Off-spectral analysis of Bergman kernels Haakan Hedenmalm and Aron - PowerPoint PPT Presentation

Off-spectral analysis of Bergman kernels Haakan Hedenmalm and Aron Wennman (KTH, Stockholm) 29 November 2018 KTH

Subharmonic potentials We let C be the complex plane. We let Q : C → R be a C 2 -smooth potential, with sufficient growth at infinity: Q ( z ) τ Q := lim inf log | z | > 0 . | z |→ + ∞ CONES OF SUBHARMONIC FUNCTIONS Let SH ( C ) denote the cone of subharmonic functions. Moreover, for w 0 ∈ C and 0 ≤ τ < + ∞ , let � � SH τ, w 0 ( C ) := q ∈ SH ( C ) : q ( z ) ≤ τ log | z − w 0 | + O ( 1 ) as z → w 0 , which we may refer to as the ( τ, w 0 ) -pinched subharmonic functions. Informally, these are the subharmonic functions whose Laplacian has a point mass of magnitude at least τ at the point w 0 .

Exponentially varying weights and associated Bergman spaces Exponentially varying weights and Bergman spaces For a positive real parameter m , let A 2 mQ denote the space of entire functions with finite norm � � f � 2 | f ( z ) | 2 e − 2 mQ ( z ) d A ( z ) < + ∞ . mQ := C Here, d A denotes the area element, normalized so that the unit disk D gets unit area. Nested subspaces of functions vanishing at a point Let A 2 mQ , n , w 0 denote the subspace mQ : f ( z ) = O ( z − w 0 ) n as z → w 0 A 2 f ∈ A 2 � � mQ , n , w 0 := . This is a closed subspace of A 2 mQ .

Bergman kernels Associated with the Hilbert space A 2 mQ is the inner product � g ( z ) e − 2 mQ ( z ) d A ( z ) . � f , g � mQ := f ( z )¯ C The Bergman kernel The Bergman kernel for A 2 mQ is the function K m ( · , ζ ) ∈ A 2 mQ with f ( ζ ) = � f , K m ( · , ζ ) � mQ , ζ ∈ C . Correspondingly, the Bergman kernel for the subspace A 2 mQ , n , w 0 is denoted by K m , n , w 0 ( · , ζ ) . Root functions The root function of of order n at w 0 , denoted k m , n , w 0 is the solution to the optimization problem Re f ( n ) ( w 0 ) : f ∈ A 2 � � max mQ , n , w 0 , � f � mQ ≤ 1 provided the maximum exists as a positive number. Otherwise, we declare k m , n , w 0 = 0.

The Bergman kernel in C 2 for a tubular domain Consider the tubular domain in C 2 given by ( z , w ) ∈ C 2 : | w | < e − mQ ( z ) � � Ω mQ := . Then the Bergman kernel K Ω mQ for the space A 2 (Ω mQ ) is such that K Ω mQ (( z , 0 ) , ( ζ, 0 )) = const K mQ ( z , ζ ) , where the constant depends on normalizations. Pseudoconvexity We use the defining function ρ = log | w | + mQ ( z ) , in which case (when w � = 0) the associated quadratic form of ( a , b ) ∈ C 2 on the complex 2 w + m ∂ Q a tangent plane ∂ z b = 0 is L ( ρ )( a , b ) = m ∂ 2 Q z | a | 2 . ∂ z ∂ ¯ This quadratic form is positive semidefinite if and only if ∆ Q ≥ 0, and consequently Ω mQ is pseudoconvex if and only if ∆ Q ≥ 0. Moreover, Ω mQ is locally pseudoconvex wherever ∆ Q ≥ 0.

