Katos Inequality for Magnetic Relativistic Schr odinger Operators - PowerPoint PPT Presentation

Kato’s Inequality for Magnetic Relativistic Schr¨ odinger Operators Takashi Ichinose (Kanazawa) Operator Theory and Krein Spaces ( dedicated to the memory of Hagen Neidhardt ) Vienna,19–20 December, 2019

Contents 1. Introduction 2. Results 3. Notes/Ideas for Proof

1. Introduction Original Kato’s inequallity (1972) reads as (i) If u ∈ L 1 loc ( R d ) such that − ∆ u ∈ L 1 loc ( R d ) , then the distributional inequality holds: Re [( sgn u )( − ∆) u ] ≥ ( − ∆) | u | Here ( sgn u )( x ) := u ( x ) / | u ( x ) | , if u ( x ) ̸ = 0; = 0 , if u ( x ) = 0 . (ii) More generally, let A ∈ C 1 ( R d ; R d ) . If u ∈ L 1 loc ( R d ) such that ( − i ∇ − A ( x )) 2 u ∈ L 1 loc ( R d ) , then it holds: Re [( sgn u )( − i ∇ − A ( x )) 2 u ] ≥ ( − ∆) | u | One of the typical applications is: Under the same hypothesis for A ( x ) & V ∈ L 2 loc ( R d ) , V ( x ) ≥ 0 a.e. H NR := ( − i ∇ − A ( x )) 2 + V is essentially selfadjoint on C ∞ 0 ( R d ) . ⇒

Now, consider the magnetic relativ. Schr¨ od. ops. ( m ≥ 0 ) corresponding to classical √ ( ξ − A ( x )) 2 + m 2 + V ( x ) [ A ( x ) , V ( x ) : vector&scalar potentials, Hamiltonian symbol m ≥ 0 ]. In literature there are 3 kinds. One, H A,m , is def. by operator-theoretical square root of the nonnegative selfadjoint, magnetic nonrelativistic Schr¨ odinger operator ( − i ∇ + A ( x )) 2 + m 2 : √ ( − i ∇ + A ( x )) 2 + m 2 . H A,m := (0) The other two are pseudo-differential operators defined by oscillatory integrals as (with f ∈ C ∞ 0 ( R d ) ) R d × R d e i ( x − y ) · ( ξ + A ( x + y ∫ ∫ √ ( H (1) ξ 2 + m 2 f ( y ) dydξ , )) 1 A,m f )( x ) := 2 (1) (2 π ) d R d × R d e i ( x − y ) · ( ξ + ∫ 1 ∫ ∫ √ ξ 2 + m 2 f ( y ) dydξ. ( H (2) 1 0 A ((1 − θ ) x + θy ) dθ ) A,m f )( x ) := (2 π ) d (2) (1) is through Weyl quantization with mid-point prescription and (2) a modification of (1) by Iftimie, M˘ antoiu and Purice. All the 3 operators differ in general, though coincide for uni- √ form magnetic field, and in particular for A ≡ 0 , H 0 ,m = H (1) 0 ,m = H (2) − ∆ + m 2 . 0 ,m = In this Lec. we treat mainly H A,m in (0) with assumption: A ( x ) ∈ L 2 loc ( R d ; R d ) . We may assume that d ≥ 2 , since for d = 1 gauge tranform can remove any magnetic vector potential.

