Optimal potentials for Schr odinger operators Giuseppe Buttazzo - PowerPoint PPT Presentation

Optimal potentials for Schr¨ odinger operators Giuseppe Buttazzo Dipartimento di Matematica Universit` a di Pisa buttazzo@dm.unipi.it http://cvgmt.sns.it “Modelisation with optimal transport” Grenoble, October 3–4, 2013

Work in collaboration with: Augusto Gerolin, Ph. D. student at Dipartim. di Matematica - Universit` a di Pisa, gerolin@mail.dm.unipi.it Berardo Ruffini, Ph. D. student at Scuola Normale Superiore di Pisa, berardo.ruffini@sns.it Bozhidar Velichkov, Ph. D. student at Scuola Normale Superiore di Pisa, b.velichkov@sns.it 1

We consider the Schr¨ odinger operator − ∆+ V ( x ) in a given bounded set Ω. The opti- mization problems we deal with are of the form � � min F ( V ) : V ∈ V , where F is a suitable cost functional and V is a suitable admissible class. We limit our- selves to the case V ≥ 0. The cost functionals we want to include in our framework are of the following types. 2

Integral functionals Given a right-hand side f ∈ L 2 (Ω) we consider the solution u V of the elliptic PDE u ∈ H 1 − ∆ u + V ( x ) u = f ( x ) in Ω , 0 (Ω) . The integral cost functionals we consider are of the form � � � F ( V ) = Ω j x, u V ( x ) , ∇ u V ( x ) dx where j is a suitable integrand that we as- sume convex in the gradient variable and bounded from below as j ( x, s, z ) ≥ − a ( x ) − c | s | 2 3

with a ∈ L 1 (Ω) and c smaller than the first eigenvalue of − ∆ on H 1 0 (Ω). In particular, the energy E f ( V ) defined by � 1 2 |∇ u | 2 +1 � � 2 V ( x ) u 2 − f ( x ) u E f ( V ) = inf dx u ∈ H 1 Ω 0 (Ω) belongs to this class since, integrating by parts its Euler-Lagrange equation, we have E f ( V ) = − 1 � Ω f ( x ) u V dx 2 which corresponds to the integral functional above with j ( x, s, z ) = − 1 2 f ( x ) s. 4

Spectral functionals For every admissible potential V ≥ 0 we consider the spectrum λ ( V ) of the Schr¨ odinger operator − ∆+ V ( x ) on H 1 0 (Ω). If Ω is bounded or has finite measure, or if the potential V satisfies some suitable inte- gral properties, the operator − ∆+ V ( x ) has a compact resolvent and so its spectrum λ ( V ) is discrete: � � λ ( V ) = λ 1 ( V ) , λ 2 ( V ) , . . . , where λ k ( V ) are the eigenvalues counted with their multiplicity. 5

The spectral cost functionals we consider are of the form � � F ( V ) = Φ λ ( V ) where Φ : R N → R is a given function. For instance, taking Φ( λ ) = λ k we obtain F ( V ) = λ k ( V ) . We say that Φ is continuous (resp. lsc) if λ n k → λ k ∀ k = ⇒ Φ( λ n ) → Φ( λ ) � � resp. Φ( λ ) ≤ lim inf Φ( λ n ) . n 6

Optimization problems for changing sign po- tentials have been recently considered by Carlen- Frank-Lieb for the cost F ( V ) = λ 1 ( V ). They prove the inequality: � 1 p + d � � p . 2 λ 1 ( V ) ≥ − c p,d R d V dx − Our goal is to obtain similar inequalities for more general cost functionals and integral constraints on the potential; on the other hand, we limit ourselves to the case of non- negative potentials. 7

The γ -convergence We denote by M + 0 (Ω) the class of capaci- tary measures on Ω, i.e. the Borel (not nec- essarily finite) measures µ on Ω such that µ ( E ) = 0 for any set E ⊂ Ω of capacity zero. For any capacitary measure µ ∈ M + 0 (Ω), we define the Sobolev space � � � R d | u | 2 dµ < + ∞ H 1 u ∈ H 1 ( R d ) : µ = , 8

which is a Hilbert space when endowed with the norm � u � 1 ,µ , where � � � R d |∇ u | 2 dx + R d u 2 dx + R d u 2 dµ. � u � 2 1 ,µ = ∈ H 1 If u / µ , we set � u � 1 ,µ = + ∞ . In particular the measure  0 if cap( E ∩ K ) = 0  ∞ K ( E ) = + ∞ if cap( E ∩ K ) > 0  is a capacitary measure and, taking K = Ω c , the space H 1 µ (Ω) becomes in this case the usual Sobolev space H 1 0 (Ω). 9

Definition We say that a sequence ( µ n ) of capacitary measures γ -converges to the ca- pacitary measure µ if the sequence of func- tionals � · � 1 ,µ n Γ -converges to the functional � · � 1 ,µ in L 2 (Ω) , i.e. the following two con- ditions are satisfied: • for every u n → u in L 2 (Ω) we have � u � 2 n →∞ � u n � 2 1 ,µ ≤ lim inf 1 ,µ n ; • for every u ∈ L 2 (Ω) , there exists u n → u in L 2 (Ω) such that � u � 2 n →∞ � u n � 2 1 ,µ = lim 1 ,µ n . 10

For every µ ∈ M 0 (Ω) and f ∈ L 2 (Ω) we may consider the PDE formally written as  − ∆ u + µu = f  u ∈ H 1 0 (Ω)  whose precise meaning has to be given in the weak form � � � Ω ∇ u ∇ ϕ dx + Ω uϕ dµ = Ω fϕ dx for every ϕ ∈ H 1 0 (Ω) ∩ L 2 µ (Ω). The resolvent operator R µ : L 2 (Ω) → L 2 (Ω) associates to every f ∈ L 2 ( D ) the unique solution u of the PDE above. 11

