Abstract
In \(L_2(\mathbb{R}^d)\), we consider an elliptic differential operator \(\mathcal{A}_\varepsilon \! = \! - \operatorname{div} g(\mathbf{x}/\varepsilon) \nabla + \varepsilon^{-2} V(\mathbf{x}/\varepsilon)\), \( \varepsilon > 0\), with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian \(\mathcal{A}_\varepsilon\), analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator \(\mathcal{A}_1\) are studied (the so called high-energy homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in \(L_2(\mathbb{R}^d)\)-norm for small \(\varepsilon\) are obtained.
DOI 10.1134/S1061920823040064
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Introduction
0.1. Periodic homogenization
The study of the wave propagation in periodic structures is of significant interest both for applications and from the theoretical point of view. Direct numerical simulations of such processes may be difficult. One of the approaches to study these problems is the application of homogenization theory. The aim of homogenization is to describe the macroscopic properties of inhomogeneous media by taking into account the properties of the microscopic structure. An extensive literature is devoted to homogenization problems. First of all, we mention the books [1–3].
Let us discuss a typical problem of homogenization theory. Let \(\Gamma\) be a lattice in \(\mathbb{R}^d\), and let \(\Omega\) be the cell of \(\Gamma\). For any \(\Gamma\)-periodic function \(F(\mathbf{x})\), we denote \(F^\varepsilon (\mathbf{x}) := F(\varepsilon^{-1} \mathbf{x})\), where \(\varepsilon > 0\) is a (small) parameter. In \(L_2(\mathbb{R}^d)\), consider a differential operator (DO) formally given by
where \(g(\mathbf{x})\) is a Hermitian \(\Gamma\)-periodic \((d \times d)\)-matrix-valued function, bounded and positive definite. The operator (0.1) models the simplest cases of microinhomogeneous media with \(\varepsilon \Gamma\)-periodic structure. Let \(u_\varepsilon(\mathbf{x})\) be a (weak) solution of the elliptic equation
where \(f \in L_2(\mathbb{R}^d)\). For \(\varepsilon \to 0\), the solution \(u_\varepsilon\) converges to the solution \(u_0\) of the “homogenized” equation:
The operator \(\widehat{\mathcal{A}}^{\mathrm{hom}}= - \operatorname{div} g^0 \nabla\) is called the effective operator for \(\widehat{\mathcal{A}}_\varepsilon\). The matrix \(g^0\) is determined by a well-known procedure (see, e.g., [1, Chap. 2, § 3], [4, Chap. 3, § 1]) that requires solving an auxiliary boundary value problem on the cell \(\Omega\). Besides finding the effective coefficients, the following questions are of great interest. What is the type of convergence \(u_\varepsilon \to u_0\)? What is an estimate for \(u_\varepsilon - u_0\)?
0.2. Operator error estimates in homogenization
M. Birman and T. Suslina (see [4]) suggested the operator-theoretic (spectral) approach to homogenization problems in \(\mathbb{R}^d\), based on the scaling transformation, the Floquet–Bloch theory, and the analytic perturbation theory.
Let \(u_\varepsilon\) be the solution of equation (0.2), and let \(u_0\) be the solution of equation (0.3). In [4], it was proved that
Since \(u_\varepsilon = (\widehat{\mathcal{A}}_\varepsilon + I)^{-1} f\) and \(u_0 = (\widehat{\mathcal{A}}^{\mathrm{hom}}+ I)^{-1} f\), estimate (0.4) can be rewritten in operator terms:
Parabolic equations were studied in [5, 6]. In operator terms, the following approximation for the parabolic semigroup \(e^{-\tau \widehat{\mathcal{A}}_\varepsilon}\), \(\tau > 0\), was obtained:
Estimates (0.5), (0.6) are order-sharp; the constants \(C\) are controlled explicitly in terms of the problem data. These estimates are called operator error estimates in homogenization. More accurate approximations for the resolvent and the exponential with correctors taken into account were found in [7–10].
A different approach to operator error estimates (the so called shift method) for the elliptic and parabolic problems was suggested by V. Zhikov and S. Pastukhova in the papers [11–13]. See also the survey [14].
The situation with homogenization of nonstationary Schrödinger-type equations and hyperbolic equations is quite different. The papers [15–20] were devoted to such problems. In operator terms, the behavior of the operator-functions \(e^{-i \tau \widehat{\mathcal{A}}_{\varepsilon}}\) and \(\cos(\tau \widehat{\mathcal{A}}_{\varepsilon}^{1/2})\), \(\widehat{\mathcal{A}}_{\varepsilon}^{-1/2} \sin(\tau \widehat{\mathcal{A}}_{\varepsilon}^{1/2})\) (where \(\tau \in \mathbb{R}\)) for small \(\varepsilon\) was studied. For these operator-functions, it is impossible to obtain approximations in the operator norm on \(L_2 (\mathbb{R}^d)\), and we are forced to consider the norm of operators acting from the Sobolev space \(H^q (\mathbb{R}^d)\) (with a suitable \(q\)) to \(L_2 (\mathbb{R}^d)\). In [15], the following sharp-order estimates were proved:
In [16], the result for the operator \(\widehat{\mathcal{A}}_{\varepsilon}^{-1/2} \sin(\tau \widehat{\mathcal{A}}_{\varepsilon}^{1/2})\) was obtained:
Moreover, in [16], an approximation of the operator \(\widehat{\mathcal{A}}_{\varepsilon}^{-1/2} \sin(\tau \widehat{\mathcal{A}}_{\varepsilon}^{1/2})\) for a fixed \(\tau\) in the \((H^2 \to H^1)\)-norm with error of order \(O (\varepsilon)\) (with a corrector taken into account) was obtained. Next, in [17–20], it was shown that these results are sharp with respect to the norm type as well as with respect to the dependence on \(\tau\) (for large \(\tau\)). On the other hand, it was shown that under some additional assumptions (e.g., if the matrix \(g(\mathbf{x})\) has real entries) estimates (0.7)–(0.9) can be improved:
More accurate approximations of the operators \(e^{-i \tau \widehat{\mathcal{A}}_{\varepsilon}}\), \(\cos(\tau \widehat{\mathcal{A}}_{\varepsilon}^{1/2})\), \(\widehat{\mathcal{A}}_{\varepsilon}^{-1/2} \sin(\tau \widehat{\mathcal{A}}_{\varepsilon}^{1/2})\) with correctors taken into account were found in [21, 22, 23].
Note that in [4–10, 15–23], a much broader class of operators than (0.1) (including matrix DOs) was studied. In particular, operators of the form
were considered. Here \(\check{g}(\mathbf{x})\) is a \(\Gamma\)-periodic positive definite and bounded \((d \times d)\)-matrix-valued function with real entries, \(V(\mathbf{x})\) is a \(\Gamma\)-periodic real-valued function, \(V \in L_p(\Omega)\) with a suitable \(p\) (and it is assumed that \(\inf \operatorname{spec} \mathcal{A}_1 = 0\)). For operator (0.12), it is impossible to find an operator \(\mathcal{A}^{\mathrm{hom}}\) with constant coefficients such that the corresponding operator-functions converge to the operator-functions of \(\mathcal{A}^{\mathrm{hom}}\). However, some approximations can be found if we “border” operator-functions of \(\widehat{\mathcal{A}}^{\mathrm{hom}}\) by appropriate rapidly oscillating factors. In particular, an analog of (0.5) is as follows:
where \(\omega (\mathbf{x})\) is a positive \(\Gamma\)-periodic solution of the equation
satisfying the normalization condition \(\| \omega \|_{L_2(\Omega)}^2 = | \Omega |\), and \(\widehat{\mathcal{A}}^{\mathrm{hom}}\) is the effective operator for operator (0.1) with the matrix \(g(\mathbf{x}) = \check{g} (\mathbf{x}) \omega^2 (\mathbf{x})\).
