1 Introduction: Iwatsuka Hamiltonians, edge currents and magnetic inverse problems↩︎

Quantum Hall Hamiltonians describe the motion of a charged particle constrained to a bounded or unbounded subdomain of the plane, subject to a constant transverse magnetic field with strength \(b \ge 0\). Confined quantum Hall systems, such as motion in a half-plane or a strip are particularly interesting as a current flowing along an edge is created. Confinement may be obtained by Dirichlet boundary conditions (hard edge) or an electrostatic potential barrier (soft edge), but in any case the edges of the confinement induce edge currents. These edge currents are carried by states with energy localized between any two Landau levels \((2n-1)b\), \(n=1,2,\ldots\), see e.g. [1]–[4].

In the present article we are interested in edge currents created by purely magnetic barriers. Namely, we consider a two-dimensional Schrödinger operator with a non-constant magnetic field \(b(x,y)=b(x)\), \((x,y) \in \mathbb{R}^2\), depending only on \(x\). When the real-valued function \(b\) is bounded and has different limits as \(x\) goes to \(\pm \infty\), it was shown in [5] by Iwatsuka that the spectrum is absolutely continuous. Later on, the transport properties of these so-called Iwatsuka Hamiltonians were investigated in [6] by physicists Reijniers and Peeters. When \(b(x)\) assumes constant value \(b_\pm\) for \(\pm x > 0\), \(0<b_- < b_+<\infty\), they argued that this discontinuity in the magnetic field at \(x=0\) creates an effective edge and that currents flow along the edge. This is the magnetic analog of the barriers created by Dirichlet boundary conditions along \(x = 0\) or a confining electrostatic potential filling the half-space \(x < 0\) described in the paragraph above.

Besides, when \(0 < b_- \le b(x) \le b_+\) for \(x \in (-\epsilon,\epsilon)\), where \(\epsilon \in [0,b_-^{-1/2})\), and \(b(x) = b_\pm\) for \(\pm x > \epsilon\), the existence of currents flowing in the \(y\)-direction was rigorously established in [7]. These currents are carried by states with energy concentration in the energy bands of the Iwatsuka Hamiltonian, and they are well-localized in \(x\) to a region of size \(b_-^{-1/2}\), centered at \(x=0\). The study of the case of a jump in the magnetic field at \(x = 0\) corresponding to \(\epsilon = 0\) in [7], was extended to \(b_+=-b_-=b>0\) in [8]. It shows that the magnetic field creates an effective barrier near \(x = 0\) that causes edge currents to flow along it consistent with the heuristic approach conducted in [6]. Moreover, partially motivated by [6], Dombrowski, Germinet and Raikov studied the edge conductance for generalized Iwatsuka models in [9].

In this article we examine the inverse problem of determining the vector potential \(A(x,y)=(0,a(x))\), where \(a(x)=\int_0^x b(s) \mathrm{d}s\), \(x \in \mathbb{R}\), of the Iwatsuka Hamiltonian defined on the dense domain \(C_0^\infty(\mathbb{R}^2) \subset L^2(\mathbb{R}^2)\) by \((-i \nabla -A)^2\), from knowledge of the edge currents carried by quantum states with energy localized in the first band \((b_-,b_+)\). More precisely, these edge currents are generated upon triggering the dynamic quantum system governed by the Iwatsuka Hamiltonian, by a suitable set of initial states with energy concentration within \((b_-,b_+)\).

Inverse coefficient problems for the magnetic Schrödinger operator have attracted a great deal of attention over the last years. For instance, in [10], using the Bukhgeim-Klibanov method, see [11], the time-independent divergence-free magnetic potential was Lipschitz stably retrieved from a finite number of partial boundary observations (over the entire course of time) of the solution. The authors proceed by suitably changing the initial state of the corresponding dynamic Schrödinger equation and then measuring the solution on a sub-boundary fulfilling the geometric optics condition for the observability derived by Bardos, Lebeau and Rauch in [12]. The case of non-zero divergence magnetic vectors was treated with the same approach in [13].

In [14], [15] the magnetic field of the Schrödinger operator was identified by the Dirichlet-to-Neumann map. These results are based on a different approach using geometric optics solutions. The stability issue in the same problem was treated in [16] but for dynamic magnetic Schrödinger equation in a bounded domain, and in [17] for the same equation on a Riemannian manifold. As for the determination of the magnetic vector potential of the Schrödinger operator by spectral data, it was established in [18]. All the above mentioned inverse results were derived for magnetic Schrödinger systems in a bounded domain, and we refer the reader to [19] for the study of the inverse problem of retrieving the magnetic field of the Schrödinger equation in an unbounded cylindrical domain.

It is worth noticing that the analysis of the inverse problem under study in this manuscript is fundamentally different from the ones used for solving the magnetic inverse problems of [10], [11], [14]–[19]. This is mostly due to the fact that our data are not naturally related to the Neumann boundary data or the spectral data used by [10], [11], [14]–[19]. Nevertheless, due to its translational invariance in the \(y\) direction, the Iwatsuka Hamiltonian admits a fiber decomposition. The fibers are Sturm-Liouville operators with a zero-th order perturbation expressed in terms of the unknown function \(a\). But despite of this, and since the fibers are defined on the real line here, it is still unclear whether the analysis conducted in [20] on the half-line could be adapted to the inverse problem under investigation in this manuscript.As for inverse spectral problems for one-dimensional Schrödinger operators defined either on the real-line or on the half-line, we refer the reader to [21] and the references therein, where the continuous and bounded from below real-valued electric potential was identified by the Krein spectral shift function.

The remaining part of the text is structured as follows. Section 2 contains the main definitions and results of this article. In Section 3 we rigorously define the data used for solving the inverse problem under examination in this article, and we briefly comment on them. Finally, in Section 4, we give the proof of the main results stated below in Theorems 1 and 2, and in Corollary 3.

2 Definitions and results↩︎

2.1 Iwatsuka Hamiltonians↩︎

Let \(b \in L^\infty(\mathbb{R})\) be a non-decreasing function satisfying \[\label{a1} \lim_{x \to \pm \infty} b(x)=b_\pm,\tag{1}\] where \(b_\pm\) are two positive real numbers such that \[\label{a2} 0 < b_-<b_+<3b_-.\tag{2}\] Notice for further use that 1 2 yields \[\label{a2b} b_- \le b(x) \le b_+,\;x \in \mathbb{R}.\tag{3}\] Next, we put \[\label{a3} a(x):=\int_0^x b(s) \mathrm{d}s,\;x \in \mathbb{R},\tag{4}\] and we introduce the magnetic potential \(A(x,y)=(A_1(x,y), A_2(x,y))\) with \(A_1(x,y)=0\) and \(A_2(x,y)=a(x)\). In the present article we consider the two-dimensional magnetic Hamiltonian \((-i \nabla -A)^2\), defined on \(C_0^\infty(\mathbb{R}^2)\) by \[\label{a4} H :=-\partial_x^2 + (-i \partial_y - a)^2.\tag{5}\] Since \(A \in L_{\mathrm loc}^4(\mathbb{R}^2,\mathbb{R}^2)\) and \(b=\partial_x A_2- \partial_y A_1 \in L_{\mathrm loc}(\mathbb{R}^2,\mathbb{R})\), the operator \(H\) is essentially self-adjoint on the dense domain \(C_0^\infty(\mathbb{R}^2) \subset L^2(\mathbb{R}^2)\) by [22], and we still denote by \(H\) its unique self-adjoint extension in \(L^2(\mathbb{R}^2)\).

