# Asymptotics for Optimal Design Problems for the Schrödinger Equation with a Potential

Authors: Alden Waters; Ekaterina Merkurjev
Journal: Journal of Optimization
 (2018)

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## Abstract

We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrödinger equation with a potential inL∞Ω, with Ω⊂Rd, 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.


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## Body

## 1. Introduction

LetΩ⊂Rd be a bounded domain with boundary ∂Ω. Let T>0 and ω be a measureable subset of Ω. We consider the Schrödinger equation with Dirichlet boundary conditions:(1)i∂tu=Δu-Vxuu0,x=u0xut,xx∈∂Ω=0.Here, u:R×Ω↦C, V(x)∈L∞(Ω), and u0(x)=u(0,x)∈H01(Ω)∩H2(Ω). In some instances, we will require higher regularity, but this will be specified when necessary.Let-Δ indicate the Laplacian on the space C0∞(Ω)⊂L2(Ω). This operator is a symmetric operator acting on L2(Ω) associated with the quadratic form (2)Q0:H01Ω→0,∞Q0f=∫Ω∇fx·∇fx¯dx.In particular we recall that the quadratic form is closable with respect to the norm (3)QDf=Q0f+fL2Ω1/2.The domain of the closure QD(f) is the Sobolev space H01(Ω). We can define the Dirichlet Laplacian -ΔD via this extension procedure and moreover,(4)Dom-ΔD1/2=DomQ=H01Ω.If Ω is a bounded domain with boundary ∂Ω of class C2 then(5)Dom-ΔD=H01Ω∩H2Ω.All of the functions of this operator are interpreted via the Hilbert space functional calculus. In particular,exp⁡(itΔD) 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 V is new and relies on applications of advanced spectral theory.If we consider the Schrödinger equation on a bounded domainΩ of Rd with Dirichlet boundary conditions, then observing the restriction of the solutions to a measurable subset ω of Ω during a time interval [0,T] with T>0 is known asobservability. Equation (1) isobservable on ω in time T if there exists C>0 such that(6)C∂tu0,xL2Ω2≤∫0T∫ω∂tut,x2dtdx.In previous literature, the above inequality is called the observability inequality when V=0.It is well known that if the pair(ω,T) satisfies the observability inequality (6), then the energy of the solutions can be estimated in terms of the energy which is localized in ω×(0,T). The search is then for the conditions on ω for which one can find the largest possible nonnegative constant for which inequality (6) holds.We denote theobservability constant by CTV(χω) to be the largest constant such that (6) holds. The constant can also be formulated as(7)CTVχω=inf⁡∫0T∫ω∂tut,x2dxdt∂tu0,xL2Ω2∣u0,x∈H01∩H2Ω.The study of the observability constantCTV(χω) is important, since it gives an account for the well-posedness of the inverse problem of reconstructing u from measurements over [0,T]×ω. In addition, we denote CT0(χω) 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 V.We now connect the theory to a possible real-life application. Assume thatΩ⊂Rd 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 Ω. 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 ω of Ω such that ω=LΩ, where L∈(0,1). The subset ω represents the sensors in Ω, and they are able to measure restrictions of the solutions of (1) to ω.Therefore, one and the most obvious way to model the problem ofbest observability is that of finding the optimal set which maximizes the functional χω→CTV(χω) over the set (8)ML=ω⊂Ω∣ωismeasurableandofLebesguemeasureω=LΩ.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 constantC∞V(χω). We analyze the largest possible C∞V(χω), over all ω∈ML, and we develop conditions analogous to the quantum unique ergodicity conditions in [6] for this constant to hold.(2)
We demonstrate the conditions on the existence of a positive asymptotic observability constantC∞V(χω) for an arbitrary subset ω of Ω and T>0, under certain requirements on the potential. Our results are supported by numerical experiments.Remark 1.
The paper [6] 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 M, such that (M,g) has a boundary, and use the Laplace Beltrami-operator Δg, and many of the same results would still hold. However, we let Ω be a subdomain of Rd for simplicity.


