Generalized Schrödinger semigroups on infinite weighted graphs

Transcription

1 Generalized Schrödinger semigroups on infinite weighted graphs Institut für Mathematik Humboldt-Universität zu Berlin QMATH 12 Berlin, September 11, 2013

2 B.G, Ognjen Milatovic, Francoise Truc: Generalized Schrödinger semigroups on infinite graphs. Preprint (2013). Partially: B.G., Matthias Keller, Marcel Schmidt: A Feynman-Kac-Ito formula for magnetic Schrödinger operators on graphs. Preprint (2013).

3 Motivation Classical fact: If H is the self-adjoint operator corresponding to a regular Dirichlet form on some locally compact space X with a Borel measure m (for example the canonical Dirichlet form on a weighted graph), then for appropriate v : X R one has the Feynman-Kac formula e t(h+v) f (x) = E x [ e t 0 v(xs)ds f (X t ) ], f L 2 (X, m). (1) Generalizations of these probabilistic formulae also hold for covariant Schrödinger type operators acting on sections in Hermitian vector bundles over noncompact Riemannian manifolds On the other hand, natural extensions of the latter covariant operators ( = LHS of (1)) exist also on weighted graphs. What about the RHS and the validity of (1) in the covariant weighted discrete graph setting?

4 Motivation Classical fact: If H is the self-adjoint operator corresponding to a regular Dirichlet form on some locally compact space X with a Borel measure m (for example the canonical Dirichlet form on a weighted graph), then for appropriate v : X R one has the Feynman-Kac formula e t(h+v) f (x) = E x [ e t 0 v(xs)ds f (X t ) ], f L 2 (X, m). (1) Generalizations of these probabilistic formulae also hold for covariant Schrödinger type operators acting on sections in Hermitian vector bundles over noncompact Riemannian manifolds On the other hand, natural extensions of the latter covariant operators ( = LHS of (1)) exist also on weighted graphs. What about the RHS and the validity of (1) in the covariant weighted discrete graph setting?

5 Motivation Classical fact: If H is the self-adjoint operator corresponding to a regular Dirichlet form on some locally compact space X with a Borel measure m (for example the canonical Dirichlet form on a weighted graph), then for appropriate v : X R one has the Feynman-Kac formula e t(h+v) f (x) = E x [ e t 0 v(xs)ds f (X t ) ], f L 2 (X, m). (1) Generalizations of these probabilistic formulae also hold for covariant Schrödinger type operators acting on sections in Hermitian vector bundles over noncompact Riemannian manifolds On the other hand, natural extensions of the latter covariant operators ( = LHS of (1)) exist also on weighted graphs. What about the RHS and the validity of (1) in the covariant weighted discrete graph setting?

6 Setting I i) Let (X, b, m) be a possibly locally unbounded weighted graph, that is, X is a countable set (the vertices) b : X X [0, ) is symmetric with b(x, x) = 0 and y X b(x, y) < (the set of x, y X with x b y : b(x, y) > 0 are the neighbors). m : X (0, ) some function (the weight function) ii) Let F X be a rank-ν vector bundle, that is, F = x X F x with each F x a complex linear space with F x = C ν iii) Let (, ) x : F x F x C, x X, be a Hermitian structure on F X, that is, each (, ) x is a Hermitian product

7 Setting I i) Let (X, b, m) be a possibly locally unbounded weighted graph, that is, X is a countable set (the vertices) b : X X [0, ) is symmetric with b(x, x) = 0 and y X b(x, y) < (the set of x, y X with x b y : b(x, y) > 0 are the neighbors). m : X (0, ) some function (the weight function) ii) Let F X be a rank-ν vector bundle, that is, F = x X F x with each F x a complex linear space with F x = C ν iii) Let (, ) x : F x F x C, x X, be a Hermitian structure on F X, that is, each (, ) x is a Hermitian product

8 Setting I i) Let (X, b, m) be a possibly locally unbounded weighted graph, that is, X is a countable set (the vertices) b : X X [0, ) is symmetric with b(x, x) = 0 and y X b(x, y) < (the set of x, y X with x b y : b(x, y) > 0 are the neighbors). m : X (0, ) some function (the weight function) ii) Let F X be a rank-ν vector bundle, that is, F = x X F x with each F x a complex linear space with F x = C ν iii) Let (, ) x : F x F x C, x X, be a Hermitian structure on F X, that is, each (, ) x is a Hermitian product