2.2 Reduction to one-dimensional operators↩︎

The Schrödinger operator defined in 5 being invariant with respect to translations in the \(y\)-direction, it decomposes into a family of parameterized Hamiltonians on \(L^2(\mathbb{R})\). Let \(\mathcal{F}\) denote the partial Fourier transform with respect to \(y\), i.e. \[(\mathcal{F}u)(x,\xi) = \hat{u}(x,\xi) := (2 \pi)^{-1\slash 2}\int_{\mathbb{R}} e^{-i\xi y} u(x,y) \mathrm{d}y,\;u \in L^2(\mathbb{R}^2),\;(x,\xi) \in \mathbb{R}^2.\] The Hilbert space \(L^2(\mathbb{R}^2)\) can be expressed as a constant fiber direct integral over \(\mathbb{R}\) with fibers \(L^2 (\mathbb{R})\), i.e. \(L^2(\mathbb{R}^2)=\int_\mathbb{R}^\oplus L^2(\mathbb{R}) \mathrm{d}\xi\), and the operator \(H\) admits a partial Fourier decomposition with respect to the \(y\)-variable, with \[\mathcal{F}H \mathcal{F}^* = \int_{\mathbb{R}}^\oplus h(\xi) \mathrm{d}\xi,\] where each \(h(\xi)\), \(\xi \in \mathbb{R}\), is self-adjoint in \(L^2(\mathbb{R})\) with domain \(D(h(\xi))\) which is independent of \(\xi\), i.e. \(D(h(\xi))=D(h(0))\), according to [5]. Moreover, we have \(h(\xi)=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+q(x,\xi)\) on \(C_0^\infty (\mathbb{R})\), where \(q(x,\xi):=v(x,\xi)^2 \slash 4\) and \(v(x,\xi):=2(\xi-a(x))\) denotes the quantum velocity at frequency \(\xi\).

In light of 1 , the potential \(q(\cdot, \xi)\), \(\xi \in \mathbb{R}\), is unbounded as \(|x|\) goes to infinity, hence \(h(\xi)\) has a compact resolvent. Let \(\{ \lambda_j(\xi),\;j \in \mathbb{N}\}\), be the non decreasing sequence of the eigenvalues of the operator \(h(\xi)\), \(\xi \in \mathbb{R}\). Since all the eigenvalues \(\lambda_j(\xi)\) are simple (see [3] or [5]), we have for all \(j \ge 3\), \[\lambda_1(\xi) < \lambda_2(\xi) < \ldots < \lambda_j(\xi) < \lambda_{j+1}(\xi) < \ldots\] and the functions \(\xi \mapsto \lambda_j(\xi)\), \(j \ge 1\), are real analytic by the Kato perturbation theory, see [23]. Moreover we have \[\label{a5} (2j-1) b_- \le \lambda_j(\xi) \le (2j-1) b_+,\;\xi \in \mathbb{R},\;j \in \mathbb{N},\tag{6}\] and \[\label{a6} \lim_{\xi \to \pm \infty} \lambda_j(\xi) = (2j-1)b_\pm,\;j \in \mathbb{N},\tag{7}\] from [9]. As a consequence the spectrum of \(H\) is purely absolutely continuous (see, e.g. [24]) and \[\sigma(H) = \bigcup_{j=1}^\infty [(2j-1)b_-,(2j-1)b_+].\] Therefore, \(\sigma(H)\) has a band structure and it follows from 2 that the first band \([b_-,b_+]\) does not overlap with the remaining part of the spectrum \(\cup_{j=2}^\infty [(2j-1)b_-, (2j-1)b_+]\).

2.3 Quantum velocity↩︎

In light of [5], there exists a \(L^2(\mathbb{R})\)-orthonormal basis \(\{ \varphi_j(\cdot,\xi),\;j \in \mathbb{N}\}\) is of eigenfunctions of \(h(\xi)\), \(\xi \in \mathbb{R}\), such that \[h(\xi) \varphi_j(\cdot,\xi)=\lambda_j(\xi) \varphi_j(\cdot,\xi),\;j \in \mathbb{N}.\] Moreover, \(\varphi_j(\cdot,\xi) \in D(h(0)) = \{ u \in H^1(\mathbb{R}),\;-u'' + a^2 u \in L^2(\mathbb{R}) \}\), \(j \in \mathbb{N}\), depends analytically on \(\xi \in \mathbb{R}\) with respect to the graph norm of \(h(0)\), defined by \[\| u \|_{D(h(0))}:=\left( \| u \|_2^2 + \| h(0) u \|_2^2 \right)^{1 \slash 2},\;u \in D(h(0)),\] where \(\| \cdot \|_2\) denotes the usual norm in \(L^2(\mathbb{R})\). In what follows, all the eigenfunctions \(\varphi_j(\cdot,\xi)\), \(j \in \mathbb{R}\), are chosen to be real-valued, and since \(\varphi_1(\cdot,\xi)\) is non-degenerate, we will always assume that \[\varphi_1(x,\xi) >0,\;x \in \mathbb{R}.\]

This being said, we introduce the current operator \(\vartheta\) as \[\label{c1} \vartheta(\chi) := \int_{\mathbb{R}} \chi(\xi)^2 \langle v(\cdot,\xi) \varphi_1(\cdot,\xi), \varphi_1(\cdot,\xi) \rangle_2 \mathrm{d}\xi,\;\chi \in C_0^\infty(\mathbb{R}):=C_0^\infty(\mathbb{R},\mathbb{R}),\tag{8}\] where \(\langle \cdot , \cdot \rangle_2\) is the usual scalar product in \(L^2(\mathbb{R})\). We shall see in Section 3.2, that \(\vartheta(\chi)\) is the expectation of the second component of the velocity operator \(2(i \partial_y+a)\) expressed in the quantum state \(e^{-it H} u_{0,\chi}\), \(t \in [0,+\infty)\), where \[\label{c1b} u_{0,\chi} := \mathcal{F}^* \psi_\chi,\;\psi_\chi(x,\xi):= \chi(\xi) \varphi_1(x,\xi),\;(x,\xi) \in \mathbb{R}^2,\tag{9}\] that is to say the quantum current current carried by \(e^{-it H} u_{0,\chi}\). Since it is time-independent according to 8 , we rather call it quantum current carried by \(u_{0,\chi}\) in the sequel.

In the present article we investigate the inverse problem to know whether knowledge of the transport properties of the Iwatsuka Hamiltonian \(H\), expressed through its current operator \(\vartheta\), uniquely determine the magnetic potential \(a\) of \(H\).

