2 2 1 5 5 Schrödinger i u t + u = λ u 2 u.
u = u(t, x 1,..., x d ) : R R d C λ i = 1 := 2 + + 2 x 2 1 x 2 d d Euclid Laplace Schrödinger 3 1 1.1 N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3,... } Q := { q p, q Z, p } p R 2 π e Q R 2 R
ε-δ 1 {q n } n=1 q 1 := 1, q n+1 := 1 + 1, n = 1, 2, 3,... q n + 1 q q q = 1 + 1 q + 1, q 1 2 q = 2 1 1 1 2 1 2 2 Euclid d R d d Euclid C R 2 i R 2 1.2 X Y 1 Y X R 1+d C-
R f x {x n } n=1 x {f(x n )} n=1 f(x ) x R f x f(x + h) f(x ) lim h, h h x d dx f(x ), df dx (x ) x x f f R d R 2 x, y d(x, y) := x y = max{x y, y x} = (x y) 2 d Euclid d((x 1,..., x d ), (y 1,..., y d )) := (x 1 y 1 ) 2 + + (x d y d ) 2 1 j 1 d f x = (x 1,..., x d ) Rd j x j f(x + he j ) f(x ) lim h, h h e j R d x j j 1 x j f(x ), f x j (x ) x R d f x j (x ) j 1 f x x
f f C 1 C 1 C 1 C 2 C 3,..., C ( f ) x 1 x 1 = 2 f, x 2 1 ( f ) = 2 f x 1 x 2 x 1 x 2 Schrödinger 2 R 1+d C 2 (i) Cantor [, 1] (ii) Weierstrass R (iii) R x, y R f(x + y) = f(x) + f(y) f(x) = ax (a R ) 1.3 d Euclid R d C k R d C k C C(R d ), C k (R d ), C (R d ) C (R d ), C k (R d ) R d R d C Cc (R d )
C, C k f C(R d ) := sup x R d f(x), g C k (R d ) := sup α f x (x) α α k x R d R d R d 2 2 C, C k Cc Cc C, C k C c 1 Cc (R d ) p L p (R d ) (1 p < ) L p L p ( f L p (R d ) := f(x) p dx R d Cc (R d ) L p L p L p Cc L p L 2 Fourier L 2 Sobolev L 2 L p ) 1/p 2 Schrödinger 2 t x = (x 1,..., x d ) t x u(t, x) t x u(t, x) x u (x) u(, x) = u (x) u(t, x)
Schrödinger u = i u + F, t u(, x) = u (x). F = F (t, x) u = u (x) F, u (1) 2.1 Fourier Sobolev R d (1) R d Fourier R d f Fourier Ff = ˆf Fourier F 1 ˆf(ξ) := F 1 f(x) := 1 e ix ξ f(x) dx, (2π) d/2 R d 1 (2π) d/2 R d e ix ξ f(ξ) dξ, ξ R d x R d x ξ := d j=1 x jξ j d Euclid f Cc (R d ) ˆf C (R d ) F( f x j )(ξ) = iξ j ˆf(ξ) (j = 1, 2,..., d), F( f)(ξ) = ξ 2 ˆf(ξ)
x Fourier Fourier Fourier 1 F(fg) = (2π) ˆf ĝ, ( ˆf ĝ)(ξ) := ˆf(ξ η)ĝ(η) dη. d/2 R d Fourier L 2 Plancherel ˆf L 2 (R d ) = F 1 f L 2 (R d ) = f L 2 (R d ). C c L 2 L 2 C c F F 1 L 2 (R d ) s R d Sobolev H s (R d ) Sobolev H s C c (R d ) ( f H s := (1 + 2 ) s/2 ˆf( ) L 2 (R d ) = (1 + ξ 2 ) s ˆf(ξ) 2 dξ R d s = H s L 2 s H s f Fourier ˆf(ξ) s ξ x s Sobolev H s s L 2 k ε > H k+ d 2 +ε (R d ) C k (R d ) Sobolev s d/2 ) 1/2. 2.2 F = (1) t x Fourier û t (t, ξ) = i ξ 2 û(t, ξ), û(, ξ) = û (ξ) ξ ξ û(t, ξ) = e it ξ 2 û (ξ) Fourier u(t, x) = F 1[ e it 2 û ( ) ] (x).