## 2. Statement of the Main Theorems

Consider the eigenvalues(λj)j∈N and the corresponding eigenfunctions ϕj(x) for -Δ+V(x) on Ω. Let (λj0)j∈N and ϕj0(x) denote the eigenvalues and corresponding eigenfunctions of -Δ on Ω. For the rest of this article we drop the subscript D for the Dirichlet Laplacian.We assume theϕjx′s are orthonormal and give references to classical spectral theory results which show that they can be used as a basis for H01(Ω). The solution of (1) can then be represented as(9)ut,x=∑j=1∞cjexp⁡iλjtϕjx,where u(0,x)∈H01(Ω) is the initial data to the solution u(t,x)∈C0((0,T);H2(Ω)). The sequence (cj)j∈N∗∈l2(C) is determined in terms of u(0,x) as (10)cj=∫Ωu0,xϕjx¯dx.Moreover, (11)∂tu0,xL2Ω2=∑j=1∞λj2cj2.If(12)GTVχω=∫0T∫ω∂tut,x2dxdt,then plugging in expansion (9) yields(13)GTVχω=∫0T∫ω∑j=1∞λjcjexp⁡iλjtϕjx2dxdt=∑j,k=1∞λjλkcjc¯kαjk∫ωϕjxϕkx¯dxwith(14)αjk=∫0Texp⁡iλj-λktdt=2λj-λkexp⁡(iλj-λkT-1,whenever j≠k and αjj=T whenever j=k.We notice that the determination of the observability constant is now a difficult spectral problem involving many inner products of eigenfunctions over the setω. 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 maximizingGTV(χω) over all possible measurable subsets ω∈ML, 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 constantC∞V(χω), as in [6]. The constant is defined as(15)C∞Vχω=inf⁡limT→∞⁡1T∫0T∫ω∂tut,x2dxdt∂tu0,xL2Ω2∣u0,x∈H01∩H2Ω.This constant is the nonnegative constant for which the time asymptotic observability inequality(16)C∞Vχω∂tu0,xL2Ω2≤limT→∞⁡1T∫0T∫ω∂tut,x2dxdtholds for everyu(0,x)∈H01(Ω)∩H2(Ω). This is where we use the additional assumption u∈H2(Ω) so the constant is well-defined. If Ω¯ is of class C2 then this is the entire domain anyway. Shortly, we will show that the time asymptotic observability constant is equal to therandomized observability constant(17)JVχω=infj∈N⁡∫ωϕj2xdx.Note that, from the definition of the observability constant, one obtains the following inequality:(18)lim⁡supT→∞⁡CTVχωT⁡≤C∞Vχω.The randomized observability constant can be derived in the following way. We introduce a field of i.i.d random variablesβjj∈N∈0,1 which we use to multiply the values of the initial data. Then(19)inf⁡limT→∞⁡1TEGTVχω=inf⁡limT→∞⁡1TE∫0T∫ω∑j∈N∗-Δ+Vcjβjϕjx2dxdt=(20)inf⁡limT→∞⁡1TE∫0T∫ω∑j∈N∗cjλjβjϕjx2dxdtNote that (19), under the conditions (21)βjj∈N∈0,1,∑jcjλj2=1,is exactly JV(χω)=infj∈N⁡∫ωϕj2(x)dx.We now state the main theorems of the paper. All of the theorems in this section are formulated for the Schrödinger equationwith a potential, which is the main novelty. The first two results concern an expression for time asymptotic observability constant C∞V(χω).Theorem 2 (analogue toTheorem 2.6 [6]).
For every measureable subsetω of Ω,(22)C∞Vχω=inf⁡∫ω∑λ∈U∑k∈Iλckϕkx2dx∑k=1∞ck2∣cjj∈N∗∈l2C∖0,where U is the set of all distinct eigenvalues λk and I(λ)=j∈N∗∣λj=λ.The proof of the theorem is in Section4. If we set(23)JVχω=infj∈N⁡∫ωϕj2xdx,then similarly to Theorem 1 in [6], we have the following.Corollary 3 (analogous toCorollary 2.7 in [6]).
The inequalityC∞V(χω)≤JV(χω) is true for every measurable subset ω of Ω. If the domain Ω is such that every eigenvalue of -Δ+V is simple, then(24)C∞Vχω=infj∈N⁡∫ωϕj2xdx=JVχωfor every measurable subset ω of Ω. 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 ifV(x)=εV0(x) for some ε∈(0,1), then, under certain conditions on ε, we can find a positive time asymptotic observability constant for the Schrödinger equation whenever the corresponding operator without the potential (V(x)≡0) has one.For Theorem4 it is assumed that the potential has regularity V∈L∞(Ω). Let C(V0,Ω) be a constant which depends uniformly on the diameter of Ω and the L∞(Ω) norm of V0. This constant will be derived and given explicitly during the course of the proof. We prove the following.Theorem 4.
We assume that-Δ and -Δ+V(x) on Ω with Dirichlet boundary conditions both havesimple spectra, for all ε∈(0,ε0) with fixed ε0 sufficiently small. When V(x)=εV0(x), suppV0⊂ω, and ε<1/C(V0,Ω), the constant JV(χω) is such that JV(χω)>0 if and only if J0(χω)>0 for the Schrödinger equation with V(x)≡0.The proof of Theorem4 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, letM¯L be the convex closure of the set ML in the L∞ weak star topology:(25)M¯L=a∈L∞Ω,0,1∣∫Ωaxdx=LΩ.We set(26)infj∈N∗⁡∫Ωaxϕj2xdx=Jεaand also(27)infj∈N∗⁡∫Ωaxϕj02xdx=Ja.We then have the following.Theorem 5.
LetV=εV0(x) with V0(x)∈L∞(Ω), with no assumption on the support of V(x) in Ω. We assume -Δ and -Δ+V(x) with Dirichlet boundary conditions both have simple spectrum, all ε∈(0,ε0) with fixed ε0 sufficiently small. It follows that for any a∈M¯L(28)Jεa-Ja≤C1V0,Ωε2with C1(V0,Ω) a constant depending only on L∞(Ω) norm of V0 and Ω. As a consequence, we can conclude(29)maxa∈M¯L⁡Jεa-maxa∈M¯L⁡Ja≤C1V0,Ωε2The proof of Theorem5 is in Section 7. Notice that we cannot show Jε(a)=0 iff J(a) is zero because we do not have such fine control over the O(ε2) terms unless a(x)=χω(x).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 [14]. Our analysis extends their results in the linear case.


### 2.1. Comparison with Previous Literature

Letω⊂Ω be any nonempty open set and T>0; then there exists a constant KT(χω) such that for any u0∈H01(Ω) we have(30)u0L2Ω2≤KTχω∫0Texp⁡itΔu0L2ω2dtor a constant BT(χω) such that for any u0∈H01(Ω)∩H2(Ω)(31)Δu0L2Ω2≤BTχω∫0TΔexp⁡itΔu0L2ω2dtdepending on the domain of the operator.In general the work of Lebeau [15] showed that control (the dual statement to the existence of positive constant BT(χω) or KT(χω)) for the Schrödinger equation with or without the potential holds under the Geometric Control Condition (GCC):(i)
There existsL=L(Ω,ω)>0 such that every geodesic of length L on Ω intersects ω.Therefore if we letKTV(χω) denote the constant with the potential, then KTV(χω)>0 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 [9]. For the flat torus, Jaffard [16] and Haraux [17] in 2D and Komoronik [18] in higher dimensions have shown that this not necessary: observability holds for any open set ω. Their work was extended to operators with smooth potentials in [19, 20], and also for higher dimensions and time-dependent potentials in [21], and for irrational torii and general Schrördinger operators in [22]. One can see [23] 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 [24] 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 areclose.When there is no potential, our formulation of the observability constant coincides with the definition (31). Indeed, for our formulation, one can rewrite (7) as(32)CTVχω=inf⁡∫0T∫ω-Δ+Vexp⁡it-Δ+Vu02dxdt-Δ+Vu0L2Ω2∣u0∈H01∩H2ΩThe positivity of this constant isnot directly equivalent to the other two when V is nonzero, as V does not commute with exp(it(-Δ+V)). The only time the existence of the constants KTV(χω) and BTV(χω) could imply the positivity of CTV(χω) 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 ω 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 [8].Moreover, Theorem4 is proved for therandomised observability constant JV(χω) (otherwise known as the observability constant for eigenfunctions [7]). It is doubtful such a strong statement is true for the full observability constants CTV(χω) and CT0(χω) as the presence of cross terms in (14) is difficult to control when λj′s are large. Once again, as in [25], the randomized constant can be viewed as the optimistic best case scenario.Since Theorem4 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 V and ω. It is not that the eigenfunction constants cannot be close for supp(V) not contained in ω; 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 Jε(a) with ε dependent potential to the original one J(a), and there is no assumption on the support of V with respect to ω. In general, showing observability for randomized initial data (otherwise known as observability of eigenfunctions) is possible under conditions on the observation region ω which are independent of V for generic potentials, c.f. [8].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 eigenvaluesλj into the semiclassical parameter h-2. Heuristically -h2Δ+h2V as a semiclassical operator has symbol ξ2+h2V, while -h2Δ+εh2V has symbol ξ2+h2εV but in the latter case the Hamiltonian ray path x(t) over which solutions are concentrated can be made sufficiently close to that of ξ2 if ε is sufficiently small, as long as h≤1 which is proved in [26], Lemma 8.3. The methodology in [26] fails here because approximate solutions can only be constructed under a nontrapping condition.Because we are exploiting the small parameterε, 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 thath=λ-1/2, then the eigenvalue/eigenvector problem becomes (-h2Δ+h2V)u=u, with symbol ξ2+h2V(x), 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 tohigh 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.


