solutions of (1.2) on \(\hbar \mathbb {Z}^n\) with the classical (resp. very weak) solutions of (1.5) on \(\mathbb {R}^n\) as \(\hbar \rightarrow 0\). We start with the proof of Theorem 7, the semi-classical limit theorems for the classical solution. In this proof, we have used techniques used in the proof of [4, Theorem 2.9].
Proof of Theorem 7
Let u and v solve the Cauchy problems
and
respectively. Using the above Cauchy problems, denoting \(w:=u-v\), we get
Applying Theorem 4 for the above Cauchy problem and using estimate (2.9) along with the fact that \(w(0,k)\equiv 0\), we get
where the constant \(C_{a}\) is given by (2.10).
Now we will estimate the term \(\left\| \left( \mathcal {H}_{V}-\mathcal {H}_{\hbar ,V}\right) v\right\| _{C([0,T];\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s})}\). Let \(\phi \in C^{4}(\mathbb {R}^{n})\), then using the Lagrange’s form of the remainder and Taylor’s theorem, we obtain
for some \(\theta _{\xi }\in (0,1)\) depending on \(\xi \). Utilizing the above relation for \(\textbf{h}=v_{j}\) and \(\textbf{h}=-v_{j}\) for \(j=1,2,\dots ,n\), we deduce that
where \( \theta _{j,\xi },\tilde{\theta }_{j,\xi }\in (0,1),\) \(\delta _{\xi _{j}}^{2}\) is the 2nd order central difference operator and \(v_{j}\) is the \(j^{th}\) standard unit vector in \(\mathbb {Z}^{n}\). Now if we define a translation operator \(E_{\theta _{j}v_{j}}\phi :\mathbb {R}^{n}\rightarrow \mathbb {R}\) by \(E_{\theta _{j}v_{j}}\phi (\xi ):=\phi (\xi -\theta _{j,\xi }v_{j}),\) then the equation (5.6) takes the form
Now we extend this to \(\hbar \mathbb {Z}^{n}\). Consider a function \(\phi _{\hbar }:\mathbb {R}^{n}\rightarrow \mathbb {R}\) defined by \(\phi _{\hbar }(\xi ):=\phi (\hbar \xi )\). Clearly \(\phi _{\hbar }\in C^{4}(\mathbb {R}^{n})\), if we take \(\phi \in C^{4}(\mathbb {R}^{n})\). Now we have
where \(\mathcal {L}\) is the Laplacian on \(\mathbb {R}^{n}\) and \(\mathcal {L}_{1}\) is the discrete difference Laplacian on \(\mathbb {Z}^{n}\). One can quickly notice that
Therefore, the equality (5.7) becomes
Combining (1.3), (1.6) and (5.9), we get
Hence, it follows that
Since V is a non-negative polynomial, therefore, the inclusion (1.8) holds, that is,
Also, we have the usual Sobolev embeddding:
see [31, Excercise 2.6.17]. Combining the above embeddings, we get
Since \(u_{0} \in \textrm{H}_{\mathcal {H}_{V}}^{2+s} \) with \(s>2+\frac{n}{2}\), therefore, from Theorem 2, it follows that the classical solution \(v\in C([0,T];\textrm{H}_{\mathcal {H}_{V}}^{2+s})\) with \(s>2+\frac{n}{2}\). Using the embedding (5.12), we get \(v\in C^{4}(\mathbb {R}^{n})\) and using the hypothesis \(u_{0}^{(4v_{j})} \in \textrm{H}_{\mathcal {H}_{V}}^{2+s} \) for all \(j=1,\dots ,n\), we deduce that
Now from (5.11), it follows that
Using (5.4), (5.13) and (5.14), we get \(\Vert w(t,\cdot )\Vert _{\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s}}+\Vert \mathbb {D}_{(g)}w(t,\cdot )\Vert _{\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s}}\rightarrow 0\) as \(\hbar \rightarrow 0.\) Hence \(\Vert w(t,\cdot )\Vert _{\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s}} \rightarrow 0\) and \(\Vert \mathbb {D}_{(g)}w(t,\cdot )\Vert _{\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{2+s}} \rightarrow 0\) as \(\hbar \rightarrow 0\). This concludes the proof of Theorem 7. \(\square \)
The proof of Theorem 8 will be almost identical to the proof of Theorem 7 with minimal modifications.
6 Remarks
A few final comments regarding the outcomes of classical and very weak solution in the Euclidean framework are as follows:
-
1.
The use of \(\mathcal {H}_{V}\)-Fourier transform instead of \(\mathcal {H}_{\hbar ,V}\)-Fourier transfrom and following the technique of proof of Theorem 4 will give the proof of Theorem 2.
-
2.
The notion of unique very weak solution in Theorem 3 can be adapted verbatim by using the Sobolev norm \(\textrm{H}_{\mathcal {H}_{V}}^{s}\) instead of \(\textrm{H}_{\mathcal {H}_{\hbar ,V}}^{s}\) in Definition 5 and Definition 6. Consequently, the proof of Theorem 3 will follow along the lines of the proof of Theorem 5.
-
3.
Suppose \(\mathcal {M}\) is a positive self-adjoint operator on a Hilbert space \(\mathcal {H}\) with a discrete spectrum. Then Theorem 2 can also be extended for the following Cauchy problem:
$$\begin{aligned} \left\{ \begin{array}{l} \mathbb {D}_{(g)} u(t)+a(t)\mathcal {M} u(t)=f(t), \quad t \in (0, T], \\ u(0)=u_0 \in \mathcal {H}, \end{array}\right. \end{aligned}$$which is a generalization of [34].
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Acknowledgements
The first author was supported by Core Research Grant, RP03890G, Science and Engineering Research Board (SERB), DST, India. Shyam Swarup Mondal is supported by NBHM post-doctoral fellowship from the Department of Atomic Energy (DAE), Government of India at Indian Institute of Science, Bangalore. SSM also thanks Ghent Analysis & PDE Center of Ghent University for the financial support of his visit to Ghent University during which this work has been completed. M. Ruzhansky is supported by the EPSRC Grants EP/R003025 by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grantnumber 01M01021). The last author is supported by the institute assistantship from the Indian Institute of Technology Delhi, India.
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Dasgupta, A., Mondal, S.S., Ruzhansky, M. et al. Time-fractional discrete diffusion equation for Schrödinger operator. Fract Calc Appl Anal (2024). https://doi.org/10.1007/s13540-024-00323-y
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DOI: https://doi.org/10.1007/s13540-024-00323-y