2. Results We can show [joint work with Hiroshima and L˝ orinczi 2017] Thm 1 (Kato’s ineq.) Let: m ≥ 0 & A ∈ L 2 loc ( R d ; R d ) . Then: u ∈ L 2 ( R d ) & H A,m u ∈ L 1 loc ( R d ) ⇒ (distributional inequality) √ − ∆+ m 2 | u | Re [( sgn u ) H A,m u ] ≥ (3) (√ ) − ∆+ m 2 − m o r Re [( sgn u )( H A,m − m ) u ] ≥ | u | (4) Here ( sgn u )( x ) := u ( x ) / | u ( x ) | , if u ( x ) ̸ = 0; = 0 , if u ( x ) = 0 . Notes . 1 ◦ H A,m is unique selfadj.operator defined through with closure of quadratic form L 2 = ( u, ( H A,m ) 2 u ) = ( u, [( − i ∇ − A ) 2 + m 2 ] u ) C ∞ 0 ∋ u �→ Q ( u ) : = ∥ H A,m u ∥ 2 = ∥ ( − i ∇ − A ) u ∥ 2 L 2 + m 2 ∥ u ∥ 2 L 2 ] 2 + m 2 ∥ u ∥ 2 [ ≤ ∥∇ u ∥ L 2 + ∥ A ∥ L 2 ( K ) ∥ u ∥ L ∞ ( K ) L 2 < ∞ , [ K := supp u ] So H A,m becomes ess.selfadj.on C ∞ 0 ( R d ) so that H A,m has domain D [ H A,m ] := { u ∈ L 2 ( R d ); ( i ∇ + A ( x )) u ∈ L 2 ( R d ) } , which contains C ∞ 0 ( R d ) as an operator core.

2 ◦ For u ∈ L 2 , H A,m u is a distribution ( ∈ S ′ ). Indeed, if ϕ ∈ C ∞ 0 , ϕ, [( − i ∇ − A ) 2 + m 2 ] ϕ ∥ H A,m ϕ ∥ 2 ( ϕ, ( H A,m ) 2 ϕ ) ( ) = = L 2 ∥ ( − i ∇ − A ) ϕ ∥ 2 L 2 + m 2 ∥ ϕ ∥ 2 = L 2 Hence (with | K | := volume of K ) ∥ H A,m ϕ ∥ L 2 ≤ ∥ ( − i ∇ − A ) ϕ ∥ L 2 + m ∥ ϕ ∥ L 2 ≤ ∥∇ ϕ ∥ L 2 + ∥ Aϕ ∥ 2 + m ∥ ϕ ∥ L 2 ] | K |∥∇ ϕ ∥ L ∞ + ∥ | A | ∥ L 2 ( K ) ∥ ϕ ∥ L ∞ ( K ) + m ∥ ϕ ∥ L ∞ ( K ) ≤ < ∞ Therefore, if u ∈ L 2 , then for ϕ ∈ C ∞ 0 ( R d ) ∫ ⟨ H A,m u, ϕ ⟩ = ⟨ u, H A,m ϕ ⟩ = ( uH A,m ϕ )( x ) dx, which means H A,m u is a distribution, because for every compact set K in R d , we have |⟨ H A,m u, ϕ ⟩| = |⟨ H A,m u, ϕ ⟩| ≤ ∥ u ∥ 2 ∥ H A,m ϕ ∥ 2 , ∀ ϕ ∈ C ∞ 0 ( R d ) with supp ϕ ⊂ K. [ ] ≤ C K ∥ u ∥ 2 ∥∇ ϕ ∥ L ∞ ( K ) + ∥ ϕ ∥ L ∞ ( K ) The characteristic feature is: H A,m is a non-local op., not diff.op., and neither integral op. nor pseudo-diff.op. associated with a certain tractable symbol.

3 ◦ Though we know the domain of H A,m is determined as just seen above, the point which becomes crucial is in how to derive regularity of the weak solution u ∈ L 2 ( R d ) of eq. √ ( − i ∇ − A ( x )) 2 + m 2 u = f, for given f ∈ L 1 loc ( R d ) . H A,m u ≡ As easy consequence is Corollary (Diamagnetic ineq.) The same hypothesis as Thm 1 | ( f, e − t [ H A,m − m ] g ) | ≤ ( | f | , e − t [ H 0 − m ] | g | ) , f, g ∈ L 2 ( R d ) . ⇒ (5) Once Thm 1 is established, can apply to show next thm on ess. selfadj-ness of rela- tiv.Schr¨ od.op. H := H A,m + V with both vector and scalar potentials A ( x ) & V ( x ) : Thm 2 The same hypothesis as Thm 1 & V ∈ L 2 loc ( R d ) , V ( x ) ≥ 0 a.e. H = H A,m + V is ess. selfadj. on C ∞ 0 ( R d ) . ⇒

