Semigroup Growth Bounds

Transcription

1 Semigroup Growth Bounds First Meeting on Asymptotics of Operator Semigroups E.B. Davies King s College London Oxford, September 2009 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

2 Norms of Semigroup Operators There are several distinct issues for a one-parameter semigroup T t = e At acting in a Banach space B. The long time asymptotics of T t ; The short time asymptotics of T t ; The intermediate time behaviour of T t ; The spectrum of A; The behaviour of the norms of the resolvent operators R z = (zi A) 1. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

3 The Classical Bounds Every semigroup has a bound of the form T t Me at for all t 0. This implies that for all z satisfying Re (z) > a. R z M(Re (z) a) 1 The precise form of the converse was proved by Feller, Miyadera and Phillips. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

4 Dissipativity If B is a Hilbert space and Re Af, f a for all f Dom(A) then and conversely. T t e at for all t 0 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

5 The Asymptotic Growth Rate The infimum of all possible a is given by ω 0 = lim t + t 1 log( T t ). This implies that Spec(A) {z : Re (z) ω 0 } and T t e ω 0t for all t > 0. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

6 Zabczyk s Example 1 but the two sides need not be equal. Spec(T t ) { e zt : z Spec(A) }. 1 Zabczyk 1975 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

7 Zabczyk s Example 1 but the two sides need not be equal. Spec(T t ) { e zt : z Spec(A) }. There exists a one-parameter group T t acting in a Hilbert space H such that Spec(A) ir but T t = e t for all t R. 1 Zabczyk 1975 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

8 The Schrödinger Group 2 The operators T t = e i t are unbounded on L p (R n ) for all p 2 and 0 t R in spite of the fact that Spec( ) R. 2 Hormander 1960 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

9 The Schrödinger Group 2 The operators T t = e i t are unbounded on L p (R n ) for all p 2 and 0 t R in spite of the fact that Spec( ) R. The resolvents of satisfy R z c p Im (z) 1 for all z / R, where c p 1 as p 2. 2 Hormander 1960 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

10 An Indefinite ODE 3 If H = L 2 ( π, π) and 0 < ε < 2 and (Lf )(θ) = ε d { sin(θ) df } + df dθ dθ dθ then df = Lf (t) dt describes the evolution of a thin fluid layer inside a rotating cylinder. 3 Benilov, O Brien, Sazonov, Weir et al E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

11 An Indefinite ODE 4 If 0 < ε < 2 then L has purely imaginary spectrum consisting of a discrete sequence of eigenvalues. 4 Benilov, O Brien, Sazonov, Weir et al E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

12 An Indefinite ODE 4 If 0 < ε < 2 then L has purely imaginary spectrum consisting of a discrete sequence of eigenvalues. If ε > 2 then the spectrum of L includes the entire imaginary axis and probably the entire complex plane. 4 Benilov, O Brien, Sazonov, Weir et al E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

13 An Indefinite ODE 4 If 0 < ε < 2 then L has purely imaginary spectrum consisting of a discrete sequence of eigenvalues. If ε > 2 then the spectrum of L includes the entire imaginary axis and probably the entire complex plane. If 0 < ε < 2 the resolvent operators are all compact but e Lt is unbounded for all t 0. 4 Benilov, O Brien, Sazonov, Weir et al E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

14 An Example with an Oscillating Norm Let (T t f )(x) = for all f L 2 (0, ) and all t 0. a(x + t) f (x + t) a(x) E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

15 An Example with an Oscillating Norm Let (T t f )(x) = for all f L 2 (0, ) and all t 0. a(x + t) f (x + t) a(x) If c > 1 then the choice a(x) = 1 + (c 1) sin 2 (πx/2) leads to T 2n = 1 and T (2n+1) = c for all positive integers n. Study of N(t) E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

16 Definition of N(t) 5 We define N(t) to be the upper log-concave envelope of T t. In other words ν(t) = log(n(t)) is defined to be the smallest concave function satisfying ν(t) log( T t ) for all t 0. 5 The following is based on EBD 2004, inspired by L N Trefethen E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

17 Definition of N(t) 5 We define N(t) to be the upper log-concave envelope of T t. In other words ν(t) = log(n(t)) is defined to be the smallest concave function satisfying ν(t) log( T t ) for all t 0. It is immediate that N(t) is continuous for t > 0, and that 1 = N(0) lim t 0+ N(t). 5 The following is based on EBD 2004, inspired by L N Trefethen E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