Properties of the γ -convergence • The γ -convergence is equivalent to: R µ n ( f ) → R µ ( f ) for every f ∈ L 2 ( D ) . Actually, it is enough to have R µ n (1) → R µ (1). In this way, the distance d γ ( µ 1 , µ 2 ) = � R µ 1 (1) − R µ 2 (1) � L 2 (Ω) is equivalent to the γ -convergence. • The space M 0 (Ω) endowed with the dis- tance d γ is a compact metric space. • Identifying a domain A with the measure ∞ Ω \ A , the class of all smooth domains A ⊂ Ω is d γ -dense in M 0 (Ω). 12

• The measures of the form V ( x ) dx , with V smooth, are d γ -dense in M 0 (Ω). • If µ n → µ for the γ -convergence, the spec- trum of the compact resolvent operator R µ n converges to the spectrum of R µ ; then the eigenvalues of the Schr¨ odinger operator − ∆+ µ n defined on H 1 0 (Ω) converge to the corre- sponding eigenvalues of the operator − ∆+ µ . 13

The case of bounded constraints Proposition If V n → V weakly in L 1 (Ω) the capacitary measures V n dx γ -converge to V dx . As a consequence, all the optimization prob- lems of the form min { F ( V ) : V ∈ V} with F γ -l.s.c (very weak assumption) and V closed convex and bounded in L p (Ω) with p > 1, admit a solution. 14

Example If p > 1 the problem � � � Ω V p dx ≤ 1 max E f ( V ) : V ≥ 0 , has the unique solution � − 1 /p �� Ω | u p | 2 p/ ( p − 1) dx | u p | 2 / ( p − 1) , V p = where u p is the minimizer on H 1 0 (Ω) of � p − 1 1 Ω |∇ u | 2 dx +1 p − � �� � Ω | u | 2 p/ ( p − 1) dx Ω fu dx 2 2 corresponding to the nonlinear PDE − ∆ u + C | u | 2 / ( p − 1) u = f. 15

Similar results for λ 1 ( V ) (see also [Henrot Birkh¨ auser 2006]). If p < 1 the problem � � � Ω V p dx ≤ 1 max E f ( V ) : V ≥ 0 , has no solution. Indeed, take for instance f = 1; it is not difficult to construct a se- quence V n such that � Ω V p n dx ≤ 1 and E f ( V n ) → 0 . The conclusion follows since no potential V can provide zero energy. 16

An interesting case is when p = 1. The solution of � � � max E f ( V ) : V ≥ 0 , Ω V dx ≤ 1 is in principle a measure. However, it is pos- sible to prove that for every f ∈ L 2 (Ω), de- noting by w the solution of the auxiliary prob- lem � 1 Ω |∇ u | 2 dx + 1 � � � 2 � u � 2 min L ∞ (Ω) − Ω uf dx , 2 u ∈ H 1 0 (Ω) and setting M = � w � L ∞ (Ω) , ω + = { w = M } , ω − = { w = − M } , 17

we have V opt = f � � 1 ω + − 1 ω − . M Note that in particular, we deduce the con- ditions of optimality • f ≥ 0 on ω + , • f ≤ 0 on ω − , � � • f dx − f dx = M. ω + ω − 18

The case of unbounded constraints We consider now problems of the form � � � min F ( V ) : V ≥ 0 , Ω Ψ( V ) dx ≤ 1 with admissible classes of potentials unbounded in every L p . For example: Ψ( s ) = s − p , for any p > 0; • Ψ( s ) = e − αs , for any α > 0. • Theorem Let Ω be bounded, F increas- ing and γ -lower semicontinuous, Ψ strictly decreasing with Ψ − 1 ( s p ) convex for some p > 1. Then there exists a solution. 19

If Ψ( s ) = s − p with p > 0, the Examples optimal potential for the energy E f is � 1 /p �� Ω | u | 2 p/ ( p +1) dx | u | − 2 / ( p +1) V opt = where u solves the auxiliary problem � (1+ p ) /p � � � � Ω |∇ u | 2 dx + Ω | u | 2 p/ (1+ p ) dx min − Ω 2 fudx u ∈ H 1 0 (Ω) which corresponds to the nonlinear PDE − ∆ u + C p | u | − 2 / ( p +1) u = f, u ∈ H 1 0 (Ω) where C p is a constant depending on p . 20

Similarly, if Ψ( s ) = e − αs , we have V opt = 1 � �� � u 2 �� Ω u 2 dx � − log log α where u solves the auxiliary problem � Ω |∇ u | 2 dx + min u ∈ H 1 0 (Ω) 1 � � � � � Ω u 2 Ω log( u 2 ) dx − log( u 2 ) dx − Ω 2 fudx. α which corresponds to the nonlinear PDE − ∆ u + C α u − 1 αu log( u 2 ) = f, u ∈ H 1 0 (Ω) where C α is a constant depending on α . 21

PROBLEMS WITH Ω = R d When Ω = R d most of the cost function- als are not γ -lower semicontinuous; for ex- ample, if V ( x ) is any potential, with V = + ∞ outside a compact set, then, for every x n → ∞ , the sequence of translated poten- tials V n ( x ) = V ( x + x n ) γ -converges to the capacitary measure  0 if cap( E ) = 0  I ∅ ( E ) = + ∞ if cap( E ) > 0 .  Thus increasing and translation invariant func- tionals are never γ -lower semicontinuous. 22