Let us explain the method using the example of operator (0.1). The scaling transformation reduces the investigation of the behavior of the operator \((\widehat{\mathcal{A}}_\varepsilon + I)^{-1}\), \(\varepsilon \to 0\), to studying the operator \((\widehat{\mathcal{A}} + \varepsilon^2 I)^{-1}\), where \(\widehat{\mathcal{A}} = \widehat{\mathcal{A}}_1 = - \operatorname{div} g(\mathbf{x}) \nabla\). Next, by the Floquet–Bloch theory, the operator \(\widehat{\mathcal{A}}\) expands in the direct integral of the operators \(\widehat{\mathcal{A}}(\mathbf{k})\) acting in the space \(L_2 (\Omega)\). The operator \(\widehat{\mathcal{A}}(\mathbf{k})\) is defined by the differential expression \(-\operatorname{div}_\mathbf{k} g(\mathbf{x}) \nabla_\mathbf{k}\), where \(\nabla_\mathbf{k} = \nabla + i \mathbf{k}\), \(\operatorname{div}_\mathbf{k} = \operatorname{div} + i \langle \mathbf{k}, \cdot \rangle\), with periodic boundary conditions. The spectrum of the operator \(\widehat{\mathcal{A}}(\mathbf{k})\) is discrete. It turns out that the behavior of the resolvent \((\widehat{\mathcal{A}} + \varepsilon^2 I)^{-1}\) can be described in terms of the threshold characteristics of \(\widehat{\mathcal{A}}\) at the edge of the spectrum, i.e., it is sufficient to know the spectral decomposition of \(\widehat{\mathcal{A}}\) only near the lower edge of the spectrum. In particular, the effective matrix \(g^0\) is a Hessian of the first band function \(E_1(\mathbf{k})\) at the point \(\mathbf{k} = 0\).
Finally, we mention the recent paper [24], where the authors investigated the problem of convergence rates for a solution of the initial-Dirichlet boundary value problem for a wave equation; analogs of estimates (0.10), (0.11) as well as results with the Dirichlet corrector were obtained.
0.3. High-frequency homogenization
As stated above, only a small neighborhood of the bottom of the spectrum (i.e., waves with low frequencies) contributes to homogenization. However, we can consider problems of wave propagation when the frequency is proportional to \(\varepsilon^{-1}\) or \(\varepsilon^{-2}\) (the high-frequency mode). In this case, even the leading order of the asymptotics oscillates rapidly. These problems were studied in [2, Chapter 4] using WKB-ansatz.
Traditional methods of homogenization theory, related to asymptotic expansions in two scales, were applied to these problems in [25, 26]. We also cite the paper [27], where application of the results of [25] to photonic crystals was considered. In [25], an asymptotic expansion for solutions of the equation
which are perturbations of the standing waves, was obtained (the functions \(g(\mathbf{x})\), \(\rho(\mathbf{x})\) were supposed to be sufficiently smooth and \(\Gamma\)-periodic). In [26], a similar problem for travelling waves was considered.
For a nonstationary Schrödinger equation, results of this kind are called effective mass theorems (see, e.g., the course [28] and references therein). In the paper [29], homogenization of the Cauchy problem for a nonstationary Schrödinger equation with well-prepared initial data concentrating on a Bloch eigenfunction was studied using techniques of two-scale convergence and suitable oscillating test functions; a rigorous derivation of effective mass theorems was obtained (in terms of the strong two-scale convergence). In [30], the effective mass approximation and the \(k\cdot p\) multi-band models, well known in solid-state physics, were discussed. Such homogenization asymptotics were investigated by using the envelope-function decomposition. These models were proved to be close (in the strong sense) to the exact dynamics. Moreover, the position density was proved to converge weakly to its effective mass approximation.
Finally, we also mention the papers [31, 32], where asymptotics of Green’s function for different values of the spectral parameter has been studied.
Now, let us discuss error estimates for high-frequency homogenization. This topic has been studied in [33–37] in the one-dimensional case (\(d=1\)) and in [38–40] in the case of arbitrary dimension \(d\). It is well-known that the spectrum of \(\mathcal{A}\) has a band structure and may have gaps. For the sake of simplicity, we consider the case where \(d=1\) and \(\Gamma = \mathbb{Z}\); in this case we shall use the notation \(A_\varepsilon\) for operator (0.12). Let \(\sigma > 0\) be a (non-degenerate) left edge of a band with an odd number (\(\ge 3\)) in the spectrum of the operator \(A = A_1\). Then for \(A_\varepsilon\), this edge ”moves” to the point \(\varepsilon^{-2} \sigma\) (to the high-frequency (high-energy) region). Instead of (0.2), we consider the equation
where \(f \in L_2(\mathbb{R})\). It is supposed that \(\varkappa > 0\) is such that the point \(\varepsilon^{-2} \sigma - \varkappa^2\) belongs to the gap in the spectrum of the operator \(A_\varepsilon\). Similarly to (0.5), the question is reduced to studying the operator \((A_\varepsilon - (\varepsilon^{-2} \sigma - \varkappa^2)I)^{-1}\). In [33], the following result was proved:
Here \(A_\sigma^{\mathrm{hom}} = -b_\sigma \frac{d^2}{dx^2}\) is the corresponding effective operator, \(b_\sigma > 0\) is the coefficient in the asymptotics of the band function \(E(k)\) corresponding to the band for which \(\sigma\) is the left edge: \(E(k) \sim \sigma + b_\sigma k^2\), \(k \sim 0\); and \(\varphi_\sigma\) is a real-valued periodic solution of the equation \(A \varphi_\sigma = \sigma \varphi_\sigma\), normalized in \(L_2(0,1)\). Consequently, the possibility of homogenization for equation (0.13) is a threshold effect near the edge of an internal gap.
Estimate (0.14) was obtained in [33] in the case where \(V(x) = 0\). In [38], an analog of estimate (0.14) was proved for operators (0.12) in arbitrary dimension \(d \ge 1\). More accurate approximations with correctors were obtained in [34, 35, 39].
Parabolic equations in the one-dimensional case were studied in [36]. It was proved that
and a more accurate approximation with a corrector was found. Here \(\mathcal{E}_{A_\varepsilon}[\varepsilon^{-2} \sigma, \infty)\) is the spectral projection of the operator \(A_\varepsilon\) corresponding to the interval \([\varepsilon^{-2} \sigma, \infty)\). The generalization of this result for the case of arbitrary dimension was obtained in [40].
In the paper [37], operator error estimates for high-frequency homogenization of nonstationary Schrödinger equations and hyperbolic equations in the one-dimensional case (\(d=1\)) were studied. Let \(f_1, f_2 \in L_2(\mathbb{R})\). Consider the Cauchy problems
where
Here \(\{e^{ikx} \varphi_j(x, k)\}_{j=s}^{\infty}\) are the Bloch waves corresponding to the bands with the numbers \(j \ge s\); \(\widetilde{\Omega}_j = (-j \pi, -(j-1)\pi] \cup ((j-1)\pi, j \pi ]\), \(j \in \mathbb{N}\), are the Brillouin zones. The initial data of problems (0.15) are superpositions of the Bloch waves with the amplitudes, which are equal to the Fourier images \((\Phi f_1) (k)\), \((\Phi f_2) (k)\) of the functions \(f_1(x)\), \(f_2(x)\), and belong to the subspace \(\mathcal{E}_{A_\varepsilon}[\varepsilon^{-2} \sigma, \infty) L_2(\mathbb{R})\). The following approximations were found:
Here \(u_0\) and \(v_0\) are the solutions of the effective problems
Note that estimates (0.16), (0.17) can be formulated in operator terms; see [37, (6.6), (6.21)–(6.23)].
