Observability for Schrödinger equations with quadratic Hamiltonians

We start by recalling some results for bounded domains. Let $D$, $D_{0}$ be bounded smooth subdomains of $\mathbb{R}^{d}$ with $D_{0}\subset D$. We consider the controlled linear Schrödinger equation

		$\displaystyle i\partial_{t}u(t,x)=-\Delta u(t,x)+V(t,x)u(t,x)+f(t,x)\mathbf{1}_{D_{0}}(x)\quad\mathrm{in}\quad(0,T)\times D$		(1.1)
		$\displaystyle u(t,x)=0\quad\mathrm{on}\quad(0,T)\times\partial D$
		$\displaystyle u_{0}(x)=u(0,x)\quad\mathrm{in}\quad D$

here $u(t,x)$ is the state, $f(t,x)$ is the control, and $V(t,x)$ is the potential. All can be complex functions. It is known that the system is well-posed in $L^{2}(D)$ with controls in $L^{2}((0,T)\times D_{0})$ for $L^{\infty}((0,T)\times D)$ potentials.

Starting with [29] it was shown that provided the Hamiltonian flow and the observation set satisfies the Geometric Control Condition, then the solution to the Schrödinger equation (1.1) with an $L^{\infty}$ potential is controllable to another state $u_{T}=u(T,x)$. Later using Carleman estimates, [26, 1] and [27, 28] reduced the assumptions on the domain for $L^{\infty}$ time independent and time dependent potentials, respectively, c.f. [44] for a survey of such results. If a solution is observable, then often by duality the solution to the adjoint equation is controllable [30]. Interior observability typically amounts to a solution $u$ satisfying an estimate of the form

$\displaystyle\|u_{0}\|_{L^{2}(D)}\leq C_{T}\|u\|_{L^{2}((0,T)\times D_{0})}$

for some nonzero $C_{T}$, $0<T<\infty$ ($f=0$). Most of the references mentioned here establish observability estimates.

While much is known about the observability problem for Schrödinger equations on a bounded domain with $L^{\infty}$ potentials, much less is known for general operators in the free space, that is when (1.1) has $D$ replaced with $\mathbb{R}^{d}$ and $D_{0}$, the support of the control, is replaced with $\omega=\mathbb{R}^{d}\setminus\overline{\Omega}$ where $\Omega$ is a bounded domain. Correspondingly, one wants to establish observability for

		$\displaystyle i\partial_{t}u=\left(-\kappa_{1}(t)\Delta+\kappa_{2}(t)|x|^{2}\right)u\quad 0<t\leq T,\quad x\in\mathbb{R}^{d}$		(1.2)
		$\displaystyle u(0,x)=u_{0}(x).$

where $\kappa_{1}\in C^{1}([0,T]),\kappa_{2}\in C([0,T])$, and $u(0,x)$ is real valued and in a weighted $L^{2}(\mathbb{R}^{d})$ space. This amounts to establishing a bound for the solution of (1.2) as

$\displaystyle\|u_{0}\|_{L^{2}(\mathbb{R}^{d})}\leq C_{T}\|u\|_{L^{2}((0,T)\times\omega)}$

which is challenging as $\omega$ is no longer bounded. While it seems that this would be an easier problem because the background space is $\mathbb{R}^{d}$, the potential itself is also not compactly supported. The Schrödinger operator in the free space with a non-compactly supported potential behaves much differently. The only results for control and observability for this problem in arbitrary dimensions are in [35] for the case of the simple harmonic oscillator $\kappa_{1}(t)=\kappa_{2}(t)=1/2$. Our goal is then to establish observability for a more general class of time-dependent harmonic oscillators. We end up establishing an approximate observability theorem for a limited class initial data $u_{0}$, and an observability estimate for some examples of ’Gaussian-like’ initial data. Approximate observability here means that $u(t,x)$ satisfies

$\displaystyle\|u_{0}\|_{L^{2}(\mathbb{R}^{d})}-\eta\leq C_{T}(\|u\|_{L^{2}((0,T)\times\omega)}+T\eta)$

(1.3)

where $\eta>0$ can be made very small. This error term has to do with the fact that translated real Gaussians are dense in $L^{1}(\mathbb{R}^{d})$ and hence can approximate $L^{2}\cap L^{\infty}(\mathbb{R}^{d})$ functions, but are not a basis for either space. Whenever $\eta$ depending on the $L^{\infty}(\mathbb{R}^{d})$ norm of $u_{0}$ is small enough that the $\eta$ ’errors’ in (1.3) are absorbed into the $L^{2}(\mathbb{R}^{d})$ norm of $u_{0}$ we call this an $L^{2}\textendash L^{\infty}(\mathbb{R}^{d})$ observability estimate.