t u(t, x) Fourier û(t, ξ) Fourier û (ξ) 1 e it ξ 2 u Sobolev H s t u(t, ) Sobolev H s s > 2 + d 2 u(t, x) x C2 t C 1 Fourier L 2 C 2 u u(t, x) C 2 u [e it u ](x) t R t e it : u u(t, ) s Sobolev H s (R d ) u u Cc (R d ) Fourier u(t, x) = [e it 1 x y u ](x) = 2 (4πit) d/2 4it u (y) dy t R d e F Fourier x Fourier û t (t, ξ) = i ξ 2 û(t, ξ) + ˆF (t, ξ), û(t, ξ) = e it ξ 2 û (ξ) + û(, ξ) = û (ξ) e i(t t ) ξ 2 ˆF (t, ξ) dt Fourier u(t, x) = [e it u ](x) + [e i(t t ) F (t, )](x) dt
3 Schrödinger u t = i u + iλ u 2 u, u(, x) = u (x) (2) 3.1 (2) (1) F iλ u 2 u u (2) F iλ u 2 u u(t, x) = [e it u ](x) + iλ [e i(t t ) ( u 2 u)(t, )](x) dt (3) u (3) u (2) Duhamel (3) (3) 2 (2) u (2) (3) u (3) u (2) (3) u (X, d) Φ : X X < α < 1 x, y X d(φ(x), Φ(y)) αd(x, y) Φ(x) = x x X Φ 1 ( ).
x X x 1 := Φ(x ), x 2 := Φ(x 1 ),... X {x n } n= n, m (n < m) Φ d(x m, x n ) d(x m, x m 1 ) + d(x m 1, x m 2 ) + + d(x n+1, x n ) = d(φ m 1 (Φ(x )), Φ m 1 (x )) + + d(φ n (Φ(x )), Φ n (x )) { α m 1 + α m 2 + + α n} d(φ(x ), x ) = αn 1 α (1 αm n )d(φ(x ), x ) αn 1 α d(φ(x ), x ) (n ). n n {x n } n= X x X x n+1 = Φ(x n ) n x = Φ(x ) x Φ x, y X Φ 2 Φ d(x, y) = d(φ(x), Φ(y)) αd(x, y). < α < 1 d(x, y) = x y u Φ[u ] Φ[u ] : u(t, x) [e it u ](x) + iλ [e i(t t ) ( u 2 u)(t, )](x) dt u (3) Φ[u ] u Φ[u ] Φ[u ] (3) u x (3) 1 2 u (i) Φ[u ] (ii) 2
3.2 Sobolev Φ[u ] 2 (Sobolev ). s > d/2 H s (R d ) 2 H s (R d ) d, s C fg H s C f H s g H s, f, g H s (R d ) 2 Cauchy-Schwarz fg L 1 f L 2 g L 2 Young f g L 2 f L 1 g L 2 a, b, c (1 + (a + b) 2 ) c 2 max{2c 1,c}{ (1 + a 2 ) c + (1 + b 2 ) c} s ξ, η R d (1 + ξ 2 ) s/2 { 1 + ( ξ η + η ) 2} s/2 { 2 max{s 1,s/2} (1 + ξ η 2 ) s/2 + (1 + η 2 ) s/2}. (1 + ξ 2 ) s/2 ( ˆf ĝ)(ξ) (1 + ξ 2 ) R s/2 ˆf(ξ η) ĝ(η) dη { d 2 max{s 1,s/2} (1 + ξ η 2 ) s/2 ˆf(ξ η) ĝ(η) dη + ˆf(ξ } η) (1 + η 2 ) s/2 ĝ(η) dη R d R d 2 max{s 1,s/2}{ [(1 + 2 ) s/2 ˆf ] ĝ + ˆf [(1 + 2 ) s/2 ĝ ] }. L 2 Young Cauchy-Schwarz (1 + 2 ) s/2 ( ˆf ĝ) L 2 2 max{s 1,s/2}{ (1 + 2 ) s/2 ˆf L 2 ĝ L 1 + ˆf } L 1 (1 + 2 ) s/2 ĝ L 2 2 max{s 1,s/2} 2 (1 + 2 ) s/2 L 2 (1 + 2 ) s/2 ˆf L 2 (1 + 2 ) s/2 ĝ L 2. C := 1 (2π) d/2 2max{s 1,s/2} 2 (1 + 2 ) s/2 L 2 s > d/2 H s
