Canonical systems whose Weyl coefficients have regularly varying - PowerPoint PPT Presentation

Canonical systems whose Weyl coefficients have regularly varying asymptotics Matthias Langer University of Strathclyde, Glasgow based on joint work with Raphael Pruckner and Harald Woracek (Vienna)

Canonical systems Consider the 2 × 2 canonical system y ′ ( x ) = zJH ( x ) y ( x ) , x ∈ (0 , ∞ ) , (CS) where • y . . . 2-vector function • z ∈ C � 0 − 1 � • J = 1 0 • H . . . locally integrable (on [0 , ∞ )) function whose values are 2 × 2 real non-negative matrices ( H . . . ‘Hamiltonian’) H does not vanish on any set of positive measure • H is in the limit point case at ∞ , i.e. � ∞ tr H ( x ) d x = ∞ . 0

Weyl function Let W ( x, z ) be the (fundamental) solution of ∂ ∂xW ( x, z ) J = zW ( x, z ) H ( x ) , x ∈ (0 , ∞ ); W (0 , z ) = I. Note that the rows of W satisfy (CS) and � w 11 (0 , z ) � � 1 � � w 21 (0 , z ) � � 0 � = , = . w 12 (0 , z ) 0 w 22 (0 , z ) 1 Since H is in the limit point case at b , the following limit exists and is independent of τ ∈ R ∪ {∞} , w 11 ( x, z ) τ + w 12 ( x, z ) q H ( z ) := lim w 21 ( x, z ) τ + w 22 ( x, z ) , z ∈ C \ R . x →∞ The function q H is called Weyl function for the Hamiltonian H . It is also characterised by the property that, for z ∈ C \ R , � w 11 ( · , z ) � � w 21 ( · , z ) � ∈ L 2 ((0 , ∞ ) , H ( x )d x ) − q H ( z ) w 12 ( · , z ) w 22 ( · , z )

Spectral measure The Weyl function is a Nevanlinna function, i.e. Im q H ( z ) ≥ 0 , q H ( z ) = q H ( z ) when Im z > 0 . Hence it has an integral representation � � � 1 t q H ( z ) = a + bz + t − z − d µ ( t ) , z ∈ C \ R , 1 + t 2 R where a ∈ R , b ≥ 0 and � 1 µ is a Borel measure on R such that 1 + t 2 d µ ( t ) < ∞ . R The measure µ is a spectral measure since the generalised Fourier transform � ∞ � w 11 ( x, · ) � f �→ H ( x ) f ( x ) d x w 12 ( x, · ) 0 establishes a unitary equivalence of the underlying operator (or relation) corresponding to (CS) with the multiplication operator in L 2 ( R , µ ).

Inverse spectral theorem (de Branges). The mapping H �→ q H establishes a one-to-one correspondence between all Hamiltonians (up to reparameterisation) and all Nevanlinna functions. Reparameterisation: H ( x ) = η ′ ( x ) H ( η ( x )) for a strictly increasing bijection η . ˜ The restriction of the mapping H �→ q H to Hamiltonians with tr H ( x ) = 1 a.e. is a bijection. Question: how are properties of H related to properties of q H ?

Basic asymptotic properties of the Weyl function Since q H is a Nevanlinna function, ∃ c 1 , c 2 , r 0 > 0: c 1 r ≤ | q H ( ri ) | ≤ c 2 r, r ≥ r 0 . The extreme cases are well known: | q H ( ri ) | • lim sup > 0 ⇐ ⇒ q H ( ri ) ∼ ibr with b > 0 r r →∞ � h 1 0 � ∃ ε > 0 . H | (0 ,ε ) = a.e. ⇐ ⇒ 0 0 q H ( ri ) ∼ ic 1 • lim inf r →∞ r | q H ( ri ) | < ∞ ⇐ ⇒ with c > 0 r � 1 ⇐ ⇒ q H ( z ) = t − z d µ ( t ) , µ finite R � 0 0 � ⇐ ⇒ ∃ ε > 0 . H | (0 ,ε ) = a.e. 0 h 2

Asymptotics of the spectral function or the Titchmarsh–Weyl coefficient for Sturm–Liouville equations: Marchenko 1952 Hille 1963 Everitt 1972 Atkinson 1981 Bennewitz 1989 . . . Strings: Kac 1971, . . . Kasahara 1976 Kasahara, Watanabe 2010 Canonical systems: Eckhardt, Kostenko, Teschl 2018

Regularly varying functions A measurable function a : (0 , ∞ ) → (0 , ∞ ) is called regularly varying at ∞ with index α ∈ R if a ( λr ) a ( r ) = λ α . ∀ λ > 0 . lim r →∞ Examples: a ( r ) = r α (log r ) β 1 · · · (log · · · log r ) β m , • α, β 1 , . . . , β m ∈ R � �� � m log r a ( r ) = r α e • log log r , α ∈ R , a ( r ) = r α e (log r ) β cos((log r ) β ) , α ∈ R , β ∈ (0 , 1 2 ) . • for r large enough. When α = 0, a is called slowly varying.