Root functions and the Berman kernel LEMMA We have that K m , n , w 0 ( z , ζ ) K m , n , w 0 ( ζ, ζ ) 1 / 2 → k m , n , w 0 ( z ) if ζ approaches w 0 along an appropriate direction. Expansion of Bergman kernel in root functions We have that + ∞ � K m , n , w 0 ( z , ζ ) = k m , n , w 0 ( z ) k m , n , w 0 ( ζ ) . l = n

An obstacle problem Let ˆ � � Q ( z ) := sup q ( z ) q ∈ SH ( C ) , q ≤ Q on C , and, analogously, for the ( τ, w 0 ) -pinched problem, ˆ � � Q τ, w 0 ( z ) := sup q ( z ) q ∈ SH τ, w 0 ( C ) , q ≤ Q on C . Spectral droplets We put S := { z ∈ C : Q ( z ) = ˆ S τ, w 0 := { z ∈ C : Q ( z ) = ˆ Q ( z ) } , Q τ, w 0 ( z ) } , and call these spectral droplets (or spectra). The bulk of the spectral droplet S is the set bulk ( S ) := { z ∈ int ( S ) : ∆ Q ( z ) > 0 } , with the obvious modifications in the case of S τ, w 0 . Note that S τ, w 0 gets smaller as τ increases, starting with S for τ = 0.

Illustration of the obstacle problem Figure: Illustration of the spectral droplet corresponding to the potential Q ( z ) = | z | 2 − log ( a + | z | 2 ) , with a = 0 . 04. The spectrum is illustrated with a thick line, and appears as the contact set between Q (solid) and the solution ˆ Q to the obstacle function (dashed).

illustration of a compact spectral droplet. Figure: Illustration of a compact spectral droplet (shaded) with two simply connected holes. In this case there are three off-spectral components: the two holes as well as the unbounded component. If we think of this in the context of the Riemann sphere we may allow for the point at infinity to be inside the spectrum.

Admissible potentials Q Admissibility The C 2 -smooth function Q : C → R is said to be admissible if (i) τ Q > 0, (ii) Q is real-analytically smooth and strictly subharmonic in a neighborhood of ∂ S , (iii) there exists a bounded component Ω of the complement S c = C \ S which is simply connected, with real-analytically smooth Jordan curve boundary. In particular, we read off from (iii) that S c must be nontrivial, and hence subharmonic Q are excluded from consideration under admissibility of Q .

( τ, w 0 ) -admissible potentials Q Pinched admissibility The C 2 -smooth function Q : C → R is said to be ( τ, w 0 ) -admissible if (i) 0 ≤ τ ≤ τ Q , (ii) Q is real-analytically smooth and strictly subharmonic in a neighborhood of ∂ S τ, w 0 , (iii) the point w 0 is an off-spectral point, i.e., w 0 / ∈ S τ, w 0 , and the component Ω τ, w 0 of the complement S c τ, w 0 containing w 0 is bounded and simply connected, with real-analytically smooth Jordan curve boundary. For an interval I ⊂ [ 0 , + ∞ [ , we speak of ( I , w 0 ) -admissibility if we have ( τ, w 0 ) -admissibility for each τ ∈ I , while the associated domains Ω τ, w 0 change smoothly as τ moves in the interval I . Here, subharmonic potentials Q are allowed, because S c τ, w 0 is automatically nontrivial for τ > 0.

Some notation Conformal mappings Let ϕ w 0 denote the conformal mapping Ω → D with ϕ w 0 ( w 0 ) = 0 and ϕ ′ w 0 ( w 0 ) > 0, provided w 0 ∈ Ω . In the pinched situation, we denote by ϕ τ, w 0 the conformal mapping Ω τ, w 0 → D with ϕ τ, w 0 ( w 0 ) = 0 and ϕ ′ τ, w 0 ( w 0 ) > 0. Complexification of Q We let Q w 0 be the function which is bounded and holomorphic in Ω and whose real part equals Q along ∂ Ω , while Im Q w 0 ( w 0 ) = 0. Analogously, in the pinched situation, we Q τ, w 0 for the bounded holomorphic function in Ω τ, w 0 whose real part equals Q along ∂ Ω τ, w 0 , while Im Q w 0 ( w 0 ) = 0. These functions are tacitly extended holomorphically across the corresponding boundary curves.