2.4 Main results↩︎

We denote by \(\mathbb{B}\) the (convex) set of Iwatsuka magnetic fields, i.e. the set of non-decreasing functions \(b \in L^\infty(\mathbb{R})\) satisfying the condition 1 . Given \(b\) and \(\tilde{b}\) in \(\mathbb{B}\), we introduce \[b_\varepsilon:=(1-\varepsilon) b+\varepsilon\tilde{b},\;\varepsilon\in (0,1).\] Evidently, \(b_\varepsilon\in \mathbb{B}\) for all \(\varepsilon\in (0,1)\) and we denote by \(\vartheta_\varepsilon\) the current operator associated with \(b_\varepsilon\). Otherwise stated, \(\vartheta_\varepsilon(\chi)\), \(\varepsilon\in (0,1)\) and \(\chi \in C_0^\infty(\mathbb{R})\), is the quantum current carried by the state \(u_{0,\varepsilon,\chi}\), characterized by \[\hat{u}_{0,\varepsilon,\chi}(\cdot,\xi)=\chi(\xi) \varphi_{\varepsilon,1}(\cdot,\xi),\;\xi \in \mathbb{R}.\] Here, \(\{ \varphi_{\varepsilon,j}(\cdot,\xi),\;j \in \mathbb{N}\}\) is a \(L^2(\mathbb{R})\)-orthonormal basis of eigenfunctions of the operator \[\label{m0} h_\varepsilon(\xi):= -\frac{\mathrm{d}^2}{\mathrm{d}x^2} + q_\varepsilon(\cdot,\xi),\;\xi \in \mathbb{R},\tag{10}\] where \(q_\varepsilon(\cdot,\xi):=v_\varepsilon(\cdot,\xi)^2 \slash 4\), \(v_\varepsilon(x,\xi):=2 (\xi-a_\varepsilon(x))\) and \(a_\varepsilon(x):=\int_0^x b_\varepsilon(s) \mathrm{d}s\).

Our first identification result is as follows.

In contrast to Theorem 1 where an infinite number of current operators \(\vartheta_\epsilon\), \(\epsilon \in (0,1]\), are supposed to be known, the following result assumes knowledge of one current operator only, but additional spectral data is needed for the identification of the Iwatsuka Hamiltonian. More precisely, Theorem 2, below, aims to recover the unknown magnetic field \(\tilde{b} \in \mathbb{B}\) from combined knowledge of 1) the quantum current \(\tilde{\vartheta}(\chi)\) carried by the state \(\tilde{u}_{0,\chi}(x,y)=\mathcal{F}^* (\chi \tilde{\varphi}_{1}(x,\cdot))(y)\), \((x,y) \in \mathbb{R}^2\), for all \(\chi \in C_0^\infty(\mathbb{R})\), where \(\{ \tilde{\varphi}_{j}(\cdot,\xi),\;j \in \mathbb{N}\}\) is a \(L^2(\mathbb{R})\)-orthonormal basis of eigenfunctions of the operator \[\tilde{h}(\xi):= -\frac{\mathrm{d}^2}{\mathrm{d}x^2} + (\xi-\tilde{a}(x))^2,\;\xi \in \mathbb{R},\] and, 2) \(\tilde{\varphi}_1(\cdot,\pm \xi_0)\) for some arbitrarily fixed \(\xi_0 \in (0,\infty)\).

The following statement is a byproduct of Theorem 2, obtained upon replacing ?? by a more natural condition that can be checked through direct observation of the quantum state of the system governed by \(\tilde{H}:=-\partial_x^2 + (-i \partial_y - \tilde{a}(x))^2\).

Notice that since \(\tilde{\vartheta}(\chi)\) is the net quantum current (carried by \(\tilde{u}_{0,\chi}\)) flowing across the horizontal line \(y=y_0\), it can be measured at the exact same place (and at the same time \(t_0\)) where the quantum state \(e^{-i t_0 \tilde{H}} \tilde{u}_{0,\chi}(\cdot,y_0)\) appearing in Corollary 3 is observed.

3 Preliminaries: definition of the current operator \(\vartheta\)↩︎

In this section we study the transport properties of quantum devices described by the system \[\label{v1} \left\{ \begin{array}{ll} (-i \partial_t + H) u(x,y,t)=0, & (x,y,t) \in \mathbb{R}^2 \times (0,\infty) \\ u(x,y,0)=u_0(x,y), & (x,y) \in \mathbb{R}^2, \end{array} \right.\tag{11}\] where \(H\) is the self-adjoint realization introduced in Section 2.1 of the Iwatsuka Hamiltonian defined by 5 on \(C_0^\infty(\mathbb{R}^2)\), and \(u_0\) is taken in \(D(H)\), the domain of \(H\). More precisely, we aim to relate the current operator \(\vartheta\) defined in 9 to the second component of the quantum velocity operator associated with \(H\).

3.1 The forward problem: Energy concentration and fast decaying property↩︎

For all \(u \in L^2(\mathbb{R}^2)\), we have \[\hat{u}(x,\xi)=\sum_{j=1}^\infty u_j(\xi) \varphi_j(x,\xi),\;(x,\xi) \in \mathbb{R}^2,\] where \(u_j(\xi):= \langle \hat{u}(\cdot,\xi), \varphi_j(\cdot,\xi) \rangle_2\), since \(\{ \varphi_j(\cdot,\xi),\;j \in \mathbb{N}\}\), \(\xi \in \mathbb{R}\), is an orthonormal basis of \(L^2(\mathbb{R})\). Thus, for all \(\lambda \in (0,\infty)\) and all \(u \in D(H)\), it holds true that \[\begin{align} \| (\lambda + iH) u \|_{L^2(\mathbb{R}^2)}^2 & = \sum_{j=1}^\infty \int_\mathbb{R}\left\lvert\lambda + i \lambda_j(\xi)\right\rvert^2 \left\lvert u_j(\xi)\right\rvert^2 \mathrm{d}\xi \\ & = \sum_{j=1}^\infty \int_\mathbb{R}(\lambda^2 + \lambda_j(\xi)^2) \left\lvert u_j(\xi)\right\rvert^2 \mathrm{d}\xi \\ & \ge \lambda^2 \| u \|_{L^2(\mathbb{R}^2)}^2, \end{align}\] and hence the operator \(-iH\) is dissipative in \(L^2(\mathbb{R}^2)\). Further, \(\lambda+iH=i(H-i\lambda)\) being surjective since the spectrum of \(H\) is embedded in \([b_-,\infty)\), the operator \(-iH\) is maximally dissipative in \(L^2(\mathbb{R}^2)\). Therefore, 11 admits a unique solution \(u \in C^0([0,\infty), D(H)) \cap C^1([0,\infty),L^2(\mathbb{R}^2))\), which is expressed as \[\label{v1b} u(x,y,t)=e^{-i t H} u_0(x,y),\;(x,y) \in \mathbb{R}^2,\;t \in [0,\infty),\tag{12}\] from [25].

For \(\chi \in C_0^\infty(\mathbb{R})\), let \(u_{0,\chi}\) be the same as in 9 . Then we have \(\hat{u}_{0,\chi} \in D(h(\xi))\) for all \(\xi \in \mathbb{R}\), and \[\int_{\mathbb{R}} \left( \| \hat{u}_{0,\chi}(\cdot,\xi) \|_2^2 + \| h(\xi) \hat{u}_{0,\chi}(\cdot,\xi) \|_2^2 \right) \mathrm{d}\xi = \int_{\mathbb{R}} \left( 1 + \lambda_1(\xi)^2 \right) \chi(\xi)^2 \mathrm{d}\xi < \infty,\] whence \(u_{0,\chi} \in D(H)\) by [24]. Therefore, it follows from 12 that \[\label{v1c} u_\chi(x,y,t):=e^{-it H} u_{0,\chi}(x,y),\;(x,y) \in \mathbb{R}^2,\;t \in [0,\infty),\tag{13}\] is well-defined.