## 2.1. Comparison with Previous Literature

Letω⊂Ω be any nonempty open set and T>0; then there exists a constant KT(χω) such that for any u0∈H01(Ω) we have(30)u0L2Ω2≤KTχω∫0Texp⁡itΔu0L2ω2dtor a constant BT(χω) such that for any u0∈H01(Ω)∩H2(Ω)(31)Δu0L2Ω2≤BTχω∫0TΔexp⁡itΔu0L2ω2dtdepending on the domain of the operator.In general the work of Lebeau [15] showed that control (the dual statement to the existence of positive constant BT(χω) or KT(χω)) for the Schrödinger equation with or without the potential holds under the Geometric Control Condition (GCC):(i)
There existsL=L(Ω,ω)>0 such that every geodesic of length L on Ω intersects ω.Therefore if we letKTV(χω) denote the constant with the potential, then KTV(χω)>0 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 [9]. For the flat torus, Jaffard [16] and Haraux [17] in 2D and Komoronik [18] in higher dimensions have shown that this not necessary: observability holds for any open set ω. Their work was extended to operators with smooth potentials in [19, 20], and also for higher dimensions and time-dependent potentials in [21], and for irrational torii and general Schrördinger operators in [22]. One can see [23] 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 [24] 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 areclose.When there is no potential, our formulation of the observability constant coincides with the definition (31). Indeed, for our formulation, one can rewrite (7) as(32)CTVχω=inf⁡∫0T∫ω-Δ+Vexp⁡it-Δ+Vu02dxdt-Δ+Vu0L2Ω2∣u0∈H01∩H2ΩThe positivity of this constant isnot directly equivalent to the other two when V is nonzero, as V does not commute with exp(it(-Δ+V)). The only time the existence of the constants KTV(χω) and BTV(χω) could imply the positivity of CTV(χω) 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 ω 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 [8].Moreover, Theorem4 is proved for therandomised observability constant JV(χω) (otherwise known as the observability constant for eigenfunctions [7]). It is doubtful such a strong statement is true for the full observability constants CTV(χω) and CT0(χω) as the presence of cross terms in (14) is difficult to control when λj′s are large. Once again, as in [25], the randomized constant can be viewed as the optimistic best case scenario.Since Theorem4 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 V and ω. It is not that the eigenfunction constants cannot be close for supp(V) not contained in ω; 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 Jε(a) with ε dependent potential to the original one J(a), and there is no assumption on the support of V with respect to ω. In general, showing observability for randomized initial data (otherwise known as observability of eigenfunctions) is possible under conditions on the observation region ω which are independent of V for generic potentials, c.f. [8].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 eigenvaluesλj into the semiclassical parameter h-2. Heuristically -h2Δ+h2V as a semiclassical operator has symbol ξ2+h2V, while -h2Δ+εh2V has symbol ξ2+h2εV but in the latter case the Hamiltonian ray path x(t) over which solutions are concentrated can be made sufficiently close to that of ξ2 if ε is sufficiently small, as long as h≤1 which is proved in [26], Lemma 8.3. The methodology in [26] fails here because approximate solutions can only be constructed under a nontrapping condition.Because we are exploiting the small parameterε, 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 thath=λ-1/2, then the eigenvalue/eigenvector problem becomes (-h2Δ+h2V)u=u, with symbol ξ2+h2V(x), 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 tohigh 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

SupposeΩ⊂Rd is a bounded domain in Rd. Then, as in the introduction the Laplace operator -Δ with Dirichlet boundary conditions can be defined as the self-adjoint operator with the quadratic form Q0(f)(33)Q0f=∇f,∇f¯L2Ωwith domain H01(Ω). Because the space H01(Ω) is compactly embedded in L2(Ω) by Rellich’s theorem, the spectrum of this operator is purely discrete and has infinity as its only possible accumulation point, c.f. [27] for a review. Hence, there exists an orthonormal basis ϕj0j∈N in L2(Ω) consisting of eigenfunctions with eigenvalues (λj), which we assume to be ordered:(34)-Δϕj0=λj0ϕj0ϕj0L2Ω=1ϕj0∣∂Ω=00≤λ10≤λ20≤⋯Recall that a linear subspaceD of the domain of a closed quadratic form Q is called a core for Q if Q is the closure of its restriction to D. We now recall the following result from [28].Theorem 6 (Thm 8.2.1 in [28]).
If0≤V∈Lloc1(Ω) and Ω⊂Rd is a domain in Rd, then the quadratic form (35)QVf=Q0f+Q1f=∫Ω∇f2+Vf2dx,which is defined on(36)DomQV=DomQ0∩DomQ1,is the form of a nonnegative self-adjoint operator H. The space Cc∞(Ω) is a core for Q.Remark 7.
We could reduce the assumption on the potential fromL∞(Ω) to Lloc1(Ω) 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 [28].Theorem 8 (Theorem 8.2.3, Corollary 4.4.3, [28]).
IfH is defined on L2(Ω) by Hf=-Δf+Vf, where V∈L∞, then H is a self-adjoint and bounded below with the same domain as H0=-Δ.We also have the following.Theorem 9 (Thm 6.3.1 in [28]).
For all bounded domainsΩ⊂Rd, the operator -Δ has an empty essential spectrum and compact resolvent. The eigenvalues λnn=1∞ of -Δ written in increasing order and repeated according to multiplicity satisfy(37)b1n2/d≤λn≤b2n2/dfor some b1,b2>0 depending only on the geometry of Ω and n≥1.As such,b1 and b2 can be made arbitrarily close to one another, if n is large, c.f. the proof of Theorem 6.3.1 in [28]. The eigenvalues of -Δ depend monotonically upon the region Ω and so can be bounded above and below by the eigenvalues of the cubes which are contained in (and, respectively, contain) Ω. It follows from Theorem 8 that H01(Ω)=DomQ(f). From this fact and the spectral theorem, we can conclude from Theorem 6 the following.Corollary 10.
ForV∈L∞(Ω), if(38)λ0≤⋯≤λj≤λj+1…with ϕjj∈N an orthonormal Hilbert basis of H01(Ω) consisting of eigenfunctions of the Dirichlet operator -Δ+V on Ω, which is associated with the eigenvalues λjj∈N, then we can write the propagated solution as(39)ut,x=∑j∈Ncjϕjt,xwith(40)cj=∫Ωu0,xϕjx¯dx.We use the basis properties in Corollary10 in the next section.


