Asymptotics for Optimal Design Problems for the Schrödinger Equation with a Potential
We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrödinger equation with a potential in , with , using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.
Let be a bounded domain with boundary . Let and be a measureable subset of . We consider the Schrödinger equation with Dirichlet boundary conditions:Here, , , and . In some instances, we will require higher regularity, but this will be specified when necessary.
Let indicate the Laplacian on the space . This operator is a symmetric operator acting on associated with the quadratic form In particular we recall that the quadratic form is closable with respect to the norm The domain of the closure is the Sobolev space . We can define the Dirichlet Laplacian via this extension procedure and moreover,If is a bounded domain with boundary of class then
All of the functions of this operator are interpreted via the Hilbert space functional calculus. In particular, is unitary, and we exploit this property to build our parametrices. The representation of the solutions presented here in the case of an added potential is new and relies on applications of advanced spectral theory.
If we consider the Schrödinger equation on a bounded domain of with Dirichlet boundary conditions, then observing the restriction of the solutions to a measurable subset of during a time interval with is known as observability. Equation (1) is observable on in time if there exists such thatIn previous literature, the above inequality is called the observability inequality when .
It is well known that if the pair satisfies the observability inequality (6), then the energy of the solutions can be estimated in terms of the energy which is localized in . The search is then for the conditions on for which one can find the largest possible nonnegative constant for which inequality (6) holds.
We denote the observability constant by to be the largest constant such that (6) holds. The constant can also be formulated as
The study of the observability constant is important, since it gives an account for the well-posedness of the inverse problem of reconstructing from measurements over . In addition, we denote as the constant associated with the Schrödinger equation without a potential. The main novelty of the paper is the analysis in the case of an added potential .
We now connect the theory to a possible real-life application. Assume that is a cavity in which signals are propagating according to (1). To measure the propagating signals, one is allowed to place a few sensors into the cavity. We now assume that, in addition to the placement of the sensors, we are allowed to choose their shape. Therefore, the problem is now of determining the best possible location and shape of the sensors, which will obtain the best observation. Of course, the best choice is to observe the solutions over the whole domain . However, in practice, the domain scanned by the sensors is usually limited, for reasons such as the cost of such an operation. To make this limitation more mathematically precise, we consider measurable subsets of fixed size, i.e., subsets of such that , where . The subset represents the sensors in , and they are able to measure restrictions of the solutions of (1) to .
Therefore, one and the most obvious way to model the problem of best observability is that of finding the optimal set which maximizes the functional over the set However, we show that this problem not only is inherently difficult to solve, but is not so relevant in practice. Thus, we consider several modifications and simplifications of the model, to be described in the next section.
Optimal observation problems are found in numerous engineering applications, thus providing the motivation for our study. Examples include acoustics, piezoelectric actuators, vibration control in mechanical structures, damage detectors, and chemical reactions [1–5]. The goal is to optimize the type and place of the sensors in order to improve the estimation of the overall behavior of the state of the system.
The main contributions of the paper are the following:(1)We present an elegant way to model the problem of best observability using the time asymptotic observability constant . We analyze the largest possible , over all , and we develop conditions analogous to the quantum unique ergodicity conditions in  for this constant to hold.(2)We demonstrate the conditions on the existence of a positive asymptotic observability constant for an arbitrary subset of and , under certain requirements on the potential. Our results are supported by numerical experiments.
Remark 1. The paper  considers a variety of boundary conditions, but we focus on how to treat the problem with a potential, so we simply impose Dirichlet boundary conditions. Different boundary conditions will be the subject of future study. One could examine the problem on a compact Riemannian manifold , such that has a boundary, and use the Laplace Beltrami-operator , and many of the same results would still hold. However, we let be a subdomain of for simplicity.
2. Statement of the Main Theorems
Consider the eigenvalues and the corresponding eigenfunctions for on . Let and denote the eigenvalues and corresponding eigenfunctions of on . For the rest of this article we drop the subscript for the Dirichlet Laplacian.
We assume the are orthonormal and give references to classical spectral theory results which show that they can be used as a basis for . The solution of (1) can then be represented aswhere is the initial data to the solution . The sequence is determined in terms of as Moreover,
Ifthen plugging in expansion (9) yieldswithwhenever and whenever .