9 Setting II x denotes the norm and operator norm on F x given by (, ) x we get the corresponding spaces Γ l p m (X, F ) of l p m-sections (depend on (, )): Γ l p m (X, F ) = { f : X F f (x) F x for all x, f l p (X, m) } Γ l 2 m (X, F ) is our Hilbert space iv) Let Φ be a unitary b-connection on F X, that is, Φ assigns to any x b y some unitary map Φ x,y : F x F y such that Φ y,x = Φ 1 x,y (depends on (, ))

10 Setting II x denotes the norm and operator norm on F x given by (, ) x we get the corresponding spaces Γ l p m (X, F ) of l p m-sections (depend on (, )): Γ l p m (X, F ) = { f : X F f (x) F x for all x, f l p (X, m) } Γ l 2 m (X, F ) is our Hilbert space iv) Let Φ be a unitary b-connection on F X, that is, Φ assigns to any x b y some unitary map Φ x,y : F x F y such that Φ y,x = Φ 1 x,y (depends on (, ))

11 Operators Define a sesqui-linear form Q Φ,0 in Γ l 2 (X, F ) with m D(Q Φ,0 ) := compactly supported sections in F X, and Q Φ,0 (f 1, f 2 ) := 1 ( ) b(x, y) f 1 (x) Φ y,x f 1 (y), f 2 (x) Φ y,x f 2 (y) 2 Q Φ,0 x b y is densely defined, symmetric, closable, and nonnegative: Q Φ,0 := Q Φ,0 Let V be a potential on F X, that is, V is a pointwise self-adjoint section in End(F ) X (depends on (, )) V determines a maximally defined quadratic form Q V in Γ l 2 m (X, F ) If V = V + V for some potentials V ± 0 such that Q V is Q Φ,0 -bounded with bound < 1, then Q Φ,V := Q Φ,0 + Q V is densely defined, symmetric, closed and bounded from below H Φ,V := the corresponding (Schrödinger type!) operator x

12 Question Precise version of our initial question: Is there a probabilistic representation of the Schrödinger SG (e th Φ,V ) t 0 L (Γ l 2 m (X, F ))?

13 Underlying processes I i) Let H be the operator corresponding to the canonical regular Dirichlet form Q in l 2 (X, m) given by (X, b, m), and let (Ω, F, (F t ) t 0, (X t ) t 0, (P x ) x X ) be the corresponding right Markoff process it is in fact a jump process let τ n : Ω [0, ] be its jump times, N(t) : Ω N 0 { } its number of jumps until t 0, and τ = sup n τ its explosion time ii) The (pathwise unitary!!!) process // Φ : [0, τ) Ω F F := 1 FX0, if N(t) = 0 // Φ t := 1 j N(t) Φ X τj 1,X τj else (x,y) X X Hom(F y, F x ) Hom(F X0, F Xt ) is called the Φ-parallel transport along the paths of X.

14 Underlying processes I i) Let H be the operator corresponding to the canonical regular Dirichlet form Q in l 2 (X, m) given by (X, b, m), and let (Ω, F, (F t ) t 0, (X t ) t 0, (P x ) x X ) be the corresponding right Markoff process it is in fact a jump process let τ n : Ω [0, ] be its jump times, N(t) : Ω N 0 { } its number of jumps until t 0, and τ = sup n τ its explosion time ii) The (pathwise unitary!!!) process // Φ : [0, τ) Ω F F := 1 FX0, if N(t) = 0 // Φ t := 1 j N(t) Φ X τj 1,X τj else (x,y) X X Hom(F y, F x ) Hom(F X0, F Xt ) is called the Φ-parallel transport along the paths of X.

15 Underlying processes II iii) For any ω Ω there is a unique solution of the initial value problem t Vt Φ (ω) = 1 FX0 (ω) 0 V Φ (ω) : [0, τ(ω)) End(F X0 (ω)) V Φ s (ω) // Φ, 1 s V (X s )// Φ s we have canonically associated the process with Φ and V. V Φ : [0, τ) Ω End(F ) ω ds in End(F X0 (ω)).