## 4. Proof of Theorem2

The basic idea is to use(41)ut,x=∑j=1∞cjexp⁡iλjtϕjxas the decomposition for the solution of (1), where (λj,ϕj(x))j∈N∗ are the eigenvalue and eigenfunction pairs for the -Δ+V operator. One can apply similar steps to [6] to prove Theorem 2. Using a standard density argument, the approximation which holds over a finite number of modes,(42)ut,x≈∑j=1Ncjexp⁡iλjtϕjx,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 Theorem2.
We start with the case when-Δ+V has simple eigenvalues. This proof is a simplification of the analogous theorem in [6] 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 ∂tu(0,x)L2(Ω)=1. Then, let(43)ΣT=1TGTχω∂tu0,xL2Ω=1TGTχωand(44)yjt,x=iλjcjexp⁡iλjtϕjx.Then, ΣT can be expressed as(45)ΣT=1T∫0T∫ω∑j=1Nyjt,x2+∑k=N+1∞ykt,x2+2∑j=1Nyjt,x∑k=N+1∞ykt,x¯dxdt.Note that(46)C∞Vχω=inf⁡limT→∞⁡ΣT.Now, we use the assumption that the spectrum of -Δ+V consists of simple eigenvalues to prove the following result.
Lemma  11.The following equation holds:(47)limT→∞⁡1T∫0T∫ω∑j=1Nyjt,x2dxdt=∑j=1Nλj2cj2∫ωϕj2xdx.
Because the sum is finite, one can invert the inf and the limit. We have(48)1T∫0T∫ω∑j=1Nyjt,x2dxdt=(49)1T∑j=1Nλj2αjj∫ωϕj2xdx+1T∑j=1N∑k=1,k≠jNλjλkcjck¯αjk∫ωϕjxϕkx¯dx,where αjk was given previously by (14). Formula (14) gives(50)limT→∞⁡αjjT=1for every j∈N. We note that(51)αjk≤2λj-λk≤2b1due to the fact that exp⁡iθ-12=(1-cos⁡θ)2+sin2⁡θ, for all θ∈R and Theorem 9. We now estimate the remainder terms of (45):(52)R=1T∫0T∫ω∑j=N+1∞yjt,x2dxdtand(53)δ=1T∫0T∫ω∑j=1Nyjt,x∑k=N+1∞ykt,x¯dxdt.Using the fact that ϕj′s form a Hilbert basis,(54)R≤1T∫0T∫Ω∑j=N+1∞yjt,x2dxdt=(55)1T∑j=N+1∞∫0Tλj2cjexp⁡iλjt2dt=1T∑j=N+1∞Tλj2cj2=∑j=N+1∞λj2cj2.To bound δ,(56)Tδ=∑j=1N∑k=N+1∞λjcjλkck¯αjk≤∑j=1∞λjcj2∑k=N+1∞λk2ck2αjk2≤∑k=N+1∞maxj⁡αjkλkck2,whenever the normalization ∑j(λjcj)2=1 is used. By Parseval’s theorem, since u,∂t2u∈L2(Ω), for every ε>0, there exists an N≥N(ε) such that(57)∑j=N+1∞λj2cj2≤ε.We conclude that, for sufficiently large N,(58)R+2δ≤ε1+4Tb1.Since ε was arbitrary and T→∞, the theorem is proved. The corollary follows since, due to the assumption that ∂tu(0,x)L2(Ω)=1, we have(59)inf∑j=1Nλj2cj2=1⁡∑j=1Nλj2cj2∫ωϕj2xdx=inf1,…,N⁡∫ωϕj2xdx.
Note that, in the case of nonsimple eigenvalues, one can group the diagonal terms to obtain the desired result. This proves Theorem2 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 operatorH=-Δ+εV0, withsimple eigenvalues λn. (Recall that this means the eigenvalues have multiplicity 1.) In the next section, more advanced results from [29] will be used to analyze the error terms.LetH0=-Δ denote the standard Laplacian with eigenvalues λn0. There exists a corresponding basis ϕn0(x) such that(60)-Δϕn0x=λn0ϕn0x.The following lemma relates the eigenvalues and eigenvectors of H to those of H0.Lemma 12.
The eigenvalues toH are given by(61)λn=λn0+ε∫V0xϕn02xdx∫ϕn02xdx+Oε2.The eigenfunctions to H are given by(62)ϕnx=ϕn0x+ε∑n≠m∫ϕn0V0ϕm0¯dxλn0-λm0ϕm0x+Oε2.Here the O terms are uniform in n depending on Ω and the L∞(Ω) norm of V0. In particular we have that(63)ϕnx-ϕn0xL2Ω≤εC2V0,Ωϕn0xL2Ωand(64)ϕnx-ϕn0x-ε∑n≠m∫ϕn0V0ϕm0¯dxλn0-λm0ϕm0xL2Ω≤ε2C3V0,Ωϕn0xL2Ωwhere C2(V0,Ω), C3(V0,Ω) depend only on the geometry of Ω and the L∞(Ω) norm of V0.We do not prove the Lemma here; it is a result of [29] (see equation (II-3.39) in Example 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 (65)ϕnx=ϕn0x+εϕn1x+ε2ϕn2x+⋯λn=λn0+ελn1+ε2λn2+⋯,then it follows by substitution that (66)-Δ+εV0xϕn0x+εϕn1x+Oε2=λn0+ελn1+Oε2ϕn0+εϕn1x+Oε2.Equating the leading order terms, (67)-Δ-λn0ϕn0x=0.At order ε, we have(68)-Δ-λn0ϕn1+V0x-λn1ϕn0x=0.The desired result for computing the first terms follows by taking the inner product of (68) with ϕj0 for j≠n. We have to have a way of encoding this inductive process of matching up the terms. In the next section we introduce the operators S and P which allow us to do just that. The terms are computed for L2(Ω) eigenfunctions, but the analysis is more sophisticated because, when computing the result of the matching over L2(ω), 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 withα>1(69)-uε′′+εx-2αuε=λεuεuε0=uε1=0the unperturbed problem is(70)-u′′=λuu0=u1=0with simple eigenvalues λ=n2π2, with n=1,2,3⋯ with corresponding normalized eigenfunctions u=21/2sin⁡(nπx),n=1,2,… The quadratic form associated with the potential Q1(f) with domain f∈L20,1:x-αf∈L20,1⊂H01(Ω) is closed in L2(0,1). The unperturbed operator isstable with respect to perturbations [29]. This is the assumption on both of the main theorems (Theorems 4 and 5). This example is from [30]. Stability of λ for the unperturbed problem means that, for ε sufficiently small, the intersection of any isolating interval for λ and the spectrum of the perturbed operator consists only of simple eigenvalues. The unperturbed/perturbed operator pair here satisfies the criterion of Theorem 5.1.12 in [29] for stability which holds provided the left hand side (86) is smaller than 1/2, which is true for sufficiently small ε. This also applies to the first numerical example in the Appendix. Usually stability is automatically satisfied when -Δ has simple spectrum and ε0 is sufficiently small, c.f. Lemma 2.1 in [31].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 ofL2(Ω) not L2(ω), 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 domainsΩ. Symmetry usually destroys the assumption of spectral simplicity, c.f. [32], and this is also discussed in Section 7.Theorem 14 (see [33] and [32]).
LetM be a compact manifold of dimension greater than 1 and C a conformal class of Riemannian metrics of fixed volume on M. Given k≥1 and d≥2, the subset of C of metrics for which the kth eigenspace is of dimension d is a submanifold of codimension of at least 1. In particular, the subset of C of metrics admitting a nonsimple eigenvalue of the Laplacian is a countable union of submanifolds of codimension of at least 1.This theorem asserts that for a given compact manifoldM “most” Riemannian metrics g on M are simple, meaning the eigenspace of the Laplace operator Δg is one-dimensional and that this set is pathwise connected. The proof naturally remains true for 0th order perturbations, like the ones we have here, c.f. [31]. 