0.4. Main results
In the present paper, we study error estimates for high-energy homogenization of nonstationary Schrödinger equations in the case of an arbitrary dimension. Let \((\mathbf{k}^\circ, \lambda_0)\) be an arbitrary point of the dispersion relation \(\mathfrak{B}_\mathcal{A}\) of the operator \(\mathcal{A} := \mathcal{A}_1\). In particular, it may be a ”regular” edge of a spectral band (see Remark 4.3 below) or a point where two branches of the dispersion relation meet (they often form the so-called Dirac cone, see [41, Sec. 5.10]). Let \(\{e^{i\left\langle \mathbf{k}^\circ, \mathbf{x} \right\rangle} \varsigma_j (\mathbf{k}^\circ, \mathbf{x}) \}_{j=1}^n\) be corresponding Bloch waves; we suppose that \(\bigl(\varsigma_j (\mathbf{k}^\circ, \cdot), \varsigma_k (\mathbf{k}^\circ, \cdot)\bigr)_{L_2(\Omega)} = \delta_{jk}\). We are interested in the behavior of the solutions \(u_{j,\varepsilon} (\mathbf{x}, \tau)\), \(\mathbf{x} \in \mathbb{R}^d\), \(\tau \in \mathbb{R}\), \(j = 1, \ldots,n\), of the following Cauchy problems for the nonstationary Schrödinger equation
as \(\varepsilon \to 0\), where \(\varsigma_j^\varepsilon(\mathbf{k}^\circ,\mathbf{x}) := \varsigma_j(\mathbf{k}^\circ,\mathbf{x}/\varepsilon)\), and \(f_j(\mathbf{x})\), \(j=1,\ldots,n\), are given functions. Main results of the paper are the following estimates:
where
and \(\mathbf{v}^{\mathrm{eff}}_{j, \varepsilon} (\mathbf{x},\tau) = (v^{\mathrm{eff}}_{j1, \varepsilon} (\mathbf{x},\tau), \ldots, v^{\mathrm{eff}}_{jn, \varepsilon} (\mathbf{x},\tau))^\mathrm{t}\) is the solution of the ”effective” system
Here \(\mathcal{A}^{\mathrm{eff}}_\varepsilon\) is an effective operator with constant coefficients (its definition is given below in (4.3), (4.4)), and \(\mathbf{e}_j\) is the element of the canonical basis in \(\mathbb{C}^n\).
0.5. Plan of the paper
The paper consists of Introduction and four more sections. In Section 1, a precise definition of the operator \(\mathcal{A}\) is given, its factorization is described. Next, in Section 2, we describe a spectral expansion of the operator \(\mathcal{A}\) (partial diagonalization via the Gelfand transformation). Then, in Section 3, spectral approximations for the operator \(\mathcal{A}\) in some neighbourhood of the point \((\mathbf{k}^\circ, \lambda_0) \in \mathfrak{B}_\mathcal{A}\) are obtained, and also the effective characteristics are calculated. Finally, in Section 4, we formulate and prove the main result of the paper.
0.6. Notation
Let \(\mathfrak{H}\) and \(\mathfrak{H}_{*}\) be complex separable Hilbert spaces. The symbols \((\cdot, \cdot)_{\mathfrak{H}}\) and \( \| \cdot \|_{\mathfrak{H}}\) denote the inner product and the norm in \(\mathfrak{H}\). The symbol \(\| \cdot \|_{\mathfrak{H} \to \mathfrak{H}_*}\) stands for the norm of a bounded linear operator from \(\mathfrak{H}\) to \(\mathfrak{H}_{*}\). Sometimes we omit the indices. By \(I = I_{\mathfrak{H}}\) we denote the identity operator in \(\mathfrak{H}\). If \(A \colon \mathfrak{H} \to \mathfrak{H}_*\) is a linear operator, then \( \operatorname{Dom} A\) and \( \operatorname{Ran} A\) stand for its domain and range, respectively. If \(\mathfrak{N}\) is a subspace in \(\mathfrak{H}\), then \(\mathfrak{N}^{\perp} := \mathfrak{H} \ominus \mathfrak{N}\). If \(P\) is the orthogonal projection of \(\mathfrak{H}\) onto \(\mathfrak{N}\), then \(P^{\perp}\) is the orthogonal projection of \(\mathfrak{H}\) onto \(\mathfrak{N}^{\perp}\). Next, if \(A\) is a selfadjoint operator in some Hilbert space, then we use the notation \( \operatorname{spec} A\) for the spectrum of \(A\).
The symbol \(\left< \cdot, \cdot \right>\) stands for the standard inner product in \(\mathbb{C}^n\); \(1\kern-3.5pt 1 = 1\kern-3.5pt 1_n\) is the identity \((n \times n)\)-matrix. For \(z \in \mathbb{C}\), by \(z^*\) we denote the complex conjugate number. If \(a\) is an \((m \times n)\)-matrix, then \(a^\mathrm{t}\) denotes the transpose matrix, and \(a^*\) stands for the adjoint \((n \times m)\)-matrix. By \(\{\mathbf{e}_j\}_{j=1}^n\) we denote the canonical basis in \(\mathbb{C}^n\).
The standard \(L_p\) classes of functions in a domain \(\mathcal{O} \subset \mathbb{R}^d\) are denoted by \(L_p (\mathcal{O})\), \(1 \le p \le \infty\); \(H^q (\mathcal{O})\) are the Sobolev classes of functions in a domain \(\mathcal{O} \subset \mathbb{R}^d\) of order \(q \in \mathbb{R}\) and integrability index \(2\). If \(f\) is a measurable function, then the operator of multiplication by the function \(f\) in the space \(L_2\) is denoted by the same symbol.
Next, \(\mathbf{x} = (x_1, \ldots , x_d) \in \mathbb{R}^d\), \(i D_j = \frac{\partial}{\partial x_j}\), \(j = 1,\ldots, d\), \(\mathbf{D} = -i \nabla = (D_1, \ldots, D_d)\).
By \(\Phi := \Phi_{\mathbf{x} \to \mathbf{k}}\) we denote the Fourier transform on \(\mathbb{R}^d\) defined on the Schwartz class by the formula
and extended by continuity up to the unitary mapping \(\Phi \colon L_2(\mathbb{R}^d) \to L_2(\mathbb{R}^d)\). For the ball of radius \(\varkappa\) centered at \(\mathbf{k}' \in \mathbb{R}^d\), we use the notation \(\mathrm{B}_{\varkappa}(\mathbf{k}')\).
1. The operator \(\mathcal{A}\)
Let \(\Gamma\) be a lattice in \(\mathbb{R}^d\) generated by a basis \(\mathbf{a}_1, \ldots , \mathbf{a}_d\):
and let \(\Omega\) be the elementary cell of the lattice \(\Gamma\):
The basis \(\mathbf{b}^1, \ldots , \mathbf{b}^d\) dual to \(\mathbf{a}_1, \ldots , \mathbf{a}_d\) is defined by the relations \(\left< \mathbf{b}^l, \mathbf{a}_j \right> = 2 \pi \delta^l_j\). This basis generates the lattice \(\widetilde \Gamma\), dual to the lattice \(\Gamma\). By \(\widetilde{\Omega}\) we denote the central Brillouin zone of the lattice \(\widetilde{\Gamma}\). This is the set with the interior
(a centrally symmetric convex polytope), which contains only one face from each pair of opposite ones. By \(\widetilde{H}^1(\Omega)\) we denote the subspace of functions in \(H^1(\Omega)\), whose \(\Gamma\)-periodic extension to \(\mathbb{R}^d\) belongs to \(H^1_{\mathrm{loc}}(\mathbb{R}^d)\).
In \(L_2(\mathbb{R}^d)\), \(d \ge 1\), we consider a selfadjoint \(\Gamma\)-periodic Schrödinger operator \(\mathcal{A}\) generated by the differential expression
with metric \(\check{g} (\mathbf{x})\) and potential \(V(\mathbf{x})\). It is supposed that
and \(V (\mathbf{x})\) is a real-valued function such that
The precise definition of the operator \(\mathcal{A}\) is given in terms of the semi-bounded closed quadratic form
Adding an appropriate constant to \(V\), we assume that \(\inf \operatorname{spec} \mathcal{A} = 0\). Under this assumption the operator \(\mathcal{A}\) admits a convenient factorization (see, e.g., [42], [4, Chap. 6, Sec. 1.1]). To describe this factorization, we consider the equation
(which is understood in the weak sense). There exists a (strictly) positive \(\Gamma\)-periodic solution \(\omega \in \widetilde{H}^1 (\Omega)\) of this equation defined up to a constant factor. This factor can be fixed so that
It turns out that \(\omega \in C^\kappa\) for some \(\kappa > 0\). Moreover, the function \(\omega\) is a multiplier in \(H^1 (\mathbb{R}^d)\) and in \(\widetilde{H}^1 (\Omega)\). The substitution \(u = \omega \psi\) transforms form (1.3) to the form
This yields the factorization
We take the representation (1.6) of the operator \(\mathcal{A}\) as the initial definition, i.e., we assume that \(\mathcal{A}\) is the operator generated by form (1.5), where \(\check{g}\) and \(\omega\) are \(\Gamma\)-periodic functions satisfying (1.2), (1.4) and the conditions \(\omega(\mathbf{x}) > 0\); \(\omega, \omega^{-1} \in L_\infty\). We can return to representation (1.1) putting \(V = - \omega^{-1} (\mathbf{D}^* \check{g} \mathbf{D} \omega)\). However, the potential \(V\) may be highly singular.