16 Feynman-Kac formula Note that for any x X, t 0, one has // Φ t (ω) Hom(F x, F Xt(ω)), Vt Φ (ω) End(F x ) for P x -a.e. ω {t < τ}. Main result: Theorem (B.G., O. Milatovic, F. Truc) Assume that V admits a decomposition V = V + V into potentials V ± 0 such that Q V is Q-bounded with bound < 1. Then Q V is Q Φ,0 -bounded with bound < 1, and for any f Γ l 2 m (X, F ), t 0, x X one has [ ] e th Φ,V f (x) = E x 1 {t<τ} Vt Φ // Φ, 1 t f (X t ). (2) Generalizes main result of B.G/M. Keller/M. Schmidt away from scalar magnetic Schrödinger SG s on (X, b, m) (the latter however deals with larger class of V s)

17 Feynman-Kac formula Note that for any x X, t 0, one has // Φ t (ω) Hom(F x, F Xt(ω)), Vt Φ (ω) End(F x ) for P x -a.e. ω {t < τ}. Main result: Theorem (B.G., O. Milatovic, F. Truc) Assume that V admits a decomposition V = V + V into potentials V ± 0 such that Q V is Q-bounded with bound < 1. Then Q V is Q Φ,0 -bounded with bound < 1, and for any f Γ l 2 m (X, F ), t 0, x X one has [ ] e th Φ,V f (x) = E x 1 {t<τ} Vt Φ // Φ, 1 t f (X t ). (2) Generalizes main result of B.G/M. Keller/M. Schmidt away from scalar magnetic Schrödinger SG s on (X, b, m) (the latter however deals with larger class of V s)

18 Some applications of Feynman-Kac formula I For any (x, y) X X, t > 0, the integral kernel of e th Φ,V is given by e th Φ,V (x, y) End(E y, E x ) e th Φ,V (x, y) = 1 [ ] m(y) Ex 1 {Xt=y}Vt Φ // t Φ, 1 ; formula for tr(e th Φ,V ) [0, ] Semigroup domination: If w : X R is such that V w (and s.t. H w is well-defined), then e th Φ,V f (x) x e t(h w) f (x), in particular min(σ(h Φ,V )) min(σ(h w))

19 Some applications of Feynman-Kac formula I For any (x, y) X X, t > 0, the integral kernel of e th Φ,V is given by e th Φ,V (x, y) End(E y, E x ) e th Φ,V (x, y) = 1 [ ] m(y) Ex 1 {Xt=y}Vt Φ // t Φ, 1 ; formula for tr(e th Φ,V ) [0, ] Semigroup domination: If w : X R is such that V w (and s.t. H w is well-defined), then e th Φ,V f (x) x e t(h w) f (x), in particular min(σ(h Φ,V )) min(σ(h w))

20 Golden-Thompson inequalty: If w : X R is such that V w (and s.t. H w is well-defined), then e th (x, x)e tw(x) m(x) < x tr(e th Φ,V ) x e th (x, x)tr Fx (e tv (x) )m(x) The latter proof is rather technical: A naive approach only gives the (nevertheless illustrative) bound tr(e th Φ,V ) rank(f ) x e th (x, x)e tw(x) m(x). Γ l 2 m (X, F ) Γ l p m (X, F ), p [2, ], smoothing properties of e th Φ,V, if V is Kato and sup x,y e th (x, y) <

21 Golden-Thompson inequalty: If w : X R is such that V w (and s.t. H w is well-defined), then e th (x, x)e tw(x) m(x) < x tr(e th Φ,V ) x e th (x, x)tr Fx (e tv (x) )m(x) The latter proof is rather technical: A naive approach only gives the (nevertheless illustrative) bound tr(e th Φ,V ) rank(f ) x e th (x, x)e tw(x) m(x). Γ l 2 m (X, F ) Γ l p m (X, F ), p [2, ], smoothing properties of e th Φ,V, if V is Kato and sup x,y e th (x, y) <

22 Thank you!