We notice that the determination of the observability constant is now a difficult spectral problem involving many inner products of eigenfunctions over the set . Moreover, it is limited in practice, since the observability constant defined by (7) describes the worst possible case, which may not occur often in applications. In order to examine the problem further, one can consider the following simplifications:(1)One can examine the problem of maximizing over all possible measurable subsets , given fixed initial data. In this case, if the optimal set exists, it depends on the initial data that is considered. This problem is still challenging, and also not relevant enough in practice, since initial data is not expected to be fixed, but uniform in nature. Therefore, we focus on the following second simplification, where all initial conditions are taken into account.(2)One can instead consider a time asymptotic observability constant , as in . The constant is defined as This constant is the nonnegative constant for which the time asymptotic observability inequality holds for every . This is where we use the additional assumption so the constant is well-defined. If is of class then this is the entire domain anyway. Shortly, we will show that the time asymptotic observability constant is equal to the randomized observability constant Note that, from the definition of the observability constant, one obtains the following inequality:
The randomized observability constant can be derived in the following way. We introduce a field of i.i.d random variables which we use to multiply the values of the initial data. ThenNote that (19), under the conditions is exactly .
We now state the main theorems of the paper. All of the theorems in this section are formulated for the Schrödinger equation with a potential, which is the main novelty. The first two results concern an expression for time asymptotic observability constant .
Theorem 2 (analogue to heorem 2.6 ). For every measureable subset of ,where is the set of all distinct eigenvalues and .
The proof of the theorem is in Section 4. If we setthen similarly to heorem 1 in , we have the following.
Corollary 3 (analogous to orollary 2.7 in ). The inequality is true for every measurable subset of . If the domain is such that every eigenvalue of is simple, thenfor every measurable subset of . This shows that the off-diagonal terms in the eigenfunction expansion contribute less in the infinite time asymptotic regime.
The more difficult problem is using known results from perturbation theory to find a nonzero observability constant. We show that if for some , then, under certain conditions on , we can find a positive time asymptotic observability constant for the Schrödinger equation whenever the corresponding operator without the potential () has one.
For Theorem 4 it is assumed that the potential has regularity . Let be a constant which depends uniformly on the diameter of and the norm of . This constant will be derived and given explicitly during the course of the proof. We prove the following.
Theorem 4. We assume that and on with Dirichlet boundary conditions both have simple spectra, for all with fixed sufficiently small. When , , and , the constant is such that if and only if for the Schrödinger equation with .
The proof of Theorem 4 is in Section 7. Moreover, in Section 7, we discuss why the assumption that the spectra are simple is spectrally sharp, as there are counter-examples to the statement of Theorem 4 for nonsimple spectra given as a result of [7–9]. There is also an appendix on convergence of numerical algorithms using these functionals.
We also consider a relaxation of the problem. In particular, let be the convex closure of the set in the weak star topology:
We setand alsoWe then have the following.
Theorem 5. Let with , with no assumption on the support of in . We assume and with Dirichlet boundary conditions both have simple spectrum, all with fixed sufficiently small. It follows that for any with a constant depending only on norm of and . As a consequence, we can conclude
The proof of Theorem 5 is in Section 7. Notice that we cannot show iff is zero because we do not have such fine control over the terms unless .
In the last section, using existing software, we show explicit computations and an explicit representation of the observability constant for a variety of potentials including a damped harmonic oscillator. While the problem for the nonlinear Schrödinger equation has been investigated from the control theory standpoint [10–13], to the best of the authors knowledge, the problem of observability for potentials has not been addressed in an explicit way using eigenfunctions and numerical methods. Observability for the linear Schrödinger equation was examined in . Our analysis extends their results in the linear case.
2.1. Comparison with Previous Literature
Let be any nonempty open set and ; then there exists a constant such that for any we haveor a constant such that for any depending on the domain of the operator.
In general the work of Lebeau  showed that control (the dual statement to the existence of positive constant or ) for the Schrödinger equation with or without the potential holds under the Geometric Control Condition (GCC):(i)There exists such that every geodesic of length on intersects .
Therefore if we let denote the constant with the potential, then as soon as the GCC is satisfied. The GCC is also necessary in the case of a smooth potential when the geodesic flow is periodic . For the flat torus, Jaffard  and Haraux  in 2D and Komoronik  in higher dimensions have shown that this not necessary: observability holds for any open set . Their work was extended to operators with smooth potentials in [19, 20], and also for higher dimensions and time-dependent potentials in , and for irrational torii and general Schrördinger operators in . One can see  for a literature review and extension to hyperbolic manifolds.
We look at the constant given by (7), which we are examining to be a different observability constant when the potential is present and this is distinct from that examined in previous literature. However it is closely related to context analyzed in  which is also done for time dependent potentials on the flat disk and other works. Therefore, the main goal here is to identify in which sense the randomized observability constant with the potential and that without are close.