The underlying idea here is the use of sophisticated Gaussian wave packets. We prove an explicit approximation theorem which decomposes $f\in L^{2}(\mathbb{R}^{d},e^{x^{2}})$ into a finite sum of Gaussians up to some quantifiable error, which is based on [5]. Thus there are two main goals here, one is to construct a nearly explicit parametrix using this decomposition and the other is to prove some localization properties of solutions to the Schrödinger equation in the form of observability estimates. While a least squares approximation may require fewer terms with a higher degree of accuracy, we give an explicit characterization of the decomposition needed to create a completely explicit parametrix with a high degree of accuracy. This parametrix in Theorem 2 was largely inspired by the Fourier integral operators developed in [38]. The derivation of the theorem constructing the parametrix is based on properties of Hermite functions and polynomials.

To see why this is important, we remark that any quantum mechanical system with a potential energy $V(x)$ has local equilibrium points which can be analysed by the model for a quadratic Hamiltonian, (1.2) with $\kappa_{1}(t)=\kappa_{2}(t)=1/2$. In other words, Taylor expanding $V(x)$ around the point $x_{0}$ gives:

$\displaystyle V(x)\approx V(x_{0})+\nabla V(x_{0})(x-x_{0})+\frac{1}{2}\nabla^{2}V(x_{0})(x-x_{0})^{2}.$

If $x_{0}$ is a critical point, then the second term vanishes. Translating $x_{0}$ to zero we obtain:

$\displaystyle V(x)\approx V(0)+\frac{1}{2}\nabla^{2}V(0)x^{2},$

and we see that the model is reduced to the one of the harmonic oscillator. The Hamiltonian associated to this operator is $\frac{1}{2}(|p|^{2}+|x|^{2})$. The Hamiltonian ray path for this operator is computed by solving the system of ODE’s

$\displaystyle\frac{dx(t)}{dt}=p(t)\quad\frac{dp(t)}{dt}=-x(t).$

The key point is that this operator has a cusp at zero, e.g. one point for which the Hamiltonian ray path is such that $\frac{dx(t)}{dt}=0$ which can affect the validity of a Gaussian wavepacket construction. Making sense of this phenomenon and the regions where a solution to a much more general time-dependent classical problem (1.2) are concentrated is one of the goals of this paper. The examples presented here all represent toy models of time dependent metrics and potentials $V(t,x)$ in the free space background case around spatial equilibrium. Establishing parametrices and observability for such potentials is key to understanding the behavior of free space quantum mechanical waves.

The use of Gaussian wavepackets has been around since the work of [16] and [11]. For the semi-classical Schrödinger equation in the free space

		$\displaystyle ih\partial_{t}u=(-h^{2}\Delta+V)u\quad\mathrm{in}\quad(0,T)\times\mathbb{R}^{d}$		(1.4)
		$\displaystyle u(0,x)=u_{0}(x)$

where $u_{0}\in C^{\infty}(\mathbb{R}^{d})$, the leading order term of $u$ up to a high degree of accuracy in $h$ has been shown to take the form of a Gaussian Fourier Integral Operator (FIO). This description holds for very general time-dependent Hamiltonians see [8, 9, 10, 31] and also [25], and finite times $T$. However for the classical Schrödinger equation the situation is a bit different as we do not have the added advantage of calculating errors in terms of the scale of the semi-classical parameter $h$. For control theory results in the semi-classical case, we direct the reader to [32, 33, 34].

We still have the property in the classical case presented here that the FIOs which are constructed for the non-autonomous problem are Gaussian distributions when the Hamiltonian is quadratic c.f. [38] for a full treatment. Hence the reduction to the model in (1.2). A Gaussian FIO applied to a Gaussian function is again, another Gaussian, which motivates our choice of approximation. The FIO construction now allows us to create a parametrix solution to the Schrödinger equation consisting of a finite collection of tractable propagated wavepackets whose properties, while technical, are not impossible to describe. Some same principles from the compact case related to the Geometric Control Condition still carry over. Moreover, because the representation used here is very close to the classical Hermite functions, the analysis of inner products on $L^{2}(\omega)$ spaces has a direct correlation to spectral theory analysis. Using separation of variables it is usually desirable to prove $\langle\psi_{n},\psi_{m}\rangle_{L^{2}(\omega)}$ is small in order that the $\|\psi_{n}\|^{2}_{L^{2}(\omega)}$ terms dominate for some $\psi_{n}$ a spectral basis for the underlying space ($\mathbb{R}^{d}$, in this case and $\omega=\mathbb{R}^{d}\setminus\overline{\Omega}$ with $\Omega$ a bounded domain). We see the analogue of this idea directly for the dynamic wavepackets in Lemma 6 and the proof of Theorem 4 in the text. Unfortunately this means we have to restrict the class of initial data for the proofs.