18 Normalization We replace T t by T t e ω 0t or, equivalently, normalize our problem by assuming that ω 0 = 0. This implies that Spec(A) {z : Re (z) 0}. It also implies that T t 1 for all t 0. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

19 Normalization We replace T t by T t e ω 0t or, equivalently, normalize our problem by assuming that ω 0 = 0. This implies that Spec(A) {z : Re (z) 0}. It also implies that T t 1 for all t 0. N(t) is increasing function of t but it increases sub-exponentially as t +. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

20 The Legendre Transform We study the function N(t) via a transform, defined for all ω > 0 by M(ω) = sup{ T t e ωt : t 0}. M(ω) is a monotonic decreasing function of ω which satisfies lim ω + Hence M(ω) 1 for all ω > 0. M(ω) = lim sup T t. t 0 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

21 The Legendre Transform We study the function N(t) via a transform, defined for all ω > 0 by M(ω) = sup{ T t e ωt : t 0}. M(ω) is a monotonic decreasing function of ω which satisfies lim ω + Hence M(ω) 1 for all ω > 0. M(ω) = lim sup T t. t 0 N(t) = inf{m(ω)e ωt : 0 < ω < } for all t > 0 by the theory of the Legendre transform. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

22 Theorem If a > 0, b R and a R a+ib = c 1 then { (a ω)c/a if 0 < ω r = a(1 1/c) M(ω) M(ω) := 1 otherwise. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

23 Theorem If a > 0, b R and a R a+ib = c 1 then { (a ω)c/a if 0 < ω r = a(1 1/c) M(ω) M(ω) := 1 otherwise. Proof. The formula implies that R a+ib = 0 T t e (a+ib)t dt c/a 0 N(t)e at dt for all ω such that 0 < ω < a. 0 M(ω)e ωt at dt = M(ω)(a ω) 1 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

24 The following theorem implies that if the resolvent norm is significantly larger than 1/a for some large a then N(t) must grow rapidly for small t > 0. Theorem If a R a+ib = c 1 and r = a(1 1/c) then for all t 0. N(t) min{e rt, c} E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

25 The following theorem implies that if the resolvent norm is significantly larger than 1/a for some large a then N(t) must grow rapidly for small t > 0. Theorem If a R a+ib = c 1 and r = a(1 1/c) then for all t 0. Proof. This uses N(t) min{e rt, c} N(t) = inf{m(ω)e ωt : ω > 0} inf{ M(ω)e ωt : ω > 0}. Schrodinger operators E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

26 Schrödinger Operators with Non-Negative Potentials Theorem Let H = + V, acting in L 1 (R n ), where V 0. Then (e Ht f )(x) = K(t, x, y)f (y) dy R n where 0 K(t, x, y) (4πt) n/2 e x y 2 /4t. This can be proved by functional integration or the Trotter product formula. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

27 Schrödinger Operators with Potentials Bounded Below Theorem Let H = + V, acting in L 1 (R n ), where V is continuous and bounded below, with c = inf{v (x) : x R N }. Then c = min{ω : e Ht e ωt for all t 0}. Note that the situation is quite different in L 2 (R n ). E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

28 Polynomial Growth of L 1 Norms Theorem (Murata 1984, 1985 and Davies-Simon 1991.) Let N 3. There exists a Schrödinger semigroup e Kt acting in L 1 (R N ) and positive constants c 1, c 2, σ 1 and σ 2 such that c 1 (1 + t) σ 1 e Kt c 2 (1 + t) σ 2 for all t 0, even though K is non-negative considered as an operator acting in L 2 (R N ). The constants σ 1 and σ 2 are more or less equal. The proof involves zero energy resonances. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

29 The Explicit Example The potential is given by V (x) = { c x 2 if x 1 0 otherwise. where 0 < c < (n 2)2. 4 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19

30 The Explicit Example The potential is given by V (x) = { c x 2 if x 1 0 otherwise. where 0 < c < (n 2)2. 4 The zero energy resonance is of the form { x α 1 β x 0 < η(x) = α 2 if x 1 1 β otherwise for certain positive constants α 1, α 2 and β. E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September / 19