When there is no potential, our formulation of the observability constant coincides with the definition (31). Indeed, for our formulation, one can rewrite (7) asThe positivity of this constant is not directly equivalent to the other two when is nonzero, as does not commute with . The only time the existence of the constants and could imply the positivity of directly is when the potential is positive. However, in the important aforementioned literature [8, 9, 15, 21, 22, 24], there are several cases in which conditions that ensure the positivity of these constants are equivalent-manifolds with periodic geodesic flow, flat tori, and the Euclidean disk. In all these cases, the geometric conditions on the observation set do not depend on the presence of the potential, regardless of whether or not this potential is positive or not, c.f. the introduction to .
Moreover, Theorem 4 is proved for the randomised observability constant (otherwise known as the observability constant for eigenfunctions ). It is doubtful such a strong statement is true for the full observability constants and as the presence of cross terms in (14) is difficult to control when are large. Once again, as in , the randomized constant can be viewed as the optimistic best case scenario.
Since Theorem 4 is only true in the case of sufficiently small and regular potentials of compact support, this shows that even in the case of randomised initial data the observability constants (eigenfunction observability constants) can be very close for strong conditions relating and . It is not that the eigenfunction constants cannot be close for not contained in ; it is just that the current technique gives much less information about controlling the constants in terms of each other. Hence, Theorem 5 has a weaker formulation of the relationship of the relaxed constant with dependent potential to the original one , and there is no assumption on the support of with respect to . In general, showing observability for randomized initial data (otherwise known as observability of eigenfunctions) is possible under conditions on the observation region which are independent of for generic potentials, c.f. .
The main tools in this article are opposite those of the general tract of semiclassical analysis papers. Previous techniques take advantage of the spectral theorem to turn the high frequency eigenvalues into the semiclassical parameter . Heuristically as a semiclassical operator has symbol , while has symbol but in the latter case the Hamiltonian ray path over which solutions are concentrated can be made sufficiently close to that of if is sufficiently small, as long as which is proved in , emma 8.3. The methodology in  fails here because approximate solutions can only be constructed under a nontrapping condition.
Because we are exploiting the small parameter , we use classical perturbation theory techniques rather than semiclassical analysis. Here we see that classical perturbation theory gives new information in the case when the eigenvalues are simple, which cannot be explained by entirely semiclassical techniques. Moreover the results are applicable to any eigenfunction/eigenvalue pair, not just the high frequency ones.
However, in this particular case examined in this article, if we rescale so that , then the eigenvalue/eigenvector problem becomes , with symbol , which in the case of the two-dimensional flat disk, and the surface of a sphere, can be solved almost explicitly using semiclassical methods to a high degree of success, c.f. [7, 8], corresponding to high frequency eigenvalues in this scenario. In other geometries this is not the case, and these are the settings which we seek to begin to resolve in this article.
3. Review of Spectral Theory
Suppose is a bounded domain in . Then, as in the introduction the Laplace operator with Dirichlet boundary conditions can be defined as the self-adjoint operator with the quadratic form with domain . Because the space is compactly embedded in by Rellich’s theorem, the spectrum of this operator is purely discrete and has infinity as its only possible accumulation point, c.f.  for a review. Hence, there exists an orthonormal basis in consisting of eigenfunctions with eigenvalues , which we assume to be ordered:
Recall that a linear subspace of the domain of a closed quadratic form is called a core for if is the closure of its restriction to . We now recall the following result from .
Theorem 6 (hm 8.2.1 in ). If and is a domain in , then the quadratic form which is defined onis the form of a nonnegative self-adjoint operator . The space is a core for .
Remark 7. We could reduce the assumption on the potential from to using the above theorem in many of the following sections.
We also require the following useful result on self-adjoint operators from the same monograph .
Theorem 8 (heorem 8.2.3, orollary 4.4.3, ). If is defined on by , where , then is a self-adjoint and bounded below with the same domain as .
We also have the following.
Theorem 9 (hm 6.3.1 in ). For all bounded domains , the operator has an empty essential spectrum and compact resolvent. The eigenvalues of written in increasing order and repeated according to multiplicity satisfyfor some depending only on the geometry of and .
As such, and can be made arbitrarily close to one another, if is large, c.f. the proof of heorem 6.3.1 in . The eigenvalues of depend monotonically upon the region and so can be bounded above and below by the eigenvalues of the cubes which are contained in (and, respectively, contain) . It follows from Theorem 8 that . From this fact and the spectral theorem, we can conclude from Theorem 6 the following.