What the theorems here say is that initial data which can be approximated by Gaussians (including some piecewise $C^{1}$ compactly supported functions which have been extended from characteristic functions) are $L^{2}\textendash L^{\infty}(\mathbb{R}^{d})$ observable far away from the ’hole’, $\Omega$ and only initial data which is very nearly a Gaussian is observable when the support of the bulk of the ’mass’ ($L^{2}(\mathbb{R}^{d})$ norm) of the initial data sits over the ’hole’. This makes sense intuitively as initial data which is Gaussian propagates as another Gaussian. Far away from the ’hole’ we see most of the mass of the Gaussian if it started off there, and near to the complement of $\Omega$, the ’hole’, we recover almost nothing if the mass started over $\Omega$. We give some explicit examples of such initial data and $\kappa_{1}(t)$ and $\kappa_{2}(t)$ in Section 7. The various values of $\kappa_{1}(t)$ and $\kappa_{2}(t)$ can make the behaviour of the position and the spread of the parametrix quite different even though the overall shape is still Gaussian in the spacial variable. It is curious that in the case of the $1d$ harmonic oscillator $\kappa_{1}=\kappa_{2}=1/2$ that observability on these types of domains $\omega$ is true for all $f\in L^{2}(\mathbb{R})$ if $T\geq\pi$ because the solutions are periodic in time, c.f, Theorem 2.2 [3]. Therefore in this case our construction still has a gap to be filled because the parametrix here is only valid until $T=\frac{\pi}{2}$ for $\kappa_{1}=\kappa_{2}=1/2$. While this parametrix could be extended to $T\geq\pi$ the problem is still that the class of data which is approximately observable depends on location of its support. This gap in our proof techniques cannot easily be closed because it would involve analyzing lower bounds on $\mathrm{erfc}(z)$ for complex $z$. However there are no known results on observability when $\kappa_{1}(t)$ and $\kappa_{2}(t)$ both depend on time.

For the free Schrödinger equation and classical harmonic oscillator, the basis of Hermite functions and principles of spectral decomposition have already resulted in our Theorem 3 with $\eta=0$ in [36, 35, 21, 41, 37]. However one does not expect these techniques to extend themselves to parametrices or observability in the case of time dependent operators as these are purely time separable techniques. Indeed, to this end some counterexamples are shown to established principles for the time independent simple harmonic oscillator in Example 2. Therefore the theorems here to try new techniques which may be applicable to a more general class of operators which are not accessible with spectral theory directly.

In this section we construct explicit examples of FIO solutions to (2.1) which are only abstractly constructed in [38]. Our treatment is similar to [15], c.f. also [18].

Theorem 1.3 in [38] then gives that the FIO is unitary when the flow is non-zero. This statement also follows by direct calculation in our case. Notice that the second assumption in the definition is equivalent to $a(t)^{2}<\infty$.

We would like to use the following Hermite inspired expansion for all $f(x)$ in $L^{2}(\mathbb{R},e^{x^{2}}\,dx):$

$\displaystyle f(x)=\sum\limits_{n=0}^{\infty}b_{n}\frac{d^{n}}{dx^{n}}e^{-x^{2}}$

where

$\displaystyle b_{n}=\frac{1}{2^{n}n!\sqrt{\pi}}\int\limits_{\mathbb{R}}f(x)e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}\,dx.$

Indeed, recall that the Hermite polynomials are defined as

$\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}.$

(4.1)

Then the set of Hermite functions

$\displaystyle\left\{h_{n}(x)=\frac{1}{\sqrt{2^{n}n!\sqrt{\pi}}}H_{n}(x)e^{-x^{2}/2}:n\in\mathbb{N}\right\}$

(4.2)

is a well known orthonormal basis of $L^{2}(\mathbb{R})$. For any $g\in L^{2}(\mathbb{R})$ it is possible to write

$\displaystyle g(x)=\sum\limits_{n=0}^{\infty}d_{n}h_{n}(x)$

where

$\displaystyle d_{n}=\int\limits_{-\infty}^{\infty}g(x)\overline{h_{n}(x)}\,dx.$

Applying the representation with $g(x)=f(x)e^{x^{2}/2}$ gives the desired formula for the $b_{n}$, and moreover we have then the following relationship:

$\displaystyle b_{n}=d_{n}\frac{(-1)^{n}}{\sqrt{2^{n}n!\sqrt{\pi}}}.$

(4.3)

Now we want to approximate the derivatives of $e^{-x^{2}}$ in $L^{2}(\mathbb{R})$. This proposition is loosely based on Proposition 2 in [5], which in which the errors are done in a different norm only for compactly supported functions.