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. LetX be an arbitrary Hilbert space, as in [29], and R(A) be the range of the bounded operator A. The monograph [29] by Kato computes perturbation theory results for generic bounded operators A, and since our operator satisfies the conditions in [29] for a Type (A) holomorphic operator in the parameter ε (Theorem 2.6 of [29]), the perturbation theory derived in the book applies.LetP be the projection operator and λ be one of the eigenvalues of H0=-Δ, H=-Δ+εV0=-Δ+V, and let λk, Pk,k=1,2,…, be the eigenvalues and eigenprojections of H0=-Δ different from λ and P under consideration. Let x1,..,xm denote a basis of M=R(P) and xk1,..,xkmk denote a basis of Mk=R(Pk) for each k. The union of the vectors xj and xkj forms a basis of X consisting of eigenvectors of H0=-Δ and is adapted to X=M⊕M1⊕⋯ of X. The adjoint basis of X⋆ is adapted to X⋆=M⋆⊕M1⋆⊕⋯, where M⋆=R(P⋆), M1⋆=R(P1⋆), etc. Let e1,…,em denote the adjoint basis of M∗ and ek1,…,ekmh denote the basis of Mk∗ for k=1,2,….For anyu∈X,(71)Pu=∑j=1mu,ejxjPku=∑j=1mku,ekjxkj,∀k=1,2…We define the operator S as the value of the reduced resolvent of H0=-Δ, such that SP=PS=0, and (H0-λ)S=S(H0-λ)=1-P. Pk′s are the orthogonal projections such that(72)P=∑k=1∞Pk,and moreover, by definition, P2=P. For λ in our particular eigenspace, it follows that one can write the operator S explicitly as(73)Su=∑kλk-λ-1Pku=∑k,jλk-λ-1u,ekjxkj.using the definitions (I-5.32) and Section II.2 in [29]. If we expand λε, which is an eigenvalue of H, in a perturbation series as(74)λε=λ+ελ^1+ε2λ^2+⋯,one obtains the following expressions for the expansions of the eigenvalues λ^n (II-(2.35) [29]):(75)λ^1=1m∑jV0xj,ej.(76)λ^2=-1m∑i,j,kλk-λ-1V0x1,ekjV0xkj,ei.Suppose that the eigenvalue of λ of H0 is simple, implying that m=1. To derive an expansion for a particular eigenvector eigenvalue pair, one can set x1=ϕj0(x) and e1=ϕ¯j0(x) as in the last section. (Now j just refers to the index of the eigenfunction, a distinct index from the one above.) The operators Pj and Sj can be written as(77)Pju=u,ϕj0L2Ωϕj0,Su=∑j≠kPkuλj0-λk0.This 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 thatm=1, a convenient form of the eigenvector ϕj of H=-Δ+εV0 corresponding to the eigenvalue λε is given by(78)ϕj=Pjεϕj0,ϕj0¯L2Ω-1Pjεϕj0,where ϕj0 is the unperturbed operator of H0 for the eigenvalue λ and ϕ¯j0 is the eigenvector of the adjoint operator H0⋆. Pj(ε) is the projection onto the jth eigenspace of H. 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 -Δ. We refer the reader to Theorem 5.1.12 in [29] and Lemma 2.1 in [31] for a precise description of ε0, the threshold required. In the case of nonsimple eigenvalues the representation above would depend on more than one ϕj0, which would be difficult to analyze. We suppress the subscript j in the operators P and S where it is understood. This gives rise to the following normalization conditions:(79)ϕj,ϕj0¯L2Ω=1,ϕj-ϕj0,ϕj0¯L2Ω=0,Pϕj-ϕj0=0.The relation (H-λε)ϕj=0 can be rewritten as(80)H0-λϕj-ϕj0+V-λε+λϕj=0,where A=H-H0=εV0(x)=V. Multiplying (80) from the left hand side by S and noting that S(H0-λ)=1-P,(81)ϕj-ϕj0+SV-λε+λϕj=0.Moreover, as Sϕj0=0 and writing ϕj=ϕj-ϕj0+ϕj0 in the last term above, one gets(82)ϕj-ϕj0=-1+SV-λε+λ-1SVϕj0=-S1+VS-λε-λSα-1Vϕj0,for sufficiently small ε, with Sα=S-αP, and α is an arbitrary scalar. Equation (82) is formula (II-3.29) in [29].One can then compute(83)ϕj-ϕj0,ϕj0L2ω=-S1+VS-λε-λSα-1Vϕj0,ϕj0L2ω.The asymptotics for the scalarλj are well worked out for small ε. Let q=V0S, s0=S, p=V0P, s=S-αP for any α, where we use the operator norm. A subscript will denote the set over which the operator norm is taken.For a linear operatorA acting on H01(Ω), we let A0 denote the norm(84)A0=supj⁡Au,ϕj0suchthatu:supj⁡u,ϕj0=1.Setp,s,q to have norm ·0 and define(85)Ψε=1-ps+qε2-4psε21/2.As a result(86)λj-λj0-ελ^1=λε-λ-ελ^1≤2pqε21-ps+qε+Ψε,which is formula (II-3.18) in [29], with the norm ·0. The expansion (83) derived above is given in section II and exercise II-3.16 in the monograph by Kato [29].Now the difficulty comes in computing inner products ofϕj-ϕj0 over the smaller sets ω where one loses the powerful orthogonality conditions. We recall the following well-known Lemma on von Neumann series.Lemma 15.
LetA:X→X be a linear operator on the Banach space X. We then have(87)∑j=0∞Aju=Id-A-1u∀u∈Xprovided(88)AjuX≤δ1juX∀j∈Nwith δ1∈(0,1/2).c.f.Lemma 2.1 in [34].In order to compute (83), we want to use Lemma 15 to essentially find a convergent von-Neumann series for(89)1+VS-λε-λSα-1with (86) so that we may obtain precise bounds on the rate of decay of the inner products ϕj-ϕj0,ϕj0L2(ω). These arguments are rather delicate as we are not integrating over the whole Ω. We let M0 denote a generic constant that depends on the volume of Ω and V0L∞(Ω) 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 foruω∈L2(ω) with supp(uω)⊆ω(90)VSuω,uωL2ω=∑j≠kuω,ϕk0L2ΩVϕk0,uωL2ωλj0-λk0≤εM0uωL2ω2.LetAv be the linear operator defined as multiplication by(91)λj0-λj-V.Lemma 17.
With no assumptions on the support of the potential, we have the following estimate foruω∈L2(ω) with supp(uω)⊆ω for all N≥0, N∈N,(92)AvSNuω,uωL2ω≤εM0NuωL2ω2.Lemma 18.
Ifsupp(V0)⊂ω and α∈(0,1), then the operator S(1-AvS-α(λj-λj0)P)-1V is bounded L2(Ω)↦L2(ω).Lemma 19.
There is a choice ofε sufficiently small, such that, for all δ∈(0,1/2), the following inequality holds:(93)ϕj0-ϕj,ϕj0L2ω=S1+VS-λε-λSα-1Vϕj0,ϕj0L2ω≤δϕj0L2ω2.Proof of Lemma16.
By the Cauchy Schwartz inequality, we have(94)VSuω,uωL2ω≤∑j≠kuω,ϕjL2Ωuω,VϕjL2ωλj-λk≤∑j≠kuω,ϕjL2Ω2λj-λk21/2∑j≠kuω,VϕjL2ω21/2We know from Theorem 3.4 from [28] that(95)λj0-λk0>C,j≠kwhere C depends on Ω independent of the index set. The constant exists because all of the eigenvalues are simple and isolated. However in practice for dimensions higher than 2 the size of C may be difficult to ascertain. The desired result follows immediately from Parseval’s theorem, noting that uω and Vuω are L2(Ω) functions. We remark that this is where we use the assumption suppV⊂ω later for the main proof as the inner product <uω,ϕj> is over L2(Ω) which cannot be bounded by uωL2(Ω) unless uω has compact support in ω.Proof of Lemma17.
The bound for this inner product is constructed inductively as(96)AvSNuω=∑m0≠juω,ϕm0L2Ωλj0-λm0∑m1≠jAvϕm0,ϕm1L2Ωλj0-λm1⋯∑mN-1≠jAvϕmN-2,ϕmN-1L2Ωλj0-λmN-1AvϕmN-1We then use the proof of Lemma 16, but with (94) applied to each of the cross terms(97)∑mi-1≠jAvϕmi-2,ϕmi-1L2Ωλj0-λmi-1i=2,…,Nto reach the desired conclusion, noting that AvL∞(Ω) is almost equivalent to VL∞(Ω). Alternatively we know, for bounded operators A,B with A:X→X,B:X→X, X a Hilbert space, that ABop≤AopBop which when applied to A=V and B=S from the previous lemma gives the result as well.Proof of Lemma18.
The mappingAvS satisfies all the properties of Lemma 15 with X the space L2(Ω) restricted to the functions with compact support in ω, by Lemma 17. The space X is L02(ω), which is a Hilbert space (although most people are more familiar with H01(ω)). Therefore since Vu∈X, for all u∈L2(Ω), we are done.Proof of Lemma19.
The inner product using (83) and Lemmas 17 and 15 with AvS+α(λj-λj0)P and δ1=εM0 is bounded as(98)ϕj0L2ω2εM0+εM02+⋯=εM01-εM0ϕj0L2ω2.If εM0 is chosen to be sufficiently small, one obtains(99)εM01-εM0<δwith δ∈(0,1/2), implying ε<δ/2M0. We use the fact suppV0⊂ω, which makes the function Vϕj∀j have support in ω.Remark 20.
Some of the analysis could be extended to the case of nonsimple eigenvalues and other Hermitian operators using perturbation theory found in [29], but we focus on simple eigenvalues for ease and clarity.


