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
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