II Spectral aan expansions

Consider µ H D Peet A acts to giefgdbj DIA.fin.snAssetSpeet functie

5 A tr fl 0 AsoMaintert invariante verder NIA PerthAlso there is a topological spectral actin

Stopt try floydwe uit assume throughout that f II a Laplaceband

flink fgteit dtto

Prod Swa Ho index 1A

Proof Mckean Singer try et TAK index D A

while Hot fglt dt is

Attend two interesting expansion for 5 A

Asymptotic expansion in 1 tellen ettectiebehnI for energie 1

Taylor expansion in A tells about propagationinteracties

anet in physics applications

p

Asymptotic in A

Suppose te e tttm n ze tv

Proper A NE Eijblaaitput Hot GIANT OM

where fp G thup dv.ba4Pvp

K w0t It A Hp tin summade then

3mn14 3,18 f log HADIEËNADYh 20

where fl INN

leut actionprinciple EONforD

Suppose f AD o t Aert ItThen the quadratic form QIAN fLADYis degenerate GIA XI 0 Htt

Pet lonsider

fAD D D f Ali

f X A D 0

by assumption Sina KAJEN'MDylan

Can we write

f AD AI L A ITA Zijnin terms of some Gettadij 7 operateT where CA g f AYA Ian

Eiras km A w 1 IFA zon LipkaIFAzen

Prof dien Shipka zonTotzer 8

Suppose A H D finitely Summer and I a

Kale transform as h Then there is a

Taylor expansionNA 2 Moi 0in A Aini R

hiv

where Hij 4 Arg DK dir eigensinsand

HR 4 0 11AM

Cort w

A Eet HIJ Ali D Ah rij Rn O times

Again there is a degeneracyH A i ftp.ovAtrj AEON for Dthen Q A A HijA ftp.dj D degenerate

in the tense that QLA.CA oV XtA

Prof Note that Dit ij P f hijThen

QAIDA Aijf t 0

Aij X it

HA X Hd by

en Can we write

GLA nl CA TIAN Innfor some operate T and Innerpydnt

0 1,111

Examples d the spectral adv

N Riem spit mld dim M 4

n f Im Boex 0 À

htt 1 EN Kok 1

Mx F E Ap Ap

A n f 2 Tg d x t 011

Here Ao Br t m to and

NZa 1 Ligt 2hij 2 It the E

III d g ÉN tot t.ttt ftp.frd t t.tn 17404

NCtornsCn_41C6ayrat.Ioohm.Vassileuehbfsonabri.Iochv knikker 08

A u 8T d 4 rijmt F 011

product geometry D Daalt Jo Dguncanny

t.hl a1trkfdast.snwhere kw u x hunpootTate LH e

trekt zie BIE eBIJ

BA trest'En prijkt 011 th

treft u 21 t.fr ebord

Now note thata xjke hdn.rs en

Expansion in powers of At

A te 4 Dmtype

µ trysynth1 ULFT

welnu it 4,1 p opFm for finasteride dan