2. Spectral decomposition of the operator \(\mathcal{A}\)
We need to describe the spectrum of the operator (1.6). For this, let us introduce the objects associated with the spectral resolution of operator (1.6). Put
In \(L_2(\Omega)\), consider the family of quadratic forms
The operator generated by form (2.1) is denoted by \(\mathcal{A} (\mathbf{k})\). Formally, we can write
The parameter \(\mathbf{k} \in \mathbb{R}^d\) is called the quasimomentum. Let \(E_l (\mathbf{k})\), \(l \in \mathbb{N}\), be consecutive (counted with multiplicities) eigenvalues of the operator \(\mathcal{A} (\mathbf{k})\), and let \(\varphi_l (\cdot, \mathbf{k})\), \(l \in \mathbb{N}\), be the corresponding normalized eigenfunctions:
The functions \(E_l (\mathbf{k})\) are called band functions; they are \(\widetilde{\Gamma}\)-periodic. Next, \(\varphi_l (\mathbf{x}, \mathbf{k})\) are \(\Gamma\)-periodic in \(\mathbf{x}\), and the functions \(e^{i\left\langle \mathbf{k},\mathbf{x} \right\rangle } \varphi_l (\mathbf{x}, \mathbf{k})\) can be chosen to be \(\widetilde{\Gamma}\)-periodic in \(\mathbf{k}\).
Remark 2.1.
Multiplying (2.2) by \(\omega(\mathbf{x})\) from the left and putting \(\phi_l(\mathbf{x}, \mathbf{k}) := \omega(\mathbf{x})^{-1} \varphi_l (\mathbf{x}, \mathbf{k})\), we arrive at the following equation for \(\phi_l(\mathbf{x}, \mathbf{k})\):
Separating the real and imaginary parts in (2.3), we obtain a system of two equations with real-valued coefficients and identical principal parts. In [43, Chap. VII, § 3, Theorem 3.1], it was proved that solutions of such systems with Dirichlet conditions belong to the Hölder class as functions of \(\mathbf{x}\). However, the proof carries over to the case of periodic boundary conditions without significant changes. This together with \(\omega \in L_\infty\) yields \(\varphi_l \in L_\infty\), \(l \in \mathbb{N}\). See also [44, § 4, Sec. 9] and [38, § 1, Sec. 1].
Initially, the Gelfand transformation \(\mathscr{G}\) is defined on functions of the Schwartz class \(v \in \mathcal{S}(\mathbb{R}^d)\) by the formula
The function \(\tilde{v} (\mathbf{x}, \mathbf{k})\) is \(\Gamma\)-periodic in \(\mathbf{x}\) and \(\widetilde{\Gamma}\)-quasiperiodic in \(\mathbf{k}\) (i.e. the function \(e^{i\left\langle \mathbf{x}, \mathbf{k} \right\rangle } \tilde{v} (\mathbf{x}, \mathbf{k})\) is \(\widetilde{\Gamma}\)-periodic in \(\mathbf{k}\)). So, it suffices to consider \(\tilde{v} (\mathbf{x}, \mathbf{k})\) for \(\mathbf{x} \in \Omega\) and \(\mathbf{k} \in \mathbb{T}^d\), where \(\mathbb{T}^d\) is the torus \(\mathbb{R}^d/\widetilde{\Gamma}\) with the induced \(\mathbb{R}^d\)-metric. Points of the torus \(\mathbf{k} \in \mathbb{T}^d\) can be realized, for example, as points in \(\widetilde{\Omega}\). The inverse transform is given by
Since \(\int_{\mathbb{T}^d} \int_{\Omega} | \tilde{v} (\mathbf{x}, \mathbf{k}) |^2 d \mathbf{x} \, d \mathbf{k} = \int_{\mathbb{R}^d} | v ( \mathbf{x} ) |^2 d \mathbf{x}\), the transformation \(\mathscr{G}\) extends by continuity up to a unitary mapping:
The relation \(v \in H^1 (\mathbb{R}^d)\) is equivalent to the fact that \(\tilde{v} (\cdot, \mathbf{k}) \in \widetilde H^1 (\Omega)\) for a.e. \(\mathbf{k} \in \mathbb{T}^d\) and
Under the Gelfand transformation \(\mathscr{G}\), the operator of multiplication by a bounded \(\Gamma\)-periodic function in \(L_2 (\mathbb{R}^d)\) turns into multiplication by the same function on the fibers of the direct integral \(\mathcal{K}\). The operator \(\mathbf{D}\) applied to \(v \in H^1 (\mathbb{R}^d)\) turns into the operator \(\mathbf{D} + \mathbf{k}\) applied to \(\tilde{v} (\cdot, \mathbf{k}) \in \widetilde H^1 (\Omega)\).
Under the Gelfand transformation \(\mathscr{G}\) the operator \(\mathcal{A}\) expands in the direct integral of the operators \(\mathcal{A} (\mathbf{k})\):
This means the following. If \(v \in \mathcal{H}^1 (\mathbb{R}^d)\), then
Conversely, if \(\tilde{v} \in \mathcal{K}\) satisfies (2.6) and the integral in (2.7) is finite, then \(v \in \mathcal{H}^1 (\mathbb{R}^d)\) and (2.7) is valid. From (2.5) it follows that the spectrum of \(\mathcal{A}\) is the union of segments (spectral bands) \( \operatorname{Ran} E_j\), \(j \in \mathbb{N}\).
Introduce the operator \(P_0\) acting as averaging over the cell \(\Omega\):
The operator \(P_0\) is the orthogonal projection of \(L_2(\Omega)\) onto the subspace of constants
The following relation is valid (see, e.g., [7, § 6, Sec. 6.1]):
Here \([P_0]\) is the projection in \(\mathcal{K}\) that acts on fibers as the operator \(P_0\). Conversely, if \(\operatorname{supp} c \subset \mathrm{B}_{\varkappa}(\mathbf{k}')\) with some \(\mathbf{k}' \in \mathbb{R}^d\) and sufficiently small \(\varkappa\), and \(c(\mathbf{k}) \in \mathfrak{N}_0\), \(\mathbf{k} \in \mathrm{B}_{\varkappa}(\mathbf{k}')\), then from (2.4) and the relation \(|\Omega||\widetilde{\Omega}| = (2 \pi)^d\) it follows that
In (2.9), the points \(\mathbf{k} \in \mathbb{T}^d\) are realized as points from a set \(\widetilde{\Omega}_{\mathbf{k}'}\) such that \(\mathrm{B}_{\varkappa}(\mathbf{k}') \subset \widetilde{\Omega}_{\mathbf{k}'}\).
Let us fix some point \(\mathbf{k}^\circ \in \mathbb{T}^d\) and a number \(s \in \mathbb{N}\). Put \(\lambda_0 := E_s(\mathbf{k}^\circ)\). Let \(n\) be the multiplicity of the eigenvalue \(\lambda_0\) of the operator \(\mathcal{A}(\mathbf{k}^\circ)\), and let \(d_0\) be the distance from the point \(\lambda_0\) to the rest of the spectrum of \(\mathcal{A}(\mathbf{k}^\circ)\). By the continuity of the band functions, we can choose \(\varkappa > 0\) such that for \(|\delta \mathbf{k}| \le \varkappa\), \(\delta \mathbf{k} := \mathbf{k} - \mathbf{k}^\circ\), there are exactly \(n\) eigenvalues (counted with multiplicities) of the operator \(\mathcal{A}(\mathbf{k})\) on the segment \([\lambda_0 - d_0/3, \lambda_0 + d_0/3]\), and
Introduce the notation \(\mathfrak{N} := \operatorname{Ker} (\mathcal{A}(\mathbf{k}^\circ) - \lambda_0 I)\). Let \(P\) be the orthogonal projection of \(L_2(\Omega)\) onto \(\mathfrak{N}\); by \(F(\mathbf{k})\) we denote the spectral projection of the operator \(\mathcal{A}(\mathbf{k})\) corresponding to the segment \([\lambda_0 - d_0/3, \lambda_0 + d_0/3]\).