For further reference we shall establish that, 1) the quantum state \(u_\chi(\cdot,\cdot,t)\), \(t \in [0,\infty)\), has energy concentration in the first spectral band \((b_-,b_+)\), of \(H\), and 2) \(\partial_t^k u(\cdot,\cdot,t)\), \(k=0,1\), together with its partial derivatives with respect to \(y\), decay faster than any polynomials in the \(y\)-direction. For this purpose we introduce the Schwartz space \(\mathcal{S}_y(\mathbb{R},L_x^2(\mathbb{R}))\) of smooth functions \(y \mapsto f(\cdot,y)\) from \(\mathbb{R}\) into \(L^2(\mathbb{R})\), whose derivatives are rapidly decreasing, as: \[\begin{align} \mathcal{S}_y(\mathbb{R},L_x^2(\mathbb{R})) := \{ f \in C_y^\infty&(\mathbb{R},L_x^2(\mathbb{R})), \\ &\forall (m,n) \in \mathbb{N}_0^2,\;\sup_{y \in \mathbb{R}} \left\lvert y\right\rvert^m \| \partial_y^n f (\cdot,y)\|_2<\infty \}. \end{align}\] Here and below, we set \(\mathbb{N}_0:=\mathbb{N}\cup \{0 \}\), where, as usual, \(\mathbb{N}:=\{1,2,3,\ldots \}\) is the set of positive integers. Next, we recall that the spectral projection of \(H\) associated with a Borel set \(I \subset \mathbb{R}\), reads \[\label{v3b} \mathbb{P}_I w(x,y)= \mathcal{F}^* \left( \sum_{j=1}^\infty \mathbb{1}_{\lambda_j^{-1}(I)} w_j \varphi_j(x,\cdot) \right)(y),\;w \in L^2(\mathbb{R}^2),\tag{14}\] where \(\mathbb{1}_I\) is the characteristic function of \(I\) and \(w_j(\xi):=\langle \hat{w}(\cdot,\xi), \varphi_j(\cdot,\xi) \rangle_2\). Then, the expected result is as follows.

3.2 Quantum velocity↩︎

Let \(u\) be given by 12 . Assume moreover that \[u \in C^1([0,\infty),\mathcal{S}_y(\mathbb{R},L_x^2(\mathbb{R}))).\] Then, the expectation of the \(y\)-component of the velocity operator of the system in the quantum state \(u\) is (well-) defined by \[\upsilon(u_0,t) := \frac{\mathrm{d}}{\mathrm{d}t} \langle y u(\cdot,t), u(\cdot,t) \rangle_{L^2(\mathbb{R}^2)},\;t \in [0,\infty),\] where \(\langle \cdot , \cdot \rangle_{L^2(\mathbb{R}^2)}\) is the usual scalar product in \(L^2(\mathbb{R}^2)\) and the notation \(y\) stands for the multiplication operator by \(y\). Otherwise stated, \(\upsilon(u_0,t)\) is the velocity, i.e. the first time derivative, of the quantum realization in the state \(e^{-i t H} u_0\), of the second component \(y\) of the position observable. Hence \(\upsilon(u_0,t)\) can be interpreted as the quantum current flowing in the \(y\)-direction, that is carried by the state \(e^{-i t H} u_0\). We refer the reader to [7]–[9] and the references therein, for an extensive mathematical study of the transport properties of Iwatsuka Hamiltonians.

Further, since \(u=e^{-it H} u_0 \in C^1([0,\infty),\mathcal{S}_y(\mathbb{R},L_x^2(\mathbb{R})))\) yields that \[\label{v1e} H u \in C^0([0,\infty),\mathcal{S}_y(\mathbb{R},L_x^2(\mathbb{R}))),\tag{15}\] we see that \[\begin{align} \upsilon(u_0,t) & = \frac{\mathrm{d}}{\mathrm{d}t} \langle y e^{-it H} u_0, e^{-it H} u_0 \rangle_{L^2(\mathbb{R}^2)}\\ & = -i \left( \langle y H e^{-it H} u_0 , e^{-it H} u_0 \rangle_{L^2(\mathbb{R}^2)} - \langle y e^{-it H} u_0 , H e^{-it H} u_0 \rangle_{L^2(\mathbb{R}^2)} \right)\\ & = -i \langle [y, H] e^{-it H} u_0 , e^{-it H} u_0 \rangle_{L^2(\mathbb{R}^2)}, \end{align}\] where \([y,H]\) denotes the commutator of \(y\) with \(H\). Next, using that \([y,H]=[y,(-i\partial_y-a)^2]\) and that \([y, (-i\partial_y-a)]=i\), we find that \([y,H]=2i(-i \partial_y -a)\), and hence that \[\label{v2} \upsilon(u_0,t) = 2 \langle (-i \partial_y - a) e^{-it H} u_0 , e^{-it H} u_0 \rangle_{L^2(\mathbb{R}^2)},\;t \in [0,\infty).\tag{16}\] Notice from 15 that \((-i \partial_y - a) e^{-it H} u_0 \in L^2(\mathbb{R}^2)\) and hence that the right-hand side of 16 is well-defined, as we have \[\| \partial_x e^{-it H} u_0 \|_{L^2(\mathbb{R}^2)}^2 + \| (-i \partial_y - a) e^{-it H} u_0 \|_{L^2(\mathbb{R}^2)}^2 = \langle H e^{-it H} u_0 , e^{-it H} u_0 \rangle_{L^2(\mathbb{R}^2)}.\] Now, the transform \(\mathcal{F}\) being unitary in \(L^2(\mathbb{R}^2)\), we deduce from the identity \[2\mathcal{F}(-i \partial_y -a(x)) \mathcal{F}^* = 2(\xi-a(x)) = v(x,\xi),\;(x,\xi) \in \mathbb{R}^2,\] and from 16 that \[\label{v3} \upsilon(u_0,t) = \langle v \mathcal{F}(e^{-it H} u_0) , \mathcal{F}(e^{-it H} u_0)\rangle_{L^2(\mathbb{R}^2)},\;t \in [0,\infty).\tag{17}\]

Let us now express \(\upsilon(u_0,t)\) when \(u_0 \in \mathbb{P}_I (L^2(\mathbb{R}^2))\) for some \(I \subset \mathbb{R}\), that is to say when \(u_0=\mathbb{P}_I u_0\). In this case, we have \[\mathcal{F}(e^{-it H} u_0)(x,\xi) = \sum_{j=1}^\infty \mathbb{1}_{\lambda_j^{-1}(I)}(\xi) e^{-it \lambda_j(\xi)} u_{0,j}(\xi) \varphi_j(x,\xi),\;(x,\xi) \in \mathbb{R}^2,\] from 14 , where \(u_{0,j}(\xi):=\langle \hat{u}_0(\cdot,\xi), \varphi_j(\cdot,\xi) \rangle_2\). Putting this into 17 , we obtain that \[\begin{align} &\upsilon(u_0,t)= \sum_{j,k=1}^\infty \int_{\lambda_j^{-1}(I) \cap \lambda_k^{-1}(I)}\label{v4} \\ &\hskip 2cm e^{-it(\lambda_j(\xi)-\lambda_k(\xi))} u_{0,j}(\xi) \overline{u_{0,k}(\xi)}\langle v(\cdot,\xi) \varphi_j(\cdot,\xi) , \varphi_k(\cdot,\xi) \rangle_2 \mathrm{d}\xi.\nonumber \end{align}\tag{18}\] Now, suppose that \(I \subset (b_-,b_+)\), in such a way that we have \[\lambda_j^{-1}(I)=\emptyset,\;j \geqslant 2,\] by virtue of 2 and 6 . Then, it follows from 18 that \[\label{v5} \upsilon(u_0,t) = \int_{\lambda_1^{-1}(I)} \left\lvert u_{0,1}(\xi)\right\rvert^2 \langle v(\cdot,\xi) \varphi_1(\cdot,\xi) , \varphi_1(\cdot,\xi) \rangle_2 \mathrm{d}\xi .\tag{19}\]

Therefore, the quantum current \(\upsilon(u_0,t)\) carried by a state \(u_0\) with energy concentration in \(I \subset (b_-,b_+)\), is independent of \(t\). For the sake of notational simplicity, we write \(\upsilon(u_0)\) instead of \(\upsilon(u_0,t)\) in the following.