## 7. Proof of Theorems4 and 5 for Convergence Estimates

Proof of Theorem4.
We recall that eigenfunctions of-Δ and -Δ+εV0 (ε sufficiently small) with Dirichlet boundary conditions are real-analytic in Ω. We can then view ∫ωϕj2(x)dx=fV(j) as a function of N taking values in (0,1] and similarly for ∫ωϕj02(x)dx=f(j). We only need to show that the following inequality is true for some order terms independent of the index j:(100)∫ωϕj2xdx=1+Oϵ∫ωϕj02xdx.These terms will bound the deviation from the original constant when including the potential term, and taking infj of the inequality gives the desired result. We need control over the order ϵ terms and show they areuniformly bounded, independent of ϕj0; e.g., the order terms are smaller than 1/2 for sufficiently small ϵ. If we can show this inequality, we will arrive at(101)12fVj≤fj≤32fVj.Taking the infimum over j gives the desired conclusion.
By the perturbation theory estimates in Section6, in Lemmas 17 and 19, by using the Lemma 15, the terms(102)2ϕj0-ϕj,ϕj0L2ω+ϕj0-ϕjL2ω2which contribute to the order terms in (100) are bounded as in Lemma 19. Therefore, it suffices to pick ε as in Lemma 19: 3δ<1/2, in order to obtain (100), with O(ϵ) terms less than 1/2.We now take a moment to remark on why the assumption of simplicity in Theorem4 is spectrally sharp, by outlining a counter-example from the details of [8]. In Remark 2.2 of [8], on the sphere S2 they construct a potential of arbitrarily small support and size and an open set ω⊂S2 such that(103)inf⁡∫ωϕ2xdxϕeigenfunctionof-Δs.t.ϕL2S2=1=0;however(104)inf⁡∫ωψ2xdxψeigenfunctionof-Δ+Vs.t.ψL2S2=1>0One can take this a step further and construct a sequence of normalized eigenfunctions of -Δ such that ϕj belongs to the eigenspace associated with the eigenvalue j(j+1) in such a way that(105)limj→∞⁡∫ωϕj2xdx=0as in [9]. One can complete this sequence to obtain an orthonormal basis of L2(S2) consisting of eigenfunctions of -Δ such that J0(χω)=0. On the other hand, JV(χω)>0. In this particular case where the spectrum is nonsimple, one cannot obtain the strong iff statement in Theorem 4 because the representation for the perturbed eigenfunctions in (78) depends onall of the 2j+1 eigenfunctions of -Δ associated with each eigenvalue j(j+1), introducing cross terms in Lemmas 16 and 17 which are computationally difficult to control. If the perturbation theory was carried out for this nonsimple case which is possible, then at most one could conclude that the constants are close.Proof of Theorem5.
Letϕ~=ϕj0-ϕj; then we can write for any j and a(x)(106)∫Ωaxϕj2dx=∫Ωaxϕj02dx+∫Ωax2ϕj0ϕ~+ϕ~2dx.As we have that using the normalisation condition ϕj0L2(Ω)2=1, and the “observation region” is the entirety of Ω(107)∫Ωax2ϕj0ϕ~+ϕ~2dx≤3∫Ωaxϕ~2dx≤3LΩC22V0,Ωε2.We obtain(108)∫Ωaxϕj2dx-∫Ωaxϕj02dx≤3LΩC22V0,Ωε2where the order terms are uniformly bounded where we have used Lemma 12. We set C1(V0,Ω)=3ΩC22(V0,Ω). Notice that the orthogonality relations imply that no additional regularity is needed on V0. Without loss of generality assume Jε(a)-J(a)>0; then we have that(109)Jεa+infj⁡∫Ωaxϕj02dx-∫Ωaxϕj2dx≤Ja.Rearranging, we obtain a bound on Jε(a)-J(a), depending on Ω and the L∞(Ω) norm of the potentials as desired, after using (107). Notice that this is probably the best control of the errors as -infj(A(j))=supj(-A(j)) for all functionals A(j).
With out loss of generality we assumemaxa∈M¯LJε(a)-maxa∈M¯LJ(a)>0, and we obtain(110)maxa∈M¯L⁡Jεa≤maxa∈M¯L⁡Jεa-Ja+maxa∈M¯L⁡Ja≤C1V0,Ωε2+maxa∈M¯L⁡Jawith constant given to us by (107) and Lemma 19.


## 8. Numerics and Examples

This section presents the results of our numerical experiments. We examine the cases of the unit interval and the unit disk. The convergence issues for the functionals in question are discussed in the Appendix.