3. Spectral approximations
3.1. Approximations for \(F(\mathbf{k})\) and \(\mathcal{A}(\mathbf{k}) F(\mathbf{k})\)
In this section, we want to find approximations for the operators \(F(\mathbf{k})\) and \(\mathcal{A}(\mathbf{k}) F(\mathbf{k})\) for \(|\delta \mathbf{k}| \le \varkappa\). For this, we shall integrate the difference of the resolvents for \(\mathcal{A}(\mathbf{k})\) and \(\mathcal{A}(\mathbf{k}^\circ)\) along an appropriate contour (see, e.g., [4, Chap. 1, Sec. 1.7, § 2, 3], [45, § 4, Sec. 4.2, the third method]). Here we apply the method of [45]. However, there is the complication that the (standard) second resolvent identity is not applicable, because, in general, the difference \(\mathcal{A}(\mathbf{k}) - \mathcal{A}(\mathbf{k}^\circ)\) makes no sense. In order to overcome this difficulty, we use the following lemma. (Here and throughout this section we drop the indices in the inner product and the norm in \(L_2(\Omega)\).)
Lemma 3.1.
We have
Proof.
Consider the form \((\mathfrak{a}(\mathbf{k})- \mathfrak{a}(\mathbf{k}^\circ))[u, v]\) on the elements \(u = (\mathcal{A}(\mathbf{k})- \zeta I)^{-1} \eta\) and \(v = (\mathcal{A}(\mathbf{k}^\circ) - \zeta^* I)^{-1} \vartheta\), where \(\eta, \vartheta \in L_2(\Omega)\), \(\zeta \in \rho(\mathcal{A}(\mathbf{k})) \cap \rho(\mathcal{A}(\mathbf{k}^\circ))\). Obviously, \((\mathcal{A}(\mathbf{k})- \zeta I)^{-1} \eta \in \operatorname{Dom} \mathcal{A}(\mathbf{k})\) and
whence
The last two terms cancel out, which yields (3.1).
Denote
Let \(\gamma\) be a contour on the complex plane that is equidistant to the interval \([\lambda_0 - d_0/3, \lambda_0 + d_0/3]\) and passes through the point \(\lambda_0 + d_0/2\). Its length is equal to
The resolvent on this contour satisfies the estimates
Passing from forms to operators, we rewrite identity (3.1) as
where
Let us estimate the norms of the operators \(\mathcal{X}(\mathbf{k}, \zeta)\), \(\mathcal{X}_0(\zeta)\), \(\mathcal{Y}(\delta \mathbf{k}, \mathbf{k}, \zeta)\) and \(\mathcal{Y}_0(\delta \mathbf{k}, \zeta)\) for \(|\delta \mathbf{k}| \le \varkappa\), \(\zeta \in \gamma\). Clearly,
Next, using the identity \(\mathcal{A}(\mathbf{k}) R(\mathbf{k}, \zeta) = I + \zeta R(\mathbf{k}, \zeta)\) and taking (3.2) into account, we get
whence
Now, iterating, we apply identity (3.3) for the resolvent \(R(\mathbf{k}, \zeta)\), contained in the terms \(\mathcal{Y}_0(\delta \mathbf{k}, \zeta^*)^* \mathcal{X}(\mathbf{k}, \zeta)\) and \(\mathcal{X}_0(\zeta^*)^* \mathcal{Y}(\delta \mathbf{k}, \mathbf{k}, \zeta)\) in (3.3). Thus, the terms of order \(|\delta \mathbf{k}|\) will not contain \(R(\mathbf{k}, \zeta)\):
Here \(\mathscr{R}_1(\delta \mathbf{k}, \mathbf{k}, \zeta)\) is defined by the expression
We have denoted
Let us estimate the norms of the operators \(\check{\mathcal{X}}_0 (\zeta)\) and \(\check{\mathcal{Y}}(\delta \mathbf{k})\). Write \(\check{\mathcal{X}}_0 (\zeta)\) as
We have
By (3.6) and the identity \(\| g^{1/2} (\mathbf{D}+\mathbf{k}^\circ) \omega^{-1} \mathcal{A}(\mathbf{k}^\circ)^{-1/2} \| = 1\),
Next, obviously,
In order to get rid of the resolvent \(R(\mathbf{k}, \zeta)\) in the terms of order \(|\delta \mathbf{k}|^2\), we apply identity (3.3) once again:
where
The operators \(\mathscr{R}_1(\delta \mathbf{k}, \mathbf{k}, \zeta)\) and \(\mathscr{R}_2(\delta \mathbf{k}, \mathbf{k}, \zeta)\) satisfy the estimates
Proposition 3.2.
The operator-valued functions \(\mathcal{X}(\mathbf{k}, z)\), \(\mathcal{Y}(\delta \mathbf{k}, \mathbf{k}, z)\) and \(\mathcal{X}_0(z)\), \(\mathcal{Y}_0(\delta \mathbf{k}, z)\), \(\check{\mathcal{X}}_0(z)\) are holomorphic in \(z\) on the domains \(\mathbb{C} \setminus \operatorname{spec} \mathcal{A}(\mathbf{k})\) and \(\mathbb{C} \setminus \operatorname{spec} \mathcal{A}(\mathbf{k}^\circ)\), respectively.
Proof.
The holomorphy of \(\mathcal{Y}(\delta \mathbf{k}, \mathbf{k}, z)\) and \(\mathcal{Y}_0(\delta \mathbf{k}, z)\) directly follows from the holomorphy of the resolvents.
Next, consider \(\mathcal{X}_0(z)\) and \(\mathcal{X}(\mathbf{k}, z)\). We have
The operator \(g^{1/2} (\mathbf{D} + \mathbf{k}^\circ) \omega^{-1} (\mathcal{A}(\mathbf{k}^\circ) + I)^{-1}\) is bounded, since
(here we have used the spectral theorem and the inequality \(t^{1/2} / (t+1) \le 1/2\), \(t \ge 0\)), and the operator-valued function \((\mathcal{A}(\mathbf{k}^\circ) + I) R_0(z)\) is holomorphic on \(\mathbb{C} \setminus \operatorname{spec} \mathcal{A}(\mathbf{k}^\circ)\), because \(\mathcal{A}(\mathbf{k}^\circ) R_0(z) = I + z R_0(z)\). Therefore, \(\mathcal{X}_0(z)\) is holomorphic. Then,
Similarly, it is easy to check that the first term is holomorphic on \(\mathbb{C} \setminus \operatorname{spec} \mathcal{A}(\mathbf{k})\). Thus, \(\mathcal{X}(\mathbf{k}, z)\) is holomorphic.
It remains to show the holomorphy of \(\check{\mathcal{X}}_0(z)\). Let us write
It is easy to check, that the operator \(g^{1/2} (\mathbf{D} + \mathbf{k}^\circ) \omega^{-1} (\mathcal{A}(\mathbf{k}^\circ) + I)^{-1/2}\) is bounded:
and the operator-valued function \((\mathcal{A}(\mathbf{k}^\circ) + I) R_0(z)\) is holomorphic on \(\mathbb{C} \setminus \operatorname{spec} \mathcal{A}(\mathbf{k}^\circ)\). This completes the proof.