3 ( ). s > d/2 u H s (R d ) T > [ T, T ] R d (3) u C([ T, T ]; H s (R d )) T s, d, λ c > T c u 2 H s C([ T, T ]; H s (R d )) [ T, T ] R d u(t, x) t [ T, T ] u(t, ) H s t u(t, ) [ T, T ] H s C([ T, T ]; H s (R d )) u C([ T,T ];H s ) := max T t T u(t, ) H s u H s T > Φ[u ] X := { u C([ T, T ]; H s (R d )) } u C([ T,T ];H s ) 2 u H s X C([ T, T ]; H s (R d )) u X Φ[u ](u) X H s e it u C([ T,T ];H s ) = max t Sobolev ū H s = u H s iλ = λ max T t T λ C 2 λ C 2 T e i(t t ) ( u 2 u)(t ) dt C([ T,T ];H s ) λ max max T t T u(t )u(t )u(t ) H s dt u(t ) 3 H s dt max T t T u(t) 3 H s λ C 2 T (2 u H s) 3, e it u H s = max u H s = u H s. t T t T e i(t t ) ( u 2 u)(t ) H s dt C Sobolev s, d T λ C 2 T 8 u 2 H s 1 Φ[u ](u) C([ T,T ];H s ) u H s + u H s = 2 u H s
Φ[u ](u) X Φ[u ](u) X u, v X Φ[u ](u) Φ[u ](v) C([ T,T ];H s ) iλ λ T max T t T u(t) 2 u(t) v(t) 2 v(t) H s. e i(t t ) ( u 2 u v 2 v)(t ) dt C([ T,T ];H s ) u 2 u v 2 v = (u v)( u 2 + v 2 ) + uv(u v) Φ[u ](u) Φ[u ](v) C([ T,T ];H s ) ( λ C 2 T u(t) 2 H s + v(t) 2 H + u(t) s H s v(t) ) H s u(t) v(t) H s max T t T λ C 2 T 3(2 u H s) 2 u v C([ T,T ];H s ). T λ C 2 T 12 u 2 H s < 1 Φ[u ](u) X T T = 1 16 λ C 2 u 2 H s 3.3 H d/2 L 2 (3) d = 1 Sobolev Strichartz C u L 2 (R) T > Schrödinger (1) u ( ) max u(t) L T t T 2 + u L 8 C u T L4 L 2 + F 8/7 L. T L4/3
1 p, q < L p T Lq u L p T Lq := { T ( ) p/q } 1/p u(t, ) L q (R) = L u(t, x) q dx p ( T,T ) dt T R Strichartz L 2 4 (L 2 ). u L 2 (R) T > [ T, T ] R d (3) u C([ T, T ]; L 2 (R)) L 8 T L4 T λ c > T c u 4 L 2 L 8 T L4 Cc (R 2 ) [ T, T ] R L 8 T L4 3 T > Φ[u ] X := { u L 8 T L 4 u L 8 T L 2C u } 4 L 2 C Strichartz 3.4 L 2 (3) T Schrödinger L 2 5 (L 2 ). u L 2 (R) u : [ T, T ] R C (3) t [ T, T ] u(t) L 2 = u L 2
u u u Cc (R) 3 u u(t) 2 L 2 t C 1 d dt u(t) 2 L = d u(t, x)u(t, x) dx = 2 dt R R { u } (t, x)u(t, x) + u(t, x) u t t (t, x) dx. 2 u (t, x) t ( d dt u(t) 2 L 2 = { (i 2 u R x + 2 iλ u 2 u ) u + u { 2 u = i R x u u } 2 u 2 (t, x) dx x 2 [{ u } ] = i x u u u (t, ) x =. i 2 u )} x 2 iλ u 2 u (t, x) dx i R { u u x x u u x x } (t, x) dx u(t) 2 L 2 (3) u L 2 u 3 u 3 u 2 u λ L 2 L 2 L 2 6 (L 2 ). u L 2 (R) R R (3) u C(R; L 2 (R))