Corollary 10. For , ifwith an orthonormal Hilbert basis of consisting of eigenfunctions of the Dirichlet operator on , which is associated with the eigenvalues , then we can write the propagated solution aswith
We use the basis properties in Corollary 10 in the next section.
4. Proof of Theorem 2
The basic idea is to useas the decomposition for the solution of (1), where are the eigenvalue and eigenfunction pairs for the operator. One can apply similar steps to  to prove Theorem 2. Using a standard density argument, the approximation which holds over a finite number of modes,is enough to describe an observability constant which is valid in the large-time regime. Then, we use previously derived facts about perturbation theory to prove the other theorems.
Proof of Theorem 2. We start with the case when has simple eigenvalues. This proof is a simplification of the analogous theorem in  which is presented for the wave equation and applicable to the Schrödinger equation with no potential. Without loss of generality, one can consider initial data such that . Then, letandThen, can be expressed asNote thatNow, we use the assumption that the spectrum of consists of simple eigenvalues to prove the following result.
Lemma 11. The following equation holds:Because the sum is finite, one can invert the inf and the limit. We havewhere was given previously by (14). Formula (14) givesfor every . We note thatdue to the fact that , for all and Theorem 9. We now estimate the remainder terms of (45):andUsing the fact that form a Hilbert basis,To bound ,whenever the normalization is used. By Parseval’s theorem, since , for every , there exists an such thatWe conclude that, for sufficiently large ,Since was arbitrary and , the theorem is proved. The corollary follows since, due to the assumption that , we haveNote that, in the case of nonsimple eigenvalues, one can group the diagonal terms to obtain the desired result. This proves Theorem 2 and Corollary 3.
5. Basic Perturbation Theory
In this section, we give an explicit example of how to calculate the eigenvalues and eigenvectors of the perturbed operator , with simple eigenvalues . (Recall that this means the eigenvalues have multiplicity 1.) In the next section, more advanced results from  will be used to analyze the error terms.
Let denote the standard Laplacian with eigenvalues . There exists a corresponding basis such thatThe following lemma relates the eigenvalues and eigenvectors of to those of .
Lemma 12. The eigenvalues to are given byThe eigenfunctions to are given byHere the terms are uniform in depending on and the norm of . In particular we have thatandwhere , depend only on the geometry of and the norm of .
We do not prove the Lemma here; it is a result of  (see equation in xample 3.6, where the constant is given explicitly); we only give an idea of why it is true. One will see that the results in the next section are more general. If we make the approximation then it follows by substitution that Equating the leading order terms, At order , we haveThe desired result for computing the first terms follows by taking the inner product of (68) with for . We have to have a way of encoding this inductive process of matching up the terms. In the next section we introduce the operators and which allow us to do just that. The terms are computed for eigenfunctions, but the analysis is more sophisticated because, when computing the result of the matching over , one loses the orthogonality of the eigenfunctions over the region of integration.
We have the following example of an operator with simple eigenvalues.
Example 13. We consider the eigenvalue problem with the unperturbed problem iswith simple eigenvalues , with with corresponding normalized eigenfunctions The quadratic form associated with the potential with domain is closed in . The unperturbed operator is stable with respect to perturbations . This is the assumption on both of the main theorems (Theorems 4 and 5). This example is from . Stability of for the unperturbed problem means that, for sufficiently small, the intersection of any isolating interval for and the spectrum of the perturbed operator consists only of simple eigenvalues. The unperturbed/perturbed operator pair here satisfies the criterion of heorem 5.1.12 in  for stability which holds provided the left hand side (86) is smaller than , which is true for sufficiently small . This also applies to the first numerical example in the Appendix. Usually stability is automatically satisfied when has simple spectrum and is sufficiently small, c.f. emma 2.1 in .
As a general remark on the example and computations above, the difficulty lies in quantifying the error terms which are usually formulated in the sense of not , which is why the next section is required.
We have the following result for more general Riemannian metrics which shows that the assumption of simple spectrum in our case covers generic domains . Symmetry usually destroys the assumption of spectral simplicity, c.f. , and this is also discussed in Section 7.
Theorem 14 (see  and ). Let be a compact manifold of dimension greater than 1 and a conformal class of Riemannian metrics of fixed volume on . Given and , the subset of of metrics for which the eigenspace is of dimension is a submanifold of codimension of at least 1. In particular, the subset of of metrics admitting a nonsimple eigenvalue of the Laplacian is a countable union of submanifolds of codimension of at least .