In this section, our goal is to find approximations for \(F(\mathbf{k})\) and \(\mathcal{A}(\mathbf{k}) F(\mathbf{k})\). Let us start with the operator \(F(\mathbf{k})\). By virtue of Riesz–Dunford operator calculus,
Substituting (3.3) into (3.15) and using relations (3.5), (3.7), (3.8), and the identity
we obtain the following result:
We also need more precise approximation for the projector \(F(\mathbf{k})\). For this, we substitute (3.9) into (3.15):
Calculate the integral in the expression for \(F_1(\delta \mathbf{k})\). Recall notation (3.4), take into account the decomposition of the resolvent
the holomorphy of the operator-valued function \(R^\perp_0(\zeta) := R_0(\zeta)P^\perp\) inside the contour \(\gamma\), the equality \( \oint _\gamma (\lambda_0 - \zeta)^{-2} d \zeta = 0\), and use the fact that integral over \(\gamma\) of a holomorphic function inside the contour is equal to zero. Therefore,
where the operator \(F^\times_1(\delta \mathbf{k})\) takes \(\mathfrak{N}^\perp\) into \(\mathfrak{N}\) and is defined by the expression
By (3.14), the remainder \(\Phi(\delta \mathbf{k}, \mathbf{k})\) satisfies the estimate
Using integral representation (3.18) and (3.5), (3.8), and the relation \(F^\times_1(\delta \mathbf{k}) = P F_1(\delta \mathbf{k}) P^\perp\), we estimate the operator \(F^\times_1(\delta \mathbf{k})\) as follows:
We also note that
Moreover, we need to consider the operator \(F^\times_1(\delta \mathbf{k}) F(\mathbf{k})\). Applying (3.17), (3.20), (3.22), (3.23), the first and the third equalities (3.24), we obtain that
The operator \(F^\times_1(\delta \mathbf{k}) F^\times_1(\delta \mathbf{k})^*\) has the form
Let us now turn to the approximation for the operator \(\mathcal{A}(\mathbf{k}) F(\mathbf{k})\). We have
By substituting (3.13) into (3.28), one obtains
where
Recall the definitions of operators (3.4), (3.10), decomposition of the resolvent (3.19), and consider first the representation for \(G_1(\delta \mathbf{k})\):
Similarly to (3.21), calculating the integral with the help of the formula for the derivative of the Cauchy integral \( f'(z) = \frac{1}{2\pi i} \oint _\gamma f(\zeta) (\zeta - z)^{-2} d \zeta\) (where \(f\) is a holomorphic function in the domain restricted by the contour \(\gamma\)), we have
Recall that \(F^\times_1(\delta \mathbf{k})\) was calculated in (3.21). By virtue of (3.29),
This together with (3.16), (3.21), the second equality (3.24), (3.25), (3.26), and (3.33) yields
if the condition in (3.35) is satisfied. Thus,
Let us now turn to expression (3.30) for \(G_2(\delta \mathbf{k})\). We write (3.31) as
where
Substituting (3.37) into (3.30), we obtain
The operator \(G^\circ_{2} (\delta \mathbf{k})\) acts on \(\mathfrak{N}\); using the elementary equality \(\frac{\zeta}{(\zeta - \lambda_0)^2} = \frac{1}{\zeta - \lambda_0} + \frac{\lambda_0}{(\zeta - \lambda_0)^2}\) and the formula for the derivative of Cauchy integral, we get
Here \((R_0^\perp)'(\lambda_0) := \left. \frac{d}{d \lambda} R_0^\perp(\lambda) \right|_{\lambda = \lambda_0}\). From the first resolvent identity it directly follows that \((R_0^\perp)'(\lambda_0) = R_0^\perp(\lambda_0)^2\). Next, we have
and
to obtain (3.41), we have used (3.27). Moreover, we need to estimate the operator \(G^\times_{2}(\delta \mathbf{k}) F(\mathbf{k})\). According to (3.5), (3.8), (3.11), (3.12), and (3.31), the operator \(T(\delta \mathbf{k}, \zeta)\) satisfies the estimate
whence, by (3.39),
Using (3.16), the first relation (3.40), and (3.42), we arrive at
At the end of this section, we give an estimate for the remainder \(\Xi(\delta \mathbf{k}, \mathbf{k})\). From (3.14), (3.32) it follows that
3.2. Approximations for the operator exponential
We put
Here the operators \(\mathfrak{G}^\circ_1 (\delta \mathbf{k})\) and \(\mathfrak{G}^\circ_2 (\delta \mathbf{k})\) were defined in (3.34) and (3.41). In this section, we want to approximate the operator \(e^{-i \tau \mathcal{A}(\mathbf{k})}P\), \(\tau \in \mathbb{R}\), by \(e^{-i \tau \mathfrak{G}^\circ(\delta \mathbf{k}) P}P\). Consider the difference
By (3.16), the last two terms admit the estimates
Next (cf. [15, the proof of Theorem 2.1]),
where \(\Sigma(\mathbf{k}, \tau) := e^{i \tau \mathfrak{G}^\circ(\delta \mathbf{k})P} F (\mathbf{k}) e^{-i \tau \mathcal{A}(\mathbf{k})} - P\). We have
Obviously, \(\Sigma(\mathbf{k}, 0) = F (\mathbf{k}) - P\), and, by (3.16), \(\| P e^{-i \tau \mathfrak{G}^\circ(\delta \mathbf{k})P} \Sigma(\mathbf{k}, 0)\| \le C_7 |\delta \mathbf{k}|\), \(|\delta \mathbf{k}| \le \varkappa\). Next,
Consider the operator \(P \bigl(\mathcal{A}(\mathbf{k}) F (\mathbf{k}) - \mathfrak{G}^\circ(\delta \mathbf{k}) P \bigr) F(\mathbf{k})\). Using the second identity in (3.24), (3.25), (3.29), (3.33), (3.38), the second and the third identities in (3.40), (3.41), and (3.45), we have
whence, by (3.26), (3.43), (3.44),
Recalling the expressions for the constants, from what has been said above we deduce the following result.
Theorem 3.3.
Let \(\tau \in \mathbb{R}\) and \(|\delta \mathbf{k}| \le \varkappa\). We have
where the constants \(C_7\) and \(C_{11}\) are defined by
3.3. Calculation of the operator \(\mathfrak{G}^\circ(\delta \mathbf{k})\) in a basis of \(\mathfrak{N}\)
Let \(\{\varsigma_p\}_{p=1}^n\) be an orthonormal basis in \(\mathfrak{N}\). In this section, our aim is to calculate the matrix elements of operator (3.45) in this basis.
First of all, obviously, \((\lambda_0 P \varsigma_p, \varsigma_l) = \lambda_0 \delta_{lp}\). Let us proceed to calculation of \((\mathfrak{G}^\circ_1 (\delta \mathbf{k}) \varsigma_p, \varsigma_l)\). We have
whence
Next, since the operator \(\bigl((\mathbf{D} + \mathbf{k}^\circ) \omega^{-1} P \bigr)^* g (\delta \mathbf{k}) \omega^{-1} P\) is adjoint to \(P \omega^{-1} (\delta \mathbf{k})^* g (\mathbf{D} + \mathbf{k}^\circ) \omega^{-1} P\), it is seen that
Thus, from (3.34), (3.46), (3.47) it follows that
where \(\mathfrak{g}^{1,lp} = (\mathfrak{g}^{1,lp}_{1}, \ldots, \mathfrak{g}^{1,lp}_{d})^\mathrm{t}\) is the column-vector with the entries
Let us proceed to calculation of the matrix elements of the operator \(\mathfrak{G}^\circ_2 (\delta \mathbf{k})\), defined by (3.41). It is convenient to write this operator as
Denote by \(\Lambda^p (\delta \mathbf{k})\) the result of the action of the operator
on the basis element \(\varsigma_p\). Obviously, \(\Lambda^p (\delta \mathbf{k}) \in \widetilde{\mathcal{H}}^1(\Omega)\) is the (weak) solution of the equation
The right-hand side of this equation is linear in \(\delta \mathbf{k}\) and has the following form:
Therefore, \(\Lambda^p (\delta \mathbf{k})\) can be written as \(\Lambda^p (\delta \mathbf{k}) = -i \sum_{r=1}^{d} (\delta k_r) \Lambda^p_r\), where \(\Lambda^p_r \in \widetilde{\mathcal{H}}^1(\Omega)\) is the solution of the equation
Next, let us calculate the inner product
Similarly to (3.46), (3.47), one obtains
Finally,
Thereby, we have obtained a formula for the matrix elements of the operator \(\mathfrak{G}^\circ_2 (\delta \mathbf{k})\):
where \(\mathfrak{g}^{2,lp}\) is \((d \times d)\)-matrix with the entries
As a result, the operator (3.45) is represented in the basis \(\{\varsigma_p\}_{p=1}^n\) by the matrix
At the end of this section, consider the action of the exponential \(e^{-i \tau \mathfrak{G}^\circ(\delta \mathbf{k})}\) on the element \(\varsigma_j\). It is easy to check that
where \( \{c_{j1}(\tau), \ldots, c_{jn}(\tau)\}^\mathrm{t} =: \mathbf{c}_j(\tau) = e^{-i \tau \mathfrak{g}(\delta \mathbf{k})} \mathbf{e}_j\) is the solution of the system
4. Main results of the paper
In this section, we formulate the main results of the paper. Let \(\varepsilon > 0\) be a small parameter. If \(F(\mathbf{x})\) is a \(\Gamma\)-periodic function, then we put \(F^\varepsilon (\mathbf{x}) := F(\varepsilon^{-1} \mathbf{x})\). In \(L_2(\mathbb{R}^d)\), we consider the operator formally defined by the differential expression
Here \(\check{g}\) and \(\omega\) are \(\Gamma\)-periodic functions satisfying conditions (1.2), (1.4), and \(\omega(\mathbf{x}) > 0\); \(\omega, \omega^{-1} \in L_\infty\). The precise definition of the operator \(\mathcal{A}_\varepsilon\) is given in terms of the corresponding quadratic form (cf. (1.5)). Operators (1.6) and (4.1) satisfy the following relation:
where \(T_\varepsilon\) is the operator of scaling transformation: \((T_\varepsilon u)(\mathbf{x}) = \varepsilon^{d/2} u(\varepsilon \mathbf{x})\).