Now, with reference to Lemma 4, we deduce from 9 and 19 that \(\vartheta(\chi)\) is the quantum current carried by \(e^{-it H} u_{0,\chi}\), \(t \in [0,\infty)\), or equivalently by \(u_{0,\chi}\): \[\vartheta(\chi) = \upsilon(u_{0,\chi}),\;\chi \in C_0^\infty(\mathbb{R}).\]

3.3 Comments on the data \(\vartheta\) and the formulation of the inverse problem↩︎

It might seem surprising at first sight that we probe the system 11 with the frequency profile \(\chi\) of the initial state \(u_{0,\chi}\), rather than with the initial state itself. But there are at least two reasons why the current operator \(\vartheta\) should not be defined as a function of the initial states \(u_0\), the first one being that this would be physically irrelevant. Indeed, since any initial state of the quantum system 11 governed by the Iwatsuka Hamiltonian \(H\), with energy concentration between \(b_-\) and \(b_+\), is expressed as \(\mathcal{F}^* (\chi \varphi_1(x,\cdot))(y)\), \((x,y) \in \mathbb{R}^2\), for some suitable \(L^2(\mathbb{R})\)-function \(\chi\), and since all the ground states \(\varphi_1(\cdot,\xi)\), \(\xi \in \mathbb{R}\), are determined by \(H\), it is clear that only the frequency profile \(\chi\) can be prescribed.

Secondly, it turns out that from a mathematical viewpoint, the inverse problem of recovering the magnetic potential \(a\) by triggering the system 11 with a suitable set of initial states \(u_0\), is pointless. This can be understood (upon using Proposition 5 below) from the following lines.

Given two magnetic fields \(b\) and \(\tilde{b}\) in \(\mathbb{B}\), we aim to compare the quantum currents \(\upsilon(u_0)\) and \(\tilde{\upsilon}(u_0)\) induced by 11 associated with, respectively, \(b\) and \(\tilde{b}\), and endowed with a non-zero initial state \(u_0 \in U\), where \(U:=\mathbb{P}_{(b_-,b_+)}(L^2(\mathbb{R}^2)) \cap \tilde{\mathbb{P}}_{(b_-,b_+)}(L^2(\mathbb{R}^2))\). Here, \(\tilde{\mathbb{P}}_{(b_-,b_+)}\) denotes the spectral projection on \((b_-,b_+)\) of the Iwatsuka Hamiltonian \(\tilde{H}\) obtained upon substituting \(\tilde{b}\) for \(b\) in 4 5 . If such a state exists, that is to say if there exists \(u_0 \in L^2(\mathbb{R}^2) \setminus \{ 0 \}\) such that \[u_0=\mathbb{P}_{(b_-,b_+)} u_0=\tilde{\mathbb{P}}_{(b_-,b_+)} u_0,\] then we have \[\label{c1c} \hat{u}_0(x,\xi)=u_{0,1}(\xi) \varphi_1(x,\xi) =\tilde{u}_{0,1}(\xi) \tilde{\varphi}_1(x,\xi),\;(x,\xi) \in \mathbb{R}^2,\tag{20}\] where \(u_{0,1}(\xi):=\langle \hat{u}_0(\cdot,\xi) , \varphi_1(\cdot,\xi) \rangle_2\), \(\tilde{u}_{0,1}(\xi):=\langle \hat{u}_0(\cdot,\xi) , \tilde{\varphi}_1(\cdot,\xi) \rangle_2\), and \(\tilde{\varphi}_1(\cdot,\xi)\) is defined as in Therorem 2. Thus, upon squaring both sides of 20 and then integrating with respect to \(x\) over \(\mathbb{R}\), we get that \(u_{0,1}(\xi)^2=\tilde{u}_{0,1}(\xi)^2\) for a.e. \(\xi \in \mathbb{R}\), and hence that \(\left\lvert u_{0,1}(\xi)\right\rvert=\left\lvert\tilde{u}_{0,1}(\xi)\right\rvert\). Therefore, the set \[\mathcal{N}(u_0):=u_{0,1}^{-1}(\{0\})=\{\xi \in \mathbb{R},\;u_{0,1}(\xi)=0 \}\] can be equivalently defined as \(\mathcal{N}(u_0):=\tilde{u}_{0,1}^{-1}(\{0\})=\{\xi \in \mathbb{R},\;\tilde{u}_{0,1}(\xi)=0 \}\), and it holds true that \(\left\lvert\varphi_1(\cdot,\xi)\right\rvert=\left\lvert\tilde{\varphi}_1(\cdot,\xi)\right\rvert\) a.e. in \(\mathbb{R}\), whenever \(\xi \in \mathbb{R}\setminus \mathcal{N}(u_0)\). As a consequence we have \[\label{c2} \varphi_1(x,\xi)=\tilde{\varphi}_1(x,\xi),\;(x,\xi) \in \mathbb{R}\times (\mathbb{R}\setminus \mathcal{N}(u_0)),\tag{21}\] since \(\varphi_1(\cdot,\xi)\) and \(\tilde{\varphi}_1(\cdot,\xi)\) are positive functions for all \(\xi \in \mathbb{R}\).

Let us now assume for a while that the function \[F(\xi) := \left\lVert\varphi_1(\cdot,\xi)-\tilde{\varphi}_1(\cdot,\xi)\right\rVert_{L^2(\mathbb{R})}^2,\;\xi \in \mathbb{R},\] is not identically zero. Since \(F\) is real analytic and the Lebesgue measure of the zero set of a non-trivial real analytic function is zero, the Lebesgue measure of \(\mathbb{R}\setminus \mathcal{N}(u_0)\) should be zero, according to 21 . This would mean that \(u_{0,1}(\xi)=0\) for a.e. \(\xi \in \mathbb{R}\), and hence that \(\hat{u}_0=0\) in \(L^2(\mathbb{R}^2)\), according to 20 , which is contradiction the fact that \(u_0\) is non-zero. Therefore, we have \(F(\xi)=0\) for all \(\xi \in \mathbb{R}\), and consequently \(\varphi_1(\cdot,\xi)=\tilde{\varphi}_1(\cdot,\xi)\) in \(L^2(\mathbb{R})\) for all \(\xi \in \mathbb{R}\).