Off-spectral expansion of the Bergman kernel THEOREM I Suppose Q an admissible potential. Then, given a positive integer κ and a positive real A , there exist a neighborhood Ω � κ � of the closure of Ω and bounded holomorphic functions B j , w 0 on Ω � κ � for j = 0 , . . . , κ , as well as domains Ω m = Ω m ,κ, A with Ω ⊂ Ω m ⊂ Ω � κ � which meet dist C ( ∂ Ω m , ∂ Ω) ≥ A m − 1 1 2 ( log m ) 2 , such that the normalized Bergman kernel at the point w 0 enjoys the expansion K m ( z , w 0 ) 1 1 4 ( ϕ ′ 2 e m Q w 0 ( z ) k m ( z , w 0 ) = K m ( w 0 , w 0 ) 1 / 2 = m w 0 ( z )) κ � � � m − j B j , w 0 ( z ) + O ( m − κ − 1 × , j = 0 as m → + ∞ , where the error term is uniform on Ω m .

The first term of the expansion In the theorem, the first term B 0 , w 0 is obtained as the unique zero-free holomorphic function on Ω which is smooth up to the boundary, positive at w 0 , with prescribed modulus on the boundary |B 0 , w 0 ( z ) | = ( 4 π ) − 1 1 4 | ∆ Q ( z ) | 4 , z ∈ ∂ Ω . As for the later terms B j , w 0 , with j = 1 , 2 , 3 , . . . , they may be obtain algorithmically. The expressions do get a bit large though.

The Gaussian wave associated with the normalized Bergman kernel Associated with the normalized Bergman kernel we have the probability wave | k m ( z , w 0 ) | 2 e − 2 mQ ( z ) . Figure: Illustration of the the probability wave of the Bergman kernel for 2 | z | − 2 − 1 w 0 = 0 and Q ( z ) = 1 8 Re ( z − 2 ) + ( 1 + 2 m ) log | z | .

Off-spectral expansion of root functions THEOREM I Suppose Q is ( I 0 , w 0 ) -admissible, where I 0 is compact. Then, given a positive integer κ and a positive real A , there exist a neighborhood Ω � κ � τ, w 0 of the closure of Ω τ, w 0 and bounded holomorphic functions B j , w 0 on Ω � κ � τ, w 0 for j = 0 , . . . , κ , as well as domains Ω τ, w 0 , m = Ω τ, w 0 , m ,κ, A with Ω τ, w 0 ⊂ Ω τ, w 0 , m ⊂ Ω � κ � τ, w 0 which meet τ, w 0 , m , Ω τ, w 0 ) ≥ A m − 1 1 dist C (Ω c 2 ( log m ) 2 , such that the root function of order n at w 0 enjoys the expansion 1 1 4 ( ϕ ′ 2 [ ϕ τ, w 0 ( z )] n e m Q w 0 ( z ) k m , n , w 0 ( z ) = m τ, w 0 ( z )) κ � � � m − j B j ,τ, w 0 ( z ) + O ( m − κ − 1 × , j = 0 on Ω τ, w 0 , m as n = τ m → + ∞ while τ ∈ I 0 , where the error term is uniform.

The first term in the expansion of root functions In the theorem, the first term B 0 ,τ, w 0 is obtained as the unique zero-free holomorphic function on Ω τ, w 0 which is smooth up to the boundary, positive at w 0 , with prescribed modulus on the boundary |B 0 , w 0 ( z ) | = ( 4 π ) − 1 1 4 | ∆ Q ( z ) | 4 , z ∈ ∂ Ω τ, w 0 . As for the later terms B j , w 0 , with j = 1 , 2 , 3 , . . . , they may be obtain algorithmically. The expressions do get a bit large.