The Schrdinger equation for the fractional Laplacian in negative - PowerPoint PPT Presentation

Introduction Homogeneous trees Hyperbolic spaces Conference Probability and Analysis (Będlewo, May 15 – 17, 2017) The Schrödinger equation for the fractional Laplacian in negative curvature Jean–Philippe Anker (Université d’Orléans) Joint work in progress with Yannick Sire (Johns Hopkins University, Baltimore) J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Equations Schrödinger equation � i ∂ t u ( t, x ) − ∆ x u ( t, x ) = F ( t, x ) u (0 , x )= f ( x ) Half wave equation � � i ∂ t u ( t, x ) + − ∆ x u ( t, x ) = F ( t, x ) u (0 , x )= f ( x ) Schrödinger equation for the fractional Laplacian � κ 2 u ( t, x ) = F ( t, x ) i ∂ t u ( t, x ) + ( − ∆ x ) (1) u (0 , x )= f ( x ) where 1 < κ < 2 (possibly also 0 < κ < 1 ) J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Equations Schrödinger equation � i ∂ t u ( t, x ) − ∆ x u ( t, x ) = F ( t, x ) u (0 , x )= f ( x ) Half wave equation � � i ∂ t u ( t, x ) + − ∆ x u ( t, x ) = F ( t, x ) u (0 , x )= f ( x ) Schrödinger equation for the fractional Laplacian � κ 2 u ( t, x ) = F ( t, x ) i ∂ t u ( t, x ) + ( − ∆ x ) (1) u (0 , x )= f ( x ) where 1 < κ < 2 (possibly also 0 < κ < 1 ) J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Equations Schrödinger equation � i ∂ t u ( t, x ) − ∆ x u ( t, x ) = F ( t, x ) u (0 , x )= f ( x ) Half wave equation � � i ∂ t u ( t, x ) + − ∆ x u ( t, x ) = F ( t, x ) u (0 , x )= f ( x ) Schrödinger equation for the fractional Laplacian � κ 2 u ( t, x ) = F ( t, x ) i ∂ t u ( t, x ) + ( − ∆ x ) (1) u (0 , x )= f ( x ) where 1 < κ < 2 (possibly also 0 < κ < 1 ) J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy Linear homogeneous equation : F = 0 Solution : � u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) = k κ t ( x, y ) f ( y ) dy Kernel estimate Dispersive estimate : � � e it ( − ∆) κ / 2 � � ∀ t ∈ R ∗ , ∀ 2 ≤ q ≤∞ L q ′ → L q Linear inhomogeneous equation : F � = 0 Duhamel formula : � t u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) + e i ( t − s )( − ∆ x ) κ / 2 F ( s, x ) ds 0 Strichartz inequality : � u ( t, x ) � L p x � � f � L 2 + � u ( t, x ) � L ˜ t L q p ′ q ′ t L ˜ x J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy Linear homogeneous equation : F = 0 Solution : � u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) = k κ t ( x, y ) f ( y ) dy Kernel estimate Dispersive estimate : � � e it ( − ∆) κ / 2 � � ∀ t ∈ R ∗ , ∀ 2 ≤ q ≤∞ L q ′ → L q Linear inhomogeneous equation : F � = 0 Duhamel formula : � t u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) + e i ( t − s )( − ∆ x ) κ / 2 F ( s, x ) ds 0 Strichartz inequality : � u ( t, x ) � L p x � � f � L 2 + � u ( t, x ) � L ˜ t L q p ′ q ′ t L ˜ x J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy Linear homogeneous equation : F = 0 Solution : � u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) = k κ t ( x, y ) f ( y ) dy Kernel estimate Dispersive estimate : � � e it ( − ∆) κ / 2 � � ∀ t ∈ R ∗ , ∀ 2 ≤ q ≤∞ L q ′ → L q Linear inhomogeneous equation : F � = 0 Duhamel formula : � t u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) + e i ( t − s )( − ∆ x ) κ / 2 F ( s, x ) ds 0 Strichartz inequality : � u ( t, x ) � L p x � � f � L 2 + � u ( t, x ) � L ˜ t L q p ′ q ′ t L ˜ x J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy Linear homogeneous equation : F = 0 Solution : � u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) = k κ t ( x, y ) f ( y ) dy Kernel estimate Dispersive estimate : � � e it ( − ∆) κ / 2 � � ∀ t ∈ R ∗ , ∀ 2 ≤ q ≤∞ L q ′ → L q Linear inhomogeneous equation : F � = 0 Duhamel formula : � t u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) + e i ( t − s )( − ∆ x ) κ / 2 F ( s, x ) ds 0 Strichartz inequality : � u ( t, x ) � L p x � � f � L 2 + � u ( t, x ) � L ˜ t L q p ′ q ′ t L ˜ x J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy Linear