Summing up, we have proved that the following equivalence holds: \[\label{c3} U \neq \{ 0 \} \Longleftrightarrow \forall \xi \in \mathbb{R},\;\varphi_1(\cdot,\xi)=\tilde{\varphi}_1(\cdot,\xi)\;in\;L^2(\mathbb{R}).\tag{22}\] Having seen this, let us suppose that \(U \neq \{ 0 \}\), and assume in addition that \[\upsilon(u_0)=\tilde{\upsilon}(u_0),\;u_0 \in U.\] Then, we have \(\varphi_1(\cdot,\xi)=\tilde{\varphi}_1(\cdot,\xi)\) for all \(\xi \in \mathbb{R}\), from 22 , and it is clear for all \(\chi \in C_0^\infty(\mathbb{R})\) that \(u_0(x,y)=\left( \mathcal{F}^*(\chi \varphi_1(x,\cdot)\right)(y) = \left( \mathcal{F}^*(\chi \tilde{\varphi}_1(x,\cdot)\right)(y) \in U\). Moreover, since \(\upsilon(u_0)=\vartheta(\chi)\) and \(\tilde{\upsilon}(u_0)=\tilde{\vartheta}(\chi)\), where \(\tilde{\vartheta}\) is the same as in Theorem 2, we get that \[\vartheta(\chi)=\tilde{\vartheta}(\chi),\;\chi \in C_0^\infty(\mathbb{R}).\] Therefore, we have \(\lambda_1=\tilde{\lambda}_1\) by Proposition 5, whence \[\left( (\xi-\tilde{a}(x))^2 - (\xi-a(x))^2 \right) \varphi_1(x,\xi)=0,\;(x,\xi) \in \mathbb{R}^2.\] Since \(\varphi_1(\cdot,\xi)\) is positive for all \(\xi \in \mathbb{R}\), this entails that \[(\xi-\tilde{a}(x))^2 = (\xi-a(x))^2,\;(x,\xi) \in \mathbb{R}^2.\] Upon differentiating the above identity with respect to \(\xi\), we get that \(\xi-\tilde{a}(x) = \xi-a(x)\) for all \((x,\xi) \in \mathbb{R}^2\), and hence that \(a=\tilde{a}\) in \(\mathbb{R}\).

4 Analysis of the inverse problem↩︎

We start with a technical result needed by the proof of Theorems 1 and 2.

4.1 Preliminaries↩︎

We aim to establish that knowledge of the current operator uniquely determines the first band function. With reference to the notations of Section 2.4, the corresponding result can be stated as follows.

4.2 Proof of Theorem 1↩︎

The proof being quite lengthy, we split it into six steps.

Let us denote by \(r_\xi\), \(\xi \in \mathbb{R}\), the resolvent operator of \(h(\xi)\), i.e., \[r_\xi(z):=(h(\xi)-z)^{-1},\;z \in \mathbb{C}\setminus \{ \lambda_j(\xi),\;j \in \mathbb{N}\}.\] Then, the spectral projection of \(h(\xi)\) associated with \(\lambda_1(\xi)\) can be expressed as \[\label{ap3} p_1(\xi)=-\frac{1}{2 i \pi} \int_{C(\lambda_1(\xi),\rho)} r_\xi(z) \mathrm{d}z,\tag{24}\] where \(\rho\) is arbitrarily fixed in \((0, 3b_- -b_+)\) and \(C(\lambda_1(\xi),\rho):=\{ \lambda_1(\xi)+ \rho e^{i \theta},\;\theta \in [0, 2 \pi) \}\) is the circle centered at \(\lambda_1(\xi)\) with radius \(\rho\), oriented counterclockwise, see e.g., [23].

With reference to ?? and the notations introduced in the following line, we have \(q_\varepsilon(x,\xi)=q(x,\xi)+ \ell_\varepsilon(x,\xi)\) for all \((x,\xi) \in \mathbb{R}^2\), where \[\label{ap3b} \ell_\varepsilon(x,\xi):= \varepsilon\omega(x,\xi) + \varepsilon^2 w(x)^2,\;\omega(x,\xi):= -v(x,\xi) w(x)\;and\;w(x):=\tilde{a}(x)-a(x).\tag{25}\] Thus, putting \(\delta:= \left\lVert\tilde{a}-a\right\rVert_{L^\infty(\mathbb{R})}\) and \[M(\xi):=\left\lVert v(\cdot,\xi)\right\rVert_{L^\infty(K)} = 2 \left\lVert\xi-a\right\rVert_{L^\infty(K)}<\infty,\] where \(K:=[-r,r]\), we infer from 25 that \[\label{ap3c} \left\lVert\ell_\varepsilon(\cdot,\xi)\right\rVert_{L^\infty(\mathbb{R})}\le \varepsilon C(\xi),\;C(\xi):=\delta (M(\xi)+\delta),\;\varepsilon\in (0,1),\;\xi \in \mathbb{R}.\tag{26}\] From this and the MinMax principle, it then follows that \[\label{ap3d} \left\lvert\lambda_1(\xi)-\lambda_{\varepsilon,1}(\xi)\right\rvert \leqslant\varepsilon C(\xi),\;\varepsilon\in (0,1),\;\xi \in \mathbb{R},\tag{27}\] where \(\lambda_{\varepsilon,j}(\xi)\), \(j \in \mathbb{N}\), denotes the \(j\)-th eigenvalue of the operator \(h_\varepsilon(\xi)\). Therefore, taking \(\varepsilon\in (0,1)\) so small that \(\varepsilon C(\xi) < 3b_- - b_+ - \rho\), we deduce from 27 that \[\label{ap3e} \overline{D}(\lambda_1(\xi),\rho) \cap \{ \lambda_{\varepsilon,j}(\xi), j \in \mathbb{N}\} = \{ \lambda_{\varepsilon,1}(\xi) \},\tag{28}\] where \(\overline{D}(\lambda_1(\xi),\rho):=\{ z \in \mathbb{C}, \left\lvert z -\lambda_1(\xi)\right\rvert \leqslant\rho \}\).

Set \(r_{\varepsilon,\xi}(z):=(h_\varepsilon(\xi)-z)^{-1}\) for all \(z \in \mathbb{C}\setminus \{ \lambda_{\varepsilon,j}(\xi), j \in \mathbb{N}\}\), and put \(\varepsilon_*:=\varepsilon_*(\xi,\rho) = \min(1, C(\xi)^{-1} (3b_- - b_+ - \rho))\). Then, with reference to 28 and the path independence of contour integration of the meromorphic function \(z \mapsto r_{\varepsilon,\xi}(z)\) around \(\lambda_{\varepsilon,1}(\xi)\), the spectral projection of \(h_\varepsilon(\xi)\) associated with \(\lambda_{\varepsilon,1}(\xi)\), \[p_{\varepsilon,1}(\xi)=-\frac{1}{2 i \pi} \int_{C(\lambda_{\varepsilon,1}(\xi),\rho)} r_{\varepsilon,\xi}(z) \mathrm{d}z,\;\] can be equivalently rewritten as \[\label{ap5} p_{\varepsilon,1}(\xi)=-\frac{1}{2 i \pi} \int_{C(\lambda_1(\xi),\rho)} r_{\varepsilon,\xi}(z) \mathrm{d}z,\;\xi \in \mathbb{R},\;\varepsilon\in (0,\varepsilon_*).\tag{29}\]