Let \(\{\varsigma_j\}_{j=1}^n\) be an (arbitrary) orthonormal basis in \(\mathfrak{N}\) and let \(f_j \in L_2(\mathbb{R}^d)\), \(j = 1, \ldots,n\). We suppose that the functions \(\varsigma_j (\mathbf{x})\), \(j=1,\ldots,n\), are \(\Gamma\)-periodically extended to \(\mathbb{R}^d\). We study the behavior of the solutions \(u_{j,\varepsilon} (\mathbf{x}, \tau)\), \(\mathbf{x} \in \mathbb{R}^d\), \(\tau \in \mathbb{R}\), \(j = 1, \ldots,n\), \(\varepsilon \to 0\), of the following Cauchy problem for the nonstationary Schrödinger equation
In \(L_2(\mathbb{R}^d;\mathbb{C}^n)\), we consider the operator
which is called the effective operator. Let \(\mathbf{v}^{\mathrm{eff}}_{j, \varepsilon} (\mathbf{x},\tau)\) be the solution of the corresponding “homogenized” problem
Here the operator \(J_j \colon \mathbb{C} \to \mathbb{C}^n\) is defined by the rule \(a \mapsto a \mathbf{e}_j\). Put
The solutions of problems (4.2), (4.5) and \(u^{\mathrm{eff}}_{j,\varepsilon}\) can be represented as follows:
where \(\tilde{J}_l \colon \mathbb{C}^n \to \mathbb{C}\) is defined by the formula \(\tilde{J}_l \mathbf{c} = \left\langle \mathbf{c}, \mathbf{e}_l\right\rangle\).
Theorem 4.1.
Let \(u_\varepsilon\) be the solution of problem (4.2), and let \(u^{\mathrm{eff}}_{j,\varepsilon}\) be defined by (4.6). Let \(\varepsilon > 0\), \(\tau \in \mathbb{R}\), \(f_j \in H^3(\mathbb{R}^d)\). We have
with the constant \(\mathcal{C}\), which depends only on \(\lambda_0\), \(\varkappa\), \(d_0\), \(\|g\|_{L_\infty}\), \(\| \omega^{-1} \|_{L_\infty}\), and \(\| \varsigma_l\|_{L_\infty}\), \(l=1,\ldots, n\).
Proof.
By (4.7), estimate (4.8) can be reformulated in the operator terms:
where \(\tau \in \mathbb{R}\), \(\varepsilon > 0\). Thus, our aim is to prove (4.9). Since the operator \((-\Delta + I)^{3/2}\) is an isometric isomorphism of the Sobolev space \(H^3(\mathbb{R}^d)\) onto \(L_2(\mathbb{R}^d)\), we have
From the unitarity of the scaling transformation it directly follows that
where
Next, we need the following operator identities:
Introduce the projection \(F_\varkappa := \Phi^* \chi_{\mathrm{B}_\varkappa(\mathbf{0})}(\mathbf{k}) \Phi\). Here \(\chi_{\mathrm{B}_\varkappa(\mathbf{0})}(\mathbf{k})\) is the characteristic function of the ball \(\mathrm{B}_\varkappa(\mathbf{0})\). Obviously, \(\varepsilon^3 (|\delta \mathbf{k}|^2 + \varepsilon^2)^{-3/2} (1 - \chi_{\mathrm{B}_\varkappa(\mathbf{0})}(\delta \mathbf{k})) \le \varkappa^{-1} \varepsilon\). Applying (4.10) and taking into account Remark 2.1, we have
Consider the operator
which, by virtue of (4.10), (4.11), can be written as
Recall that the operator \(\mathcal{A}\) is decomposed into direct integral (2.5). Using (2.8) and (2.9) (with \(\mathbf{k}' = \mathbf{k}^\circ\)), one obtains
From the inclusion \( \operatorname{Ran} \varsigma_j P_0 \subset \mathfrak{N}\) and (3.49) it is seen that
Application of Theorem 3.3 (with \(\tau\) replaced by \(\tau \varepsilon^{-2}\)) together with the equality \(\| \varsigma_j \|_{L_2(\Omega)}=1\) gives the estimate with the constants that do not depend on \(\mathbf{k}\):
This completes the proof.
Remark 4.2.
Interpolating between (4.9) and the estimate
we obtain
When applied to Cauchy problem (4.2), this leads to the estimate
for \(\varepsilon > 0\), \(\tau \in \mathbb{R}\), \(f_j \in H^q(\mathbb{R}^d)\), and \(0 \le q \le 3\).
Remark 4.3.
Suppose that \(\lambda_0\) is a spectral edge of the operator \(\mathcal{A}\) and the corresponding extremal value is attained by one band function: \(\lambda_0 = E_s(\mathbf{k}^\circ)\), \(\lambda_0 \ne E_l(\mathbf{k}^\circ)\), \(l \ne s\). In this case, \(n = 1\), and, by (3.36), (3.48), \(\mathfrak{g}^{1,11} = 0\). Let \(u_\varepsilon (\mathbf{x},\tau) := u_{1,\varepsilon} (\mathbf{x},\tau)\) be the solution of problem (4.2), and let \(\tilde{v}^{\mathrm{eff}} (\mathbf{x},\tau)\) be the solution of the problem
where \(\widetilde{\mathcal{A}}^{\mathrm{eff}} = - \operatorname{div} \mathfrak{g}^{2,11} \nabla\). Let \(\varepsilon > 0\), \(\tau \in \mathbb{R}\), \(f_1 \in H^3(\mathbb{R}^d)\). We have
References
N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging Processes in Periodic Media. Mathematical Problems in Mechanics of Composite Materials, Kluwer Acad. Publ. Group, Dordrecht, 1989.
A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Co., Amsterdam – New York, 1978.
V. V. Zhikov, S. M. Kozlov, and O. A. Olejnik, Homogenization of Differential Operators, Springer-Verlag, Berlin, 1994.
M. Sh. Birman and T. A. Suslina, “Second Order Periodic Differential Operators. Threshold Properties and Homogenization”, Algebra i Analiz, 15:5 (2003), 1–108; St. Petersburg Math. J., 15:5 (2004), 639–714.
T. A. Suslina, “On Homogenization of Periodic Parabolic Systems”, Funktsional. Analiz i ego Prilozhen., 38:4 (2004), 86–90; Funct. Anal. Appl., 38:4 (2004), 309–312.
T. A. Suslina, “Homogenization of a Periodic Parabolic Cauchy Problem”, Amer. Math. Soc. Transl. (2), 220 (2007), 201–233.
M. Sh. Birman and T. A. Suslina, “Homogenization with Corrector Term for Periodic Elliptic Differential Operators”, Algebra i Analiz, 17:6 (2005), 1–104; St. Petersburg Math. J., 17:6 (2006), 897–973.
M. Sh. Birman and T. A. Suslina, “Homogenization with Corrector for Periodic Differential Operators. Approximation of Solutions in the Sobolev Class \(H^1(\mathbb{R}^d)\)”, Algebra i Analiz, 18:6 (2006), 1–130; St. Petersburg Math. J., 18:6 (2007), 857–955.
E. S. Vasilevskaya, “A Periodic Parabolic Cauchy Problem: Homogenization with Corrector”, Algebra i Analiz, 21:1 (2009), 3–60; St. Petersburg Math. J., 21:1 (2010), 1–41.
T. A. Suslina, “Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space \(H^1({\mathbb R}^d)\)”, Math. Model. Nat. Phenom., 5:4 (2010), 390–447.
V. V. Zhikov, “On Some Estimates of Homogenization Theory”, Dokl. Ros. Akad. Nauk, 406:5 (2006), 597–601; Dokl. Math., 73 (2006), 96–99.