### 8.1. Interval[0,1]

The first experiment involvedΩ=[0,1]. The orthonormal eigenvectors of -Δ with Dirichlet boundary conditions on this domain are fn(x)=2sin⁡(nπx), for n=1,2,…, with eigenvalues of λn=π2n2, for n=1,2,…. The eigenvalues have multiplicity one.Next, we calculate the eigenvectors and eigenvalues ofH=-Δ+εV0 on the unit interval with Dirichlet boundary conditions. According to perturbation theory of Section 5, the eigenvalues of operator H are given by(111)λn=λn0+ε∫V0xϕn02xdx∫ϕn02xdx+Oε2.The eigenfunctions of H are given by(112)ϕnx=ϕn0x+ε∑n≠m∫ϕn0V0ϕm0¯dxλn0-λm0ϕm0x+Oε2.In our case, we use the potential (113)V0=x2χ0.5-δ,0.5+δ,where δ is a parameter ∈[0,0.5].Matlab was used to code the experiments. The integration of functions with explicit formulas was performed using theintegral function in Matlab. We used a mesh size of about 1000 equal increments. The first two hundred eigenfunctions were calculated.Next, we consider the problem of maximizing the functionalJNV(χω),(114)JNVχω=inf1≤j≤N⁡∫ωϕj2xdx,over all subsets satisfying ω=LΩ, for some L∈(0,1). A subset with this property is called the optimal set. According to Proposition 4.1 of [6], in the case of the -Δ operator, the supremum of J(χω)=inf1≤j≤N∫ωϕj02(x) over ML is equal to L. When L=0.5, the supremum is reached for all measurable subsets ω of [0,1] satisfying ω=0.5Ω, such that ω and its symmetric image are complementary in [0,1]. Note that, for the -Δ operator, inf1≤j≤N∫[0,0.5]ϕj02(x)dx=0.5 since(115)∫00.52sin2⁡nπxdx=0.5.∀n=1,2,3,…The more interesting case is theH=-Δ+εV0 operator, and we address the question by using L=0.5 and computing JNV(χω) for subsets satisfying ω=0.5Ω and the conditions of Proposition 4.1 of [6]. In particular, we present results for ω=[0,5] and N=200.To calculateJNV(χ[0,0.5]) for H=-Δ+εV0, integration using the left point and 1000 equal increments in [0,1] were used. The δ and ε variables were varied as shown in Table 1. The values in the table show that, in all cases, the value of JNV(χ[0,0.5]) is very close to 0.5, which is the answer for the -Δ operator.Table 1
Value ofJNV(χ[0,0.5]).
ε/δ 0.1. 0.2. 0.3. 0.4 0.475. 0.01 0.499997124 0.499984760 0.499972504 0.499968543 0.499968340 0.05 0.499985619 0.499923804 0.499862542 0.499842757 0.499841748 0.1 0.499971238 0.499847620 0.499725137 0.499685621 0.499683617 0.5 0.499856202 0.499238531 0.498627808 0.498432406 0.498422979 1 0.499712437 0.498478145 0.497260919 0.496875569 0.496858189


### 8.2. Unit Disk

The orthonormal eigenvectors of-Δ on a unit disk with Dirichlet boundary conditions are given by the triply indexed sequence(116)ϕjkm0=Rjkr2π,ifj=0,RjkrYjmθ,ifj≥1,for j=0,1,2,…, k=1,2,…, and m=1,2, where (r,θ) are polar coordinates. Here, Yj1(θ)=(1/π)cos⁡(jθ), Yj2(θ)=(1/π)sin⁡(jθ), and(117)Rjkr=2JjzjkrJj′zjk,where Jj is the Bessel function of the first kind of order j and zjk>0 is the kth zero of Jj. The eigenvalues are given by the double sequence of -zjk2. Their multiplicity is 1 if j=0 and 2 if j≥1. Examples of eigenvectors of -Δ on a unit disk with Dirichlet boundary conditions are given in Figure 1.Figure 1
Graph of Bessel Functions and the region of integration.To compute the eigenvectors and eigenvalues ofH=-Δ+εV0, we use formulas (111) and (112). In this case the corresponding functionals are mock functionals as (111) and (112) do not take into account the degeneracy of the problem which means that the eigenvalues are of multiplicity two. The correct formulae require some complicated normalization constants given by (78).In our case, we use the potentials(118)V0r=1r2χr≤δ,V0r=rχr≤δ,where δ<1.There are several important equalities to note here. For radial subsetsω of the form ω=(r,θ)∈[0,1]×[0,2π]∣θ∈ω0,(119)∫ωϕjkm0x2dx=∫01Rjkr2rdr∫ω0Yjmθ2dθ=∫ω0Yjmθ2dθ,since ∫01Rjkr2rdr=1.Matlab was used for computations, and Chebfun was utilized for the numerical computation of bessel functions and its derivatives. The integration of functions with explicit formulas was performed using theintegral function in Matlab. The integration involving bessel functions was performed using thebesselj function in Matlab. For the integration of the eigenvectors of H=-Δ+εV0, the 2D trapezoid rule was used. We used a mesh size of 301 equal increments. Twenty-five eigenfunctions were computed.Next, we consider the problem of maximizing the functionalJNV(χω),(120)JNVχω=inf1≤j≤N⁡∫ωϕj2xdx,over all subsets satisfying ω=LΩ, for some L∈(0,1); the argument of the maximum of which is called the optimal set. According to Propositions 3.9 and 4.5 of [6], for the -Δ operator, the maximum value of J(χω)=inf1≤j≤N∫ωϕj02(x) for radial subsets ω of the form ω=(r,θ)∈[0,1]×[0,2π]∣θ∈ω0 and measure Lπ is L. In the case when L=0.5, the supremum is reached for all subsets ω of the form ω=(r,θ)∈[0,1]×[0,2π]∣θ∈ω0 of measure π/2, where ωθ is any measurable subset of [0,2π] such that ω and its symmetric image are complementary in [0,2π]. To gain a better understanding of the case of the H=-Δ+εV0 operator, we used L=0.5 and tested radial subsets of measure 0.5π (or half the area of the total disk) satisfying Proposition 4.5 of [6]. In particular, we note results for ω0=[0,π/4]∪[π/2,3π/4]∪[π,5π/4]∪[3π/2,7π/4] and N=25.The results are shown in Tables2 and 3; the δ and ε variables were varied. The values in the table show that, in all cases, the value of JNV(χω) is very close to 0.5, which is the answer for the -Δ operator.Table 2
Value ofJNV on unit disk with V=1/r2.
ε / δ 0.1. 0.2. 0.3. 0.4 0.475. 0.01 0.499999996 0.499999997 0.499999763 0.499995606 0.499999756 0.05 0.5 0.499999988 0.499998816 0.499987946 0.499998898 0.1 0.5 0.499999975 0.499997638 0.499999650 0.499998085 0.5 0.5 0.499999988 0.499998824 0.499988114 0.499998896 1 0.499999999 0.499999997 0.499999764 0.499995642 0.499999756Table 3
Value ofJNV on unit disk with V=r.
ε / δ 0.1. 0.2. 0.3. 0.4 0.475. 0.01 0.5 0.5 0.499999995 0.499999759 0.499998584 0.05 0.5 0.5 0.499999975 0.499998825 0.499999896 0.1 0.5 0.5 0.499999950 0.499997720 0.499999794 0.5 0.499999999 0.499999999 0.499999748 0.499991449 0.499999063 1 0.499999997 0.499999998 0.499999496 0.499990010 0.499998365