Having established 29 , we turn now to relating \(\lambda_{\varepsilon,1}(\xi)\) to \(\lambda_1(\xi)\) with the aid the identity \(h_{\varepsilon}(\xi)=h(\xi)+\ell_\varepsilon(\cdot,\xi)\). To do that, we start from the eigenvalue equality \(h_{\varepsilon}(\xi) p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi)=\lambda_{1,\varepsilon}(\xi) p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi)\), recall that the operators \(h(\xi)\) and \(\ell_\varepsilon(\cdot,\xi)\) are self-adjoint in \(L^2(\mathbb{R})\), and obtain for all \(\xi \in \mathbb{R}\) and all \(\varepsilon\in (0,1)\), that \[\begin{align} \lambda_{1,\varepsilon}(\xi) \langle p_{\varepsilon,1}(\xi) &\varphi_1(\cdot,\xi), \varphi_1(\cdot,\xi) \rangle_2 \\ &= \langle h_{\varepsilon}(\xi) p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi) ,\varphi_1(\cdot,\xi) \rangle_2\\ & = \langle h(\xi) p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi) ,\varphi_1(\cdot,\xi) \rangle_2+\langle \ell_\varepsilon(\xi) p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi) ,\varphi_1(\cdot,\xi) \rangle_2 \\ & = \langle p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi) , h(\xi) \varphi_1(\cdot,\xi) \rangle_2+ \langle p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi) , \ell_\varepsilon(\xi)\varphi_1(\cdot,\xi) \rangle_2 \\ & = \lambda_1(\xi) \langle p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi) , \varphi_1(\cdot,\xi) \rangle_2+ \langle p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi) , \ell_\varepsilon(\xi)\varphi_1(\cdot,\xi) \rangle_2. \end{align}\] Therefore, for all \(\xi \in \mathbb{R}\) and all \(\varepsilon\in (0,1)\), we have \[\begin{align} F_\xi(\varepsilon) :&= \langle p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi) , \ell_\varepsilon(\xi)\varphi_1(\cdot,\xi) \rangle_2 \label{ap5b} \\ & = (\lambda_{1,\varepsilon}(\xi) - \lambda_1(\xi)) \langle p_{\varepsilon,1}(\xi) \varphi_1(\cdot,\xi), \varphi_1(\cdot,\xi) \rangle_2.\nonumber \end{align}\tag{30}\]

With reference to the second resolvent formula, we have \[\label{ap7} r_{\varepsilon,\xi}(z)=r_\xi(z)-r_\xi(z) \ell_\varepsilon(\cdot,\xi) r_{\varepsilon,\xi}(z),\;\varepsilon\in (0,\varepsilon_*),\;\xi \in \mathbb{R},\;z \in C(\lambda_1(\xi),\rho).\tag{31}\] Notice that for all \(\xi \in \mathbb{R}\) and all \(z \in C(\lambda_1(\xi),\rho)\), we have \[\mathrm{dist} \left( z,\{\lambda_j(\xi),\;j \in \mathbb{N}\} \right)= \min (\left\lvert\lambda_1(\xi)-z\right\rvert, \left\lvert\lambda_2(\xi)-z\right\rvert) \\ \ge \min(\rho,3b_- - b_+-\rho),\] whence \[\begin{align} \left\lVert\ell_\varepsilon(\cdot,\xi) r_\xi(z)\right\rVert_{\mathcal{B}(L^2(\mathbb{R}))}& \le \left\lVert\ell_\varepsilon(\cdot,\xi)\right\rVert_{L^\infty(\mathbb{R})} \left\lVert r_\xi(z)\right\rVert_{\mathcal{B}(L^2(\mathbb{R}))} \\ & \le \frac{\varepsilon C(\xi)}{\min(\rho,3b_- - b_+-\rho)} , \end{align}\] from 26 , where \(\mathcal{B}(L^2(\mathbb{R})\) denotes the space of linear bounded operators in \(L^2(\mathbb{R})\). Therefore, we get that \(\left\lVert\ell_\varepsilon(\cdot,\xi) r_\xi(z)\right\rVert_{\mathcal{B}(L^2(\mathbb{R}))}<1\) for all \(\xi \in \mathbb{R}\) and all \(z \in C(\lambda_1(\xi),\rho)\), provided that \[\varepsilon\in (0,\varepsilon_\star),\;\varepsilon_\star=\varepsilon_\star(\xi,\rho) := \min(\varepsilon_*, C(\xi)^{-1}\rho).\] Thus, by iterating 31 we get for all \(\xi \in \mathbb{R}\) and all \(z \in C(\lambda_1(\xi),\rho)\), that \[r_{\varepsilon,\xi}(z)= \sum_{n=0}^\infty (-1)^n r_\xi(z) \left( \ell_\varepsilon(\cdot,\xi) r_\xi(z) \right)^n,\;\varepsilon\in (0,\varepsilon_\star),\] where the series converges in \(\mathcal{B}(L^2(\mathbb{R}))\). In view of 25 , this leads to \[\label{ap8} r_{\varepsilon,\xi}(z) =\sum_{n=0}^\infty (-1)^n \theta_{n,\xi}(z) \varepsilon^n ,\;\xi \in \mathbb{R},\;\varepsilon\in (0,\varepsilon_\star),\tag{32}\] the series being convergent in \(\mathcal{B}(L^2(\mathbb{R}))\), uniformly in \(z \in C(\lambda_1(\xi),\rho)\). Here, each \(\theta_{n,\xi}(z) \in \mathcal{B}(L^2(\mathbb{R}))\), \(n \in \mathbb{N}\cup \{0 \}\), can be expressed in terms of \(r_\xi(z)\), \(\omega(\cdot,\xi)\) and \(w\) only. For instance, we get through elementary computations that \[\begin{align} \theta_{0,\xi}(z)=r_\xi(z),&\quad \theta_{1,\xi}(z)=r_\xi(z) \omega(\cdot,\xi) r_\xi(z) \label{ap4} \\ &and\quad \theta_{2,\xi}(z)=r_\xi(z) ( w^2 r_\xi(z)- (\omega(\cdot,\xi) r_\xi(z))^2).\nonumber \end{align}\tag{33}\]

By inserting 32 into 29 , we obtain with the aid of 24 that \[\begin{align} p_{\varepsilon,1}(\xi) & = \sum_{n=0}^\infty \frac{(-1)^{n+1}}{2i \pi} \left( \int_{C(\lambda_1(\xi),\rho)} \theta_{n,\xi}(z) \mathrm{d}z \right) \varepsilon^n \\ & = p_1(\xi) + \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2i \pi} \left( \int_{C(\lambda_1(\xi),\rho)} \theta_{n,\xi}(z) \mathrm{d}z \right) \varepsilon^n,\;\xi \in\mathbb{R},\;\varepsilon\in (0,\varepsilon_\star). \end{align}\] It follows readily from this and 25 that the function \(F_\xi\) defined in 30 can be brought into the form \[\label{ap8b} F_\xi(\varepsilon)=\sum_{n=1}^\infty A_n(\xi) \varepsilon^n,\;\xi \in \mathbb{R},\;\varepsilon\in (0,\varepsilon_\star),\tag{34}\] where each \(A_n : \mathbb{R}\to \mathbb{C}\), \(n \in \mathbb{N}\), is independent of \(\varepsilon\). As a matter of fact, in the special case when \(n=2\), we find with the aid of 33 by elementary calculation, that for all \(\xi \in \mathbb{R}\), \[\begin{align} A_2(\xi) & = \langle p_1(\xi) \varphi_1(\cdot,\xi) , w^2 \varphi_1(\cdot,\xi) \rangle_2 \label{ap8c} \\ & \quad + \frac{1}{2i \pi} \int_{C(\lambda_1(\xi),\rho)} \langle r_\xi(z) \omega(\cdot,\xi) r_\xi(z) \varphi_1(\cdot,\xi) , \omega(\cdot,\xi) \varphi_1(\cdot,\xi) \rangle_2 \mathrm{d}z \nonumber \\ & = \| w \varphi_1(\cdot,\xi) \|_2^2 + \frac{1}{2i \pi} \int_{C(\lambda_1(\xi),\rho)} \langle ( \omega(\cdot,\xi) r_\xi(z) )^2 \varphi_1(\cdot,\xi) , \varphi_1(\cdot,\xi) \rangle_2 \mathrm{d}z. \nonumber \end{align}\tag{35}\] In light of 34 , each \(F_\xi\), where \(\xi \in \mathbb{R}\) is fixed, can be extended to a real-analytic function at \(\varepsilon=0\).