V. V. Zhikov and S. E. Pastukhova, “On Operator Estimates for Some Problems in Homogenization Theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524.
V. V. Zhikov and S. E. Pastukhova, “Estimates of Homogenization for a Parabolic Equation with Periodic Coefficients”, Russ. J. Math. Phys., 13:2 (2006), 224–237.
V. V. Zhikov and S. E. Pastukhova, “Operator Estimates in Homogenization Theory”, Uspekhi Matem. Nauk, 71:3 (2016), 27–122; Russ. Math. Surv., 71:3 (2016), 417–511.
M. Sh. Birman and T. A. Suslina, “Operator Error Estimates in the Homogenization Problem for Nonstationary Periodic Equations”, Algebra i Analiz, 20:6 (2008), 30–107; St. Petersburg Math. J., 20:6 (2009), 873–928.
Yu. M. Meshkova, “On Operator Error Estimates for Homogenization of Hyperbolic Systems With Periodic Coeffcients”, J. Spectr. Theory, 11:2 (2021), 587–660.
T. A. Suslina, “Spectral Approach to Homogenization of Nonstationary Schrödinger-Type Equations”, J. Math. Anal. and Appl., 446:2 (2017), 1466–1523.
M. A. Dorodnyi and T. A. Suslina, “Spectral Approach to Homogenization of Hyperbolic Equations With Periodic Coefficients”, J. Differ. Equ., 264:12 (2018), 7463–7522.
M. A. Dorodnyi and T. A. Suslina, “Homogenization of Hyperbolic Equations with Periodic Coefficients in \(\mathbb {R}^d\): Sharpness of the Results”, Algebra i Analiz, 32:4 (2020), 3–136; St. Petersburg Math. J., 32:4 (2021), 605–703.
M. A. Dorodnyi, “Operator Error Estimates for Homogenization of the Nonstationary Schrödinger-Type Equations: Sharpness of the Results”, Appl. Anal., 101:16 (2022), 5582–5614.
T. A. Suslina, “Homogenization of the Schrödinger-Type Equations: Operator Estimates With Correctors”, Funktsional. Analiz i ego Prilozhen., 56:3 (2022), 93–99; Funct. Anal. Appl., 56:3 (2022), 229–234.
T. A. Suslina, “Operator-theoretic Approach to Homogenization of the Schrödinger-type Equations with Periodic Coefficients”, Uspekhi Matem. Nauk, 78:6 (2023), 47–178; Russ. Math. Surv., 78:6 (2023), to appear.
M. A. Dorodnyi and T. A. Suslina, “Homogenization of Hyperbolic Equations: Operator Estimates With Correctors”, Funktsional. Analiz i ego Prilozhen., 57:4 (2023), 123–129; Funct. Anal. Appl., 57:4 (2023), to appear.
F. Lin and Z. Shen, “Uniform Boundary Controllability and Homogenization of Wave Equations”, J. Eur. Math. Soc., 24:9 (2022), 3031–3053.
R. V. Craster, J. Kaplunov, and A. V. Pichugin, “High-Frequency Homogenization for Periodic Media”, Proc. R. Soc. A., 466:2120 (2010), 2341–2362.
D. Harutyunyan, G. W. Milton , and R. V. Craster, “High-Frequency Homogenization for Travelling Waves in Periodic Media”, Proc. R. Soc. A, 472:2191 (2016), 20160066.
L. Ceresoli et al., “Dynamic Effective Anisotropy: Asymptotics, Simulations, and Microwave Experiments with Dielectric Fibers”, Phys. Rev. B, 92:17 (2015), 174307.
G. Allaire, “Periodic Homogenization and Effective Mass Theorems for the Schrödinger Equation”, Quantum Transport. Lecture Notes in Mathematics, 1946 (2008), 1–44.
G. Allaire and A. Piatnitski, “Homogenization of the Schrödinger Equation and Effective Mass Theorems”, Comm. Math. Phys., 258:1 (2005), 1–22.
L. Barletti and N. Ben Abdallah, “Quantum Transport in Crystals: Effective Mass Theorem and \(k\cdot p\) Hamiltonians”, Comm. Math. Phys., 307:3 (2011), 567–607.
P. Kuchment and A. Raich, “Green’s Function Asymptotics Near the Internal Edges of Spectra of Periodic Elliptic Operators. Spectral Edge Case”, Math. Nachr., 285:14-15 (2012), 1880–1894.
M. Kha, P. Kuchment, and A. Raich, “Green’s Function Asymptotics Near the Internal Edges of Spectra of Periodic Elliptic Operators. Spectral Gap Interior”, J. Spectr. Theory, 7:4 (2017), 1171–1233.
M. Sh. Birman, “On Homogenization Procedure for Periodic Operators Near the Edge of an Internal Gap”, Algebra i Analiz, 15:4 (2003), 61–71; St. Petersburg Math. J., 15:4 (2004), 507–513.
T. A. Suslina and A. A. Kharin, “Homogenization with Corrector for a Periodic Elliptic Operator Near an Edge of Inner Gap”, Problemy Mat. Analiza, 41 (2009), 127–141; J. Math. Sci., 159:2 (2009), 264–280.
A. A. Mishulovich, V. A. Sloushch, and T. A. Suslina, “Homogenization of a One-Dimensional Periodic Elliptic Operator at the Edge of a spectral Gap: Operator Estimates in the Energy Norm”, Zap. Nauchn. Sem. POMI, 519 (2022), 114–151; J. Math. Sci., , to appear.
A. R. Akhmatova, E. S. Aksenova, V. A. Sloushch, and T. A. Suslina, “Homogenization of the Parabolic Equation with Periodic Coefficients at the Edge of a Spectral Gap”, Complex Var. Elliptic Equ., 67:3 (2022), 523–555.
M. A. Dorodnyi, “High-Frequency Homogenization of Nonstationary Periodic Equations”, Appl. Anal., (2023).
M. Sh. Birman and T. A. Suslina, “Homogenization of a Multidimensional Periodic Elliptic Operator in a Neighborhood of the Edge of an Internal Gap”, Zap. Nauchn. Sem. POMI, 318 (2004), 60–74; J. Math. Sci., 136:2 (2006), 3682–3690.
T. A. Suslina and A. A. Kharin, “Homogenization with Corrector for a Multidimensional Periodic Elliptic Operator Near an Edge of an Inner Gap”, Problemy Mat. Analiza, 59 (2011), 177–193; J. Math. Sci., 177:1 (2011), 208–227.
A. A. Mishulovich, “Homogenization of the Multidimensional Parabolic Equations with Periodic Coefficients at the Edge of a Spectral Gap”, Zap. Nauchn. Sem. POMI, 516 (2022), 135–175; J. Math. Sci., , to appear.
P. Kuchment, “An Overview of Periodic Elliptic Operators”, Bull. Amer. Math. Soc., 53:3 (2016), 343–414.
W. Kirsch and B. Simon, “Comparison Theorems for the Gap of Schrödinger Operators”, J. Funct. Anal., 75:2 (1987), 396–410.
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type, 2nd ed., Nauka, Moscow, 1973; English transl. of 1st ed.: Acad. Press, New York – London, 1968.
M. Sh. Birman, “The Discrete Spectrum in Gaps of the Perturbed Periodic Schrödinger Operator. II. Nonregular Perturbations”, Algebra i Analiz, 9:6 (1997), 62–89; St. Petersburg Math. J., 9:6 (1998), 1073–1095.
A. Piatnitski, V. Sloushch, T. Suslina, and E. Zhizhina, “On Operator Estimates in Homogenization of Nonlocal Operators of Convolution Type”, J. Differ. Equ., 352 (2023), 153–188.
Acknowledgments
The author is grateful to T. A. Suslina for helpful discussions and attention to the work. The author would like to thank V. A. Sloushch for useful discussions.
Funding
This work was supported by Young Russian Mathematics award and the Ministry of Science and Higher Education of the Russian Federation, agreement тДЦ 075-15-2022-287.
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Dorodnyi, M. High-Energy Homogenization of a Multidimensional Nonstationary Schrödinger Equation. Russ. J. Math. Phys. 30, 480–500 (2023). https://doi.org/10.1134/S1061920823040064
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DOI: https://doi.org/10.1134/S1061920823040064