Regular and rapid variation at 0 A function a : (0 , ∞ ) → (0 , ∞ ) is called regularly varying at 0 with index α ∈ R if r �→ 1 / a ( 1 r ) is regularly varying at ∞ with index α , or equivalently, a ( λx ) a ( x ) = λ α . ∀ λ > 0 . lim x ց 0 A function a is called rapidly varying at 0 with index ∞ if  0 , λ ∈ (0 , 1) ,   a ( λx )  lim a ( x ) = 1 , λ = 1 , x ց 0    ∞ , λ ∈ (1 , ∞ ) . Example: a ( x ) = e − 1 x .

Assumptions for the main theorem � h 1 h 3 � Let H = be a Hamiltonian as above. h 3 h 2 � h 1 0 � 0 0 � � (i) Assume that neither H = nor H = on any interval 0 h 2 0 0 of the form (0 , ε ) with ε > 0. � t (ii) Set m i ( t ) := 0 h i ( s ) d s , i = 1 , 2 , 3. (iii) By (i), m 1 m 2 is a strictly increasing bijection from (0 , ∞ ) onto itself. Hence, for every r > 0 there exists a unique ˚ x ( r ) > 0 such that x ( r )) = 1 ( m 1 m 2 )(˚ r 2 . (iv) Assume that m 1 + m 2 is regularly varying at 0 with positive index. This is satisfied, e.g. if tr H = 1 a.e.

Theorem. Let the assumptions from above be satisfied. TFAE (i) ∃ a : (0 , ∞ ) → (0 , ∞ ), regularly varying at ∞ , ∃ ω ∈ C \ { 0 } : � � ∀ r > 0 . q H ( ri ) = iω a ( r ) 1 + R ( r ) ; r →∞ R ( r ) = 0 . lim (ii) m 1 , m 2 are regularly or rapidly varying at 0 with indices ρ i and m 3 ( x ) δ := lim exists . � x ց 0 m 1 ( x ) m 2 ( x ) x ( r )) = 1 (iii) Let ˚ x ( r ) be the unique solution of ( m 1 m 2 )(˚ r 2 , r > 0. � m 1 (˚ x ( r )) The function a H ( r ) := x ( r )) , r > 0, is regularly varying at ∞ m 2 (˚ with index α , � z � α a H ( r ) � � ∀ r > 0 , z ∈ C + . q H ( rz ) = iω 1 + R ( z, r ) ; i r →∞ R ( z, r ) = 0 locally uniformly for z ∈ C + ; ∗ lim ∗ α = ρ 2 − ρ 1 ω ∈ C \ { 0 } with | arg ω | ≤ π ; 2 (1 − | α | ). ρ 2 + ρ 1

The role of ω . Assume that (i)–(iii) hold. Recall that � z � α a H ( r ) , q H ( rz ) ∼ iω r → ∞ , i with α = ρ 2 − ρ 1 | arg ω | ≤ π ρ 2 + ρ 1 ∈ [ − 1 , 1], 2 (1 − | α | ). � z � α arg ω z �→ iω i − → π | α | The coefficient ω depends explicitly on α (and hence on ρ 1 , ρ 2 ) and on � m 3 ( x ) � 1 − α 2 � δ = lim where | δ | ≤ . � x → 0 m 1 ( x ) m 2 ( x ) For fixed α the mapping δ �→ arg ω is strictly decreasing. Further, arg ω = 0 ⇐ ⇒ δ = 0.

The spectral measure � h 1 h 3 � Since H is not of the form on any interval of the form (0 , ε ), h 3 h 2 b = 0 in the integral representation of q H , i.e. � � � 1 t q H ( z ) = a + t − z − d µ ( t ) , 1 + t 2 R Assume that (i)–(iii) are satisfied and that α � = ± 1. With the help of a Tauberian theorem one can show that | ω | � � � � arg ω + π µ ([0 , t )) = α + 1 sin 2 (1 − α ) t a ( t ) 1 + o (1) , | ω | � � � � arg ω + π µ (( − t, 0]) = α + 1 sin 2 (1 + α ) t a ( t ) 1 + o (1) , as t → ∞ .