homogeneous equation : F = 0 Solution : � u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) = k κ t ( x, y ) f ( y ) dy Kernel estimate Dispersive estimate : � � e it ( − ∆) κ / 2 � � ∀ t ∈ R ∗ , ∀ 2 ≤ q ≤∞ L q ′ → L q Linear inhomogeneous equation : F � = 0 Duhamel formula : � t u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) + e i ( t − s )( − ∆ x ) κ / 2 F ( s, x ) ds 0 Strichartz inequality : � u ( t, x ) � L p x � � f � L 2 + � u ( t, x ) � L ˜ t L q p ′ q ′ t L ˜ x J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy (continued) Nonlinear equation : F ( t, x ) = � F ( u ( t, x )) with � F ( u ) power–like  u γ ( γ integer ≥ 2)   � | u | γ e.g. F ( u ) = const . ( γ > 1)   u | u | γ − 1 ( γ > 1) Main problem = local/global well–posedness ∼ existence and uniqueness of solutions Tools : Strichartz inequality Fixed point theorem Conservation law J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy (continued) Nonlinear equation : F ( t, x ) = � F ( u ( t, x )) with � F ( u ) power–like  u γ ( γ integer ≥ 2)   � | u | γ e.g. F ( u ) = const . ( γ > 1)   u | u | γ − 1 ( γ > 1) Main problem = local/global well–posedness ∼ existence and uniqueness of solutions Tools : Strichartz inequality Fixed point theorem Conservation law J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy (continued) Nonlinear equation : F ( t, x ) = � F ( u ( t, x )) with � F ( u ) power–like  u γ ( γ integer ≥ 2)   � | u | γ e.g. F ( u ) = const . ( γ > 1)   u | u | γ − 1 ( γ > 1) Main problem = local/global well–posedness ∼ existence and uniqueness of solutions Tools : Strichartz inequality Fixed point theorem Conservation law J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Standard strategy (continued) Nonlinear equation : F ( t, x ) = � F ( u ( t, x )) with � F ( u ) power–like  u γ ( γ integer ≥ 2)   � | u | γ e.g. F ( u ) = const . ( γ > 1)   u | u | γ − 1 ( γ > 1) Main problem = local/global well–posedness ∼ existence and uniqueness of solutions Tools : Strichartz inequality Fixed point theorem Conservation law J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Homogeneous trees T = T Q homogeneous tree with Q +1 ≥ 3 edges Example : Q =5 Discrete analogs of hyperbolic spaces Volume of balls of radius r ∈ N : V ( r ) = 1+ Q +1 Q − 1 ( Q r − 1) ≍ Q r Combinatorial Laplacian on (the vertices of) T : � 1 ∆ f ( x ) = f ( y ) − f ( x ) Q +1 d ( y,x )=1 J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Homogeneous trees T = T Q homogeneous tree with Q +1 ≥ 3 edges Example : Q =5 Discrete analogs of hyperbolic spaces Volume of balls of radius r ∈ N : V ( r ) = 1+ Q +1 Q − 1 ( Q r − 1) ≍ Q r Combinatorial Laplacian on (the vertices of) T : � 1 ∆ f ( x ) = f ( y ) − f ( x ) Q +1 d ( y,x )=1 J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Homogeneous trees T = T Q homogeneous tree with Q +1 ≥ 3 edges Example : Q =5 Discrete analogs of hyperbolic spaces Volume of balls of radius r ∈ N : V ( r ) = 1+ Q +1 Q − 1 ( Q r − 1) ≍ Q r Combinatorial Laplacian on (the vertices of) T : � 1 ∆ f ( x ) = f ( y ) − f ( x ) Q +1 d ( y,x )=1 J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Homogeneous trees T = T Q homogeneous tree with Q +1 ≥ 3 edges Example : Q =5 Discrete analogs of hyperbolic spaces Volume of balls of radius r ∈ N : V ( r ) = 1+ Q +1 Q − 1 ( Q r − 1) ≍ Q r Combinatorial Laplacian on (the vertices of) T : � 1 ∆ f ( x ) = f ( y ) − f ( x ) Q +1 d ( y,x )=1 J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces Schrödinger equation on T On T , the Schrödinger equation (with continuous time) � κ 2 u ( t, x ) = F ( t, x ) i ∂ t u ( t, x ) + ( − ∆ x ) (1) u (0 , x )= f ( x ) can be solved by using the Fourier transform : � t u ( t, x ) = e it ( − ∆) κ / 2 f ( x ) e i ( t − s )( − ∆ x ) κ / 2 F ( s, x ) ds + 0 � �� � � �� � homogeneous inhomogeneous where � e it ( − ∆) κ / 2 f ( x ) = y ∈ T f ( y ) k κ t ( d ( x, y )) � �� � f ∗ k κ t ( x ) J.–Ph. Anker P&A Będlewo 2017