## 8.1. Interval[0,1]

The first experiment involvedΩ=[0,1]. The orthonormal eigenvectors of -Δ with Dirichlet boundary conditions on this domain are fn(x)=2sin⁡(nπx), for n=1,2,…, with eigenvalues of λn=π2n2, for n=1,2,…. The eigenvalues have multiplicity one.Next, we calculate the eigenvectors and eigenvalues ofH=-Δ+εV0 on the unit interval with Dirichlet boundary conditions. According to perturbation theory of Section 5, the eigenvalues of operator H are given by(111)λn=λn0+ε∫V0xϕn02xdx∫ϕn02xdx+Oε2.The eigenfunctions of H are given by(112)ϕnx=ϕn0x+ε∑n≠m∫ϕn0V0ϕm0¯dxλn0-λm0ϕm0x+Oε2.In our case, we use the potential (113)V0=x2χ0.5-δ,0.5+δ,where δ is a parameter ∈[0,0.5].Matlab was used to code the experiments. The integration of functions with explicit formulas was performed using theintegral function in Matlab. We used a mesh size of about 1000 equal increments. The first two hundred eigenfunctions were calculated.Next, we consider the problem of maximizing the functionalJNV(χω),(114)JNVχω=inf1≤j≤N⁡∫ωϕj2xdx,over all subsets satisfying ω=LΩ, for some L∈(0,1). A subset with this property is called the optimal set. According to Proposition 4.1 of [6], in the case of the -Δ operator, the supremum of J(χω)=inf1≤j≤N∫ωϕj02(x) over ML is equal to L. When L=0.5, the supremum is reached for all measurable subsets ω of [0,1] satisfying ω=0.5Ω, such that ω and its symmetric image are complementary in [0,1]. Note that, for the -Δ operator, inf1≤j≤N∫[0,0.5]ϕj02(x)dx=0.5 since(115)∫00.52sin2⁡nπxdx=0.5.∀n=1,2,3,…The more interesting case is theH=-Δ+εV0 operator, and we address the question by using L=0.5 and computing JNV(χω) for subsets satisfying ω=0.5Ω and the conditions of Proposition 4.1 of [6]. In particular, we present results for ω=[0,5] and N=200.To calculateJNV(χ[0,0.5]) for H=-Δ+εV0, integration using the left point and 1000 equal increments in [0,1] were used. The δ and ε variables were varied as shown in Table 1. The values in the table show that, in all cases, the value of JNV(χ[0,0.5]) is very close to 0.5, which is the answer for the -Δ operator.Table 1
Value ofJNV(χ[0,0.5]).
ε/δ 0.1. 0.2. 0.3. 0.4 0.475. 0.01 0.499997124 0.499984760 0.499972504 0.499968543 0.499968340 0.05 0.499985619 0.499923804 0.499862542 0.499842757 0.499841748 0.1 0.499971238 0.499847620 0.499725137 0.499685621 0.499683617 0.5 0.499856202 0.499238531 0.498627808 0.498432406 0.498422979 1 0.499712437 0.498478145 0.497260919 0.496875569 0.496858189


## 8.2. Unit Disk

The orthonormal eigenvectors of-Δ on a unit disk with Dirichlet boundary conditions are given by the triply indexed sequence(116)ϕjkm0=Rjkr2π,ifj=0,RjkrYjmθ,ifj≥1,for j=0,1,2,…, k=1,2,…, and m=1,2, where (r,θ) are polar coordinates. Here, Yj1(θ)=(1/π)cos⁡(jθ), Yj2(θ)=(1/π)sin⁡(jθ), and(117)Rjkr=2JjzjkrJj′zjk,where Jj is the Bessel function of the first kind of order j and zjk>0 is the kth zero of Jj. The eigenvalues are given by the double sequence of -zjk2. Their multiplicity is 1 if j=0 and 2 if j≥1. Examples of eigenvectors of -Δ on a unit disk with Dirichlet boundary conditions are given in Figure 1.Figure 1
Graph of Bessel Functions and the region of integration.To compute the eigenvectors and eigenvalues ofH=-Δ+εV0, we use formulas (111) and (112). In this case the corresponding functionals are mock functionals as (111) and (112) do not take into account the degeneracy of the problem which means that the eigenvalues are of multiplicity two. The correct formulae require some complicated normalization constants given by (78).In our case, we use the potentials(118)V0r=1r2χr≤δ,V0r=rχr≤δ,where δ<1.There are several important equalities to note here. For radial subsetsω of the form ω=(r,θ)∈[0,1]×[0,2π]∣θ∈ω0,(119)∫ωϕjkm0x2dx=∫01Rjkr2rdr∫ω0Yjmθ2dθ=∫ω0Yjmθ2dθ,since ∫01Rjkr2rdr=1.Matlab was used for computations, and Chebfun was utilized for the numerical computation of bessel functions and its derivatives. The integration of functions with explicit formulas was performed using theintegral function in Matlab. The integration involving bessel functions was performed using thebesselj function in Matlab. For the integration of the eigenvectors of H=-Δ+εV0, the 2D trapezoid rule was used. We used a mesh size of 301 equal increments. Twenty-five eigenfunctions were computed.Next, we consider the problem of maximizing the functionalJNV(χω),(120)JNVχω=inf1≤j≤N⁡∫ωϕj2xdx,over all subsets satisfying ω=LΩ, for some L∈(0,1); the argument of the maximum of which is called the optimal set. According to Propositions 3.9 and 4.5 of [6], for the -Δ operator, the maximum value of J(χω)=inf1≤j≤N∫ωϕj02(x) for radial subsets ω of the form ω=(r,θ)∈[0,1]×[0,2π]∣θ∈ω0 and measure Lπ is L. In the case when L=0.5, the supremum is reached for all subsets ω of the form ω=(r,θ)∈[0,1]×[0,2π]∣θ∈ω0 of measure π/2, where ωθ is any measurable subset of [0,2π] such that ω and its symmetric image are complementary in [0,2π]. To gain a better understanding of the case of the H=-Δ+εV0 operator, we used L=0.5 and tested radial subsets of measure 0.5π (or half the area of the total disk) satisfying Proposition 4.5 of [6]. In particular, we note results for ω0=[0,π/4]∪[π/2,3π/4]∪[π,5π/4]∪[3π/2,7π/4] and N=25.The results are shown in Tables2 and 3; the δ and ε variables were varied. The values in the table show that, in all cases, the value of JNV(χω) is very close to 0.5, which is the answer for the -Δ operator.Table 2
Value ofJNV on unit disk with V=1/r2.
ε / δ 0.1. 0.2. 0.3. 0.4 0.475. 0.01 0.499999996 0.499999997 0.499999763 0.499995606 0.499999756 0.05 0.5 0.499999988 0.499998816 0.499987946 0.499998898 0.1 0.5 0.499999975 0.499997638 0.499999650 0.499998085 0.5 0.5 0.499999988 0.499998824 0.499988114 0.499998896 1 0.499999999 0.499999997 0.499999764 0.499995642 0.499999756Table 3
Value ofJNV on unit disk with V=r.
ε / δ 0.1. 0.2. 0.3. 0.4 0.475. 0.01 0.5 0.5 0.499999995 0.499999759 0.499998584 0.05 0.5 0.5 0.499999975 0.499998825 0.499999896 0.1 0.5 0.5 0.499999950 0.499997720 0.499999794 0.5 0.499999999 0.499999999 0.499999748 0.499991449 0.499999063 1 0.499999997 0.499999998 0.499999496 0.499990010 0.499998365



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