This theorem asserts that for a given compact manifold “most” Riemannian metrics on are simple, meaning the eigenspace of the Laplace operator is one-dimensional and that this set is pathwise connected. The proof naturally remains true for order perturbations, like the ones we have here, c.f. . We leave the question of what happens to the observability constants for metric perturbations to future work.
6. Advanced Perturbation Theory
In this section, we elaborate on advanced perturbation theory for a better understanding of the results derived in the paper. Let be an arbitrary Hilbert space, as in , and be the range of the bounded operator . The monograph  by Kato computes perturbation theory results for generic bounded operators , and since our operator satisfies the conditions in  for a Type (A) holomorphic operator in the parameter (heorem 2.6 of ), the perturbation theory derived in the book applies.
Let be the projection operator and be one of the eigenvalues of , , and let , , be the eigenvalues and eigenprojections of different from and under consideration. Let denote a basis of and denote a basis of for each . The union of the vectors and forms a basis of consisting of eigenvectors of and is adapted to of . The adjoint basis of is adapted to , where , , etc. Let denote the adjoint basis of and denote the basis of for
For any ,We define the operator as the value of the reduced resolvent of , such that , and . are the orthogonal projections such thatand moreover, by definition, . For in our particular eigenspace, it follows that one can write the operator explicitly asusing the definitions and ection II.2 in . If we expand , which is an eigenvalue of , in a perturbation series asone obtains the following expressions for the expansions of the eigenvalues (II-(2.35) ):Suppose that the eigenvalue of of is simple, implying that . To derive an expansion for a particular eigenvector eigenvalue pair, one can set and as in the last section. (Now just refers to the index of the eigenfunction, a distinct index from the one above.) The operators and can be written asThis substitution compares immediately to the results in the previous section for the expansion of the eigenvalues (61). Now we describe a more advanced decomposition of the eigenvectors.
Assuming for simplicity that , a convenient form of the eigenvector of corresponding to the eigenvalue is given bywhere is the unperturbed operator of for the eigenvalue and is the eigenvector of the adjoint operator . is the projection onto the eigenspace of . The assumption of stability here is used in a hidden way as we want to make sure the projection onto the eigenspace is well-defined. In particular the projection is defined as the integral of the resolvent over an interval containing only one eigenvalue. As such, in order for the projection to be well-defined, the eigenvalue needs to be sufficiently isolated, whence the assumption of simplicity in a perturbed neighbourhood of . We refer the reader to heorem 5.1.12 in  and emma 2.1 in  for a precise description of , the threshold required. In the case of nonsimple eigenvalues the representation above would depend on more than one , which would be difficult to analyze. We suppress the subscript in the operators and where it is understood. This gives rise to the following normalization conditions:The relation can be rewritten aswhere . Multiplying (80) from the left hand side by and noting that ,Moreover, as and writing in the last term above, one getsfor sufficiently small , with , and is an arbitrary scalar. Equation (82) is formula in .
One can then compute
The asymptotics for the scalar are well worked out for small . Let , , , for any , where we use the operator norm. A subscript will denote the set over which the operator norm is taken.
For a linear operator acting on , we let denote the norm
Set to have norm and defineAs a resultwhich is formula (II-3.18) in , with the norm . The expansion (83) derived above is given in section II and exercise II-3.16 in the monograph by Kato .
Now the difficulty comes in computing inner products of over the smaller sets where one loses the powerful orthogonality conditions. We recall the following well-known Lemma on von Neumann series.
Lemma 15. Let be a linear operator on the Banach space . We then haveprovidedwith .
c.f. emma 2.1 in .
In order to compute (83), we want to use Lemma 15 to essentially find a convergent von-Neumann series forwith (86) so that we may obtain precise bounds on the rate of decay of the inner products . These arguments are rather delicate as we are not integrating over the whole . We let denote a generic constant that depends on the volume of and We state the four necessary Lemmas first, followed by their technical proofs to see how the pieces fit together to allow us to use Lemma 15 by examining each term in the series expansion to bound (89).
Lemma 16. With no assumptions on the support of the potential, we have the following estimate for with
Let be the linear operator defined as multiplication by
Lemma 17. With no assumptions on the support of the potential, we have the following estimate for with for all , ,
Lemma 18. If and , then the operator is bounded .
Lemma 19. There is a choice of sufficiently small, such that, for all , the following inequality holds:
Proof of Lemma 16. By the Cauchy Schwartz inequality, we haveWe know from heorem 3.4 from  thatwhere