fe-pseudodifferential - PDF document

a a i = L a two a space. a a a a a a = Z a a fe-pseudodifferential a finite sketched I a a a (dr) a T(E pô* ó(d/dr+D) a e a Ã(E) (-å, 0]xM c X -^ M x a P 0. a and its index is ¢(P) — � is the the heat kernel D and ç(D) is the D A a V a a a - d*)ù e T{K - examples a a a a a 0 ë = � := a = the a = - . a a a two = a x x a a a a / = a = I(d/dr / D = 0. a a - = a a / A a è = P P P / = L e e . a = , Lagrangian L) I a - ç(D - ç(D L =: d(D L a a äD) f a{D,äD), ß Ù{D,äD). L a right-hand right D E a / = copies = x a x x Lagrangian right L L := a = m(L Z a a := V / a y) = ® y ® a L = m(l = V a V := x - 1 F = . Moreover, a a a i - is a is an = (kL e K) + m(L right-hand - ç(D - ç(D = the the L the of (V a Z a a a Z . Z = 0 L) Ù{D,äD) = 0. = Z U M Z a L x := L E) D = ç(M) = ç(M a H D_ a P_ e ç(M - ç(M L = m{L - 2I(P . n a a a a a a a a a a a ¾ôDe dr � - dr � 0� � 0� n t) = � t) t) replace ^ t) = t) - R_(s, Þ) - � = ç(D a� Tr(D_e~ � 0 C � finite = E . side a 0 := s - - = + = + B(Þ - + 2K-B- 2K) - a A D t) R t) - R_(0, t) a copies Z . / = 3 , a + 0 + ç(M — Ak—\ i = 3 . / = a / E ç(M) = m{F{L) n - \ - is the L under the F on kerD In - ç(M)] e = a a / = = M U = - ç(M = - Z + + m(L L = = L := M U Z U� a r] x N M r M r Z a � P_) 0 � r a {(P right = finite / = a a L i = a . = . . x N x N. � / = � r = . a an 1 for all + uG, u e [0,1] a � r a an r for all ç(M + . - ç(M � r = a M M IL + m(L IL + m(L i = n iö j an r all� rr IL - m{L IL IL - m(L IL - ç{M IL = m{L IL - IL - ç(M ILJ - m(L IL - ç(M IL IL - m(L IL a . x a F±ÈF~ Moreover, M dX_ a Ã(F±) F F * M i a = M {ø e = = kerB i L = iö i,j = an r for all + - = / = a N N N M / E , M a i = 2 . an r for all r� r - ç(D - m{L = ç(M) - IF(L - m{F{L + - ç(M = m(F(L - m(L - = - A a a a Ã(End(E = a a - D.(s), s, t e a a (t) a e Moreover, e a = = are 'continuous' such that the = t = a a - P - sf{D_(t)} + r� r for and L w 1 for all = sf{D + sf{D - sf{D a a a finished. a a Moreover, finds a a a a a C / and are a e X a C ^ = a 1 g e / a = G = H*(G, G a a G is a compact X such that G measurable, then H*(C = 0. finite a associated a ô = ô a = Ë Ë a a / = 3 . - a ô e = 0. e C = ô S\ a 0 H a find a a a / a fix a = a is a unique ì e C\ (K) / := ms^ô) ô f ô(kl,l K = mds^ô) = = C / = dì ô. ì = a e = ì Ë = '.— a = = = = Q, I(Q) = -P. = (Q, Q) = 0. G = = = = / £ a = = 0 = 1 = m F a ( , ) a / = / a := (Ix Lagrangian / - / Ë . u e K. l A = l, = I) = AA~ e 1 K Ë + = , G = f a a = l = m(l = m(L IL + m(L IL + m{L IL i = i ^ then L) = e K IL) = IL). IL) A = ô(L L L = = L = + + An a M) a &-ç(ëß), is Hirzebruch-Jú? X N) -M dX M dX_ a + 3 - & - S? - ç(M any i = - N and a the M = m e = m nN), (X_, nM -nM a � e L(H) A e a = P P e È � = - - - P are H, then - f is a /� 0 (Q(PQ) P(QP) = 2(Q(PQ) - - PQP) a e 1 e e / = 0 := 1 - a Z 1 0 \— y b, c, d: H -+ H / ¥ / Z Z Z + + = = c*b, b*c = d*a, a*b = = = = = = l a x x N e + is to P_) e L for� N2n. = + / - â + J_ � = J- Ð « + + å + � 0 a = e~ = = ) , = precompact, a � � � = (ä(x),D finite and, ø e V� 0 [D °-D \ éëD � � n/2 y) a all x, y e M a e M c \ë\) c i = x, y e + A t) e- E finite x + = t x x 0 cZuZ¼ = 0 x . x,y) � C Ce~ Ce' Ce~ y) - Ce~ Ce~ y))r(x)\ Ce~ R(s (R(s,t) t, x e the = t)_)l R{s � x x R(s t) t) K(t) Z a (M a L constructions e x x finite are� C for � t0, (x,y)eM WJt,x,y))\ y) -D_ÚV_(t, x, y))\ W_(t,x,y))l(x)\ Ce~ x, y) - D_W_(t, WJt,x,y))\ Ce~ y) -D_W_(t, x, y))\ W_(t,x,y))r(x)\ Ce~ x, y) -D_ÚV_(t, x the (R(s R(s (R(s,t) (s, t, x, y) e x x the t)_) = l(R(0, 0) t)_) = r(R{0, = R(s, t)_) = = x U x v(R(s R(s are uni- (s e x x = finite � ß� 0 - A = R(s . Moreover, + - + 2K(Þ) + - - + = + - - limit - B(Þ - 2K{Þ) + - - the Hence, = + = -2I(P . - B{t) - - B(t) - - first a / a x x = l[P ° - l of P_{t)) is (t e x e~ - e~ / finite l - are (t e x = lP - - - P right-hand - + - l - P find a a 0 ~ P = lP - P a - l a finite = P_(t)] / a = v x x N-i N-i-l 7 Q]v - = P_(t)](Q Q) a a finishes ç(M - ç(M - = G a D — tG (P . A a right a a {pô a := keôD is a V, i.e., L and Ö L ø e u, -P ø) - {ö, Bø) = {Iu, e L a e kerB = x x x Z L = 5 = = a = 0]x N a finishes a� a x r N) x a P_) e := D D_ r U Z . U Z (NU N), a Z , a Z U Z - l)G x right x e domB(u) e x N e {L a a h L L UZ,E be ø with ø(s) e V È V for all s e [-r, r]. Then B(u) h = two copies Z . � � r � � r E e (N_ . � -ìlh ëH L a x(NuN), x(NuN). L e L e = ö ö e = 0 - 1 ^ = ø ø_ + ø Z . ø L := + x - - M � the r� r e ø be a B(u) to the d/2 - ëK(ø)\\ C(\ë\/Vf O . ø two - ëK(ø) 0 first x - 1 = \\ø y/7/2\\ø - x C(l/y/ú)e- x M first a a are constants � 1 all r� r following holds: ø be a D corresponding eigenvalue assume a / = J á - 2/V?. ^ � � ^ — 1 * - � oo,� c r all r� w be a the c a a oo, c� 0, all r� r u e be a an � � e c a 0 1 - å, -r] x N. a = ø x N of e å, -r] y(ø Ø-) C/y/r e x e (-å ,0]xN. e - x N = - + are constants C c� all� rr u e following a normed eigenvector of D(u) corresponding to an eigenvalue ë with c the boundary values on the first component the following c(ë first a / = � all r� r u e be a an Then c� all r� r u e be a the c. the and a = ø + ø Z . N e ã) + = ö ö C{ë e~ C(ë e~ e = a 1 + 0 , = ø + ø Z . + - x + - 1 * G are � 1 for all r� r ø be a D(u) an c. Then - ëA(ø)\\ C(ë O . ø two - ëA(ø) 0 + - + x 0 - x - l N. - ëA(ø)\\ C(ë a fix d/2 � r is an r r for r� r we have equality of the = sf{D(u)}. ë e a a = r e[0, I] e - e a � r c e [u~ e [u~ C � 0 c, a ë/y/r + e e [u~ Ëv) a , u*] finitely - £ e + e-") c ë_ 0 . 0 = sf{B(u) finishes = = B - u) = sf{B & asymmetry and Riemannian & quantum Hall effect and the for , quantum Hall effect and the for & kernels and Dirac & on the families index theorem for manifolds & analysis of & boundary Dirac of the ç-function for an Dirac type, Dirac manifolds , the families of Dirac , theorem and , for manifolds , for the eta-invariant, , the , ç-invariant as a Lagrangian of & , the Maslov index, , the adiabatic approximation and conical & estimates for of the Laplace the manifolds, in & limits of the ç-invariants, the odd- Atiyah-Patodi-Singer problem, theory, the heat equation, and the Atiyah-Singer index theorem, & curvature and the Dirac com- manifolds, & theory for linear & cup & & ç-invariant of value of ç on the boundary condition and a ð & Weil index and theta & the ç-invariant, ç-invariant, , and the ç-invariant, and manifolds Maslov index, the manifolds, of the the additivity of the ç-invariant, manifolds,