Moreover, by assumption, \(\varepsilon=0\) is an accumulation point of the zeros of \(F_\xi\). Therefore, the function \(F_\xi\) is necessarily identically zero by the principle of isolated zeros, and we have \[\label{ap10} A_n(\xi)=0,\;\xi \in \mathbb{R},\;n \in \mathbb{N},\tag{36}\] according to 34 .

With reference to 35 we start by decomposing \(\omega(\cdot,\xi) r_\xi(z) \varphi_1(\cdot,\xi)\), \(\xi \in \mathbb{R}\), on the \(L^2(\mathbb{R})\)-orthonormal basis \(\{ \varphi_j(\cdot,\xi),\;j \geqslant 1 \}\). We get that \[\omega(\cdot,\xi) r_\xi(z) \varphi_1(\cdot,\xi) = \frac{\omega(\cdot,\xi)}{\lambda_1(\xi)-z} \varphi_1(\cdot,\xi)= \frac{1}{\lambda_1(\xi)-z}\sum_{j=1}^\infty \omega_j(\xi) \varphi_j(\cdot,\xi),\] where \(\omega_j(\xi):=\langle \omega(\cdot,\xi) \varphi_1(\cdot,\xi) , \varphi_j(\cdot,\xi) \rangle_2\). Therefore, we have \[\langle (\omega(\cdot,\xi) r_\xi(z))^2 \varphi_1(\cdot,\xi) , \varphi_1(\cdot,\xi) \rangle_2 =\frac{1}{\lambda_1(\xi)-z} \sum_{j =1}^\infty \frac{\left\lvert\omega_j(\xi)\right\rvert^2}{\lambda_j(\xi)-z},\] and 35 then yields that \[\begin{align} A_2(\xi) & = \| w \varphi_1(\cdot,\xi) \|_2^2 + \sum_{j =1}^\infty \frac{\left\lvert\omega_j(\xi)\right\rvert^2}{2 i \pi} \int_{C(\lambda_1(\xi),\rho)} \frac{\mathrm{d}z}{(\lambda_1(\xi)-z)(\lambda_j(\xi)-z)} \label{ap11} \\ & = \| w \varphi_1(\cdot,\xi) \|_2^2 -\sum_{j =2}^\infty \frac{\left\lvert\omega_j(\xi)\right\rvert^2}{\lambda_j(\xi)-\lambda_1(\xi)},\;\xi \in \mathbb{R}, \nonumber \end{align}\tag{37}\] by using the residue theorem.

The next step is to make the second term on the right-hand side of 37 sufficiently small relative to \(\| w \varphi_1(\cdot,\xi) \|_2^2\) by choosing \(\xi\) suitably in \(\mathbb{R}\).

We refer to ?? and notice that there exists \(\kappa \in (0,1)\) such that \[r <r(\kappa) := (1-\kappa)^{1 \slash2}\frac{(3b_- - b_+)^{1 \slash2}}{2b_+}.\]

Further, since \(\left\lvert a(x)\right\rvert \le b_+ \left\lvert x\right\rvert\) for all \(x \in \mathbb{R}\), by 3 4 , we get that \[\left\lvert v(x,\xi)\right\rvert \le 2(\left\lvert\xi\right\rvert + b_+ r),\;x \in K,\;\xi \in \mathbb{R},\] and hence that \[\label{ap13} \| v(\cdot,\xi) \|_{L^\infty(K)} \le (1 -\kappa)^{1 \slash 2} (3b_- - b_+)^{1 \slash2},\;\xi \in [-\xi(\kappa),\xi(\kappa)],\tag{38}\] where \(\xi(\kappa):=b_+ (r(\kappa) - r)\). Moreover, we see from 6 that \[\lambda_j(\xi)-\lambda_1(\xi) \ge (2j-1) b_- - b_+ \ge 3b_- - b_+,\;\xi \in \mathbb{R},\;j \ge 2,\] and consequently \[\sum_{j=2}^\infty \frac{\left\lvert\omega_j(\xi)\right\rvert^2}{\lambda_j(\xi)-\lambda_1(\xi)} \le \sum_{j=2}^\infty \frac{\left\lvert\omega_j(\xi)\right\rvert^2}{3b_- - b_+} \le \frac{\| \omega(\cdot,\xi) \varphi_1(\cdot,\xi) \|_2^2}{3b_- - b_+},\] by the Plancherel formula. It follows from this and the identity \(\omega(\cdot,\xi)=-v(\cdot,\xi) w\) for all \(\xi \in \mathbb{R}\), that \[\sum_{j= 2}^\infty \frac{\left\lvert\omega_j(\xi)\right\rvert^2}{\lambda_j(\xi)-\lambda_1(\xi)} \le \frac{\| v(\cdot,\xi) \|_{L^\infty(K)}^2}{3 b_- - b_+} \| w \varphi_1(\cdot,\xi) \|_2^2,\;\xi \in \mathbb{R}.\] Thus, taking \(\xi \in [-\xi(\kappa),\xi(\kappa)]\) in the above line, we deduce from 38 that \[\sum_{j=2}^\infty \frac{\left\lvert\omega_j(\xi)\right\rvert^2}{\lambda_j(\xi)-\lambda_1(\xi)} \le (1-\kappa) \| w \varphi_1(\cdot,\xi) \|_2^2.\] Therefore, we get from 37 that \[A_2(\xi) \ge \kappa \| w \varphi_1(\cdot,\xi) \|_2^2,\;\xi \in [-\xi(\kappa),\xi(\kappa)],\] and then from 36 that \(w \varphi_1(\cdot,\xi)=0\) in \(L^2(\mathbb{R})\). Finally, bearing in mind that \(\varphi_1(x,\xi) >0\) for a.e. \(x \in \mathbb{R}\) and all \(\xi \in [-\xi(\kappa),\xi(\kappa)]\), we obtain that \(w=0\) a.e. in \(\mathbb{R}\). This proves the desired result.

4.3 Proof of Theorem 2 and Corollary 3↩︎

The proof of Theorem 2 boils down to the following lemma.

Under the conditions of Theorem 2, it follows readily from Proposition 5 and Lemma 6 that \[\label{g3} (\pm \xi_0-a(x))^2=(\pm \xi_0-\tilde{a}(x))^2,\;x \in \mathbb{R}.\tag{41}\] Now, since \(\pm a(x) \geqslant 0\) and \(\pm \tilde{a}(x) \ge 0\) whenever \(\pm x \ge 0\), it is apparent that 41 yields that \(\tilde{a}=a\).

Finally, we prove Corollary 3 by noticing that ?? can be equivalently rewritten as \[\int_\mathbb{R}e^{i( y_0 \xi - t_0 \lambda_1(\xi))} \chi(\xi) (\varphi_1(x,\xi)-\tilde{\varphi}_1(x,\xi)) \mathrm{d}x =0,\;\chi \in C_0^\infty(\mathbb{R}),\;x \in \mathbb{R}.\] This entails that \(e^{i( y_0 \xi - t_0 \lambda_1(\xi))} (\varphi_1(\cdot,\xi)-\tilde{\varphi}_1(\cdot,\xi))=0\) for all \(\xi \in \mathbb{R}\), and hence that \(\varphi_1(\cdot,\xi)=\tilde{\varphi}_1(\cdot,\xi)\), which yields ?? .

References↩︎