3. Notes/Ideas for Proof of (2)/(3) Modify along idea/method of Kato’s original proof for magnetic non-relativ. Schr¨ od.op. 1 2 ( − i ∇ − A ( x )) 2 . However, the present case seems not so simple as to need much further modifications within “operator theory plus alpha ”. 4 ◦ H A,m C ∞ 0 ( R d ) ⊂ L 2 ( R d ) . Indeed, for ϕ ∈ C ∞ 0 with supp ϕ ⊂ K : ( compact ) ⊂ R d , ∥ H A,m ϕ ∥ L 2 ≤ C K [ ∥∇ ϕ ∥ L ∞ ( K ) + ∥ ϕ ∥ L ∞ ( K ) ] , C K : const. depending on K Therefore, for u ∈ L 2 , can define distribution H A,m u by ∫ for ϕ ∈ C ∞ 0 ( R d ) , ⟨ H A,m u, ϕ ⟩ = ⟨ u, H A,m ϕ ⟩ = ( uH A,m ϕ )( x ) dx, because, for ∀ ϕ ∈ C ∞ 0 ( R d ) with supp ϕ ⊂ K , = |⟨ H A,m u, ϕ ⟩| |⟨ u, H A,m ϕ ⟩| ≤ ∥ u ∥ 2 ∥ H A,m ϕ ∥ 2 [ ] ∥∇ ϕ ∥ L ∞ ( K ) + ∥ ϕ ∥ L ∞ ( K ) ≤ C K ∥ u ∥ 2 . √ 5 ◦ For ψ ∈ C ∞ − ∆ + m 2 , ψ ] u ∥ p ≤ C ψ ∥ u ∥ p , 1 < p < ∞ 0 , ∥ [

Then we would have √ For u ∈ C ∞ ∩ L 2 , u ε := | u | 2 + ε 2 ( ε > 0) , √ − ∆ + m 2 u ε )( x ) , a.e. Re [ u ( x )( H A,m u )( x )] ≥ u ε ( x )( or Re [ u ( x ) √ − ∆ + m 2 u ε )( x ) , a.e. u ε ( x )( H A,m u )( x )] ≥ ( Hence u δ := ρ δ ∗ u ( δ > 0) . For u ∈ L 2 , H A,m u ∈ L 1 loc . Re [ u δ ( x ) √ ( u δ ) ε ( x )( H A,m u δ )( x )] ≥ ( − ∆ + m 2 ( u δ ) ε )( x ) , a.e. Then first δ ↓ 0 , next ε ↓ 0 , if could take limit. Rather easy to see the RHS tend weakly. However, for the LHS, we encounter to establish the following very crucial claim: 6 ◦ “for u ∈ L 2 , H A,m u ∈ L 1 loc [ u δ := ρ δ ∗ u ]” ⇒ “ H A,m u δ → H A,m u in L 1 loc ” , δ ↓ 0 . The proof is a little troublesome task (at least for me!), because it turns to ask what is the domain of the operator H A,m , which is defined operator-theoretically, but not as an integral operator nor pseudo-diff. operator.

Remark on the other 2 Magnetic Relativ. Schr¨ od. Ops. (1), (2) defined by Pseudo-diff.ops. Kato’s Ineq for H (1) A,m exists already [I 89, Tsuchida-I 92]. Similarly can be shown for H (2) A,m . It is easier, partly because they can be also expressed as integral operators: | y | > 0 [ e − iy · A ( x + y ∫ ([ H (1) 2 ) u ( x + y ) − u ( x ) A,m − m ] u )( x ) = − − I {| y | < 1 } y · ( ∇ − iA ( x )) u ( x )] n ( dy ) | y | > 0 [ e − iy · ∫ 1 ∫ ([ H (2) 0 A ( x + θy ) dθ u ( x + y ) − u ( x ) A,m − m ] u )( x ) = − − I {| y | < 1 } y · ( ∇ − iA ( x )) u ( x )] n ( dy ) , where n ( dy ) = n ( y ) dy is an m -dependent measure on R d \ { 0 } having density n ( y ) . So it will be facile to treat.

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