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Talk  at QMath7, Prague, 1998.
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Adiabatic evolution, Quantum mechanics 
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% PATkshow - kshow with the current pattezn
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% PATmp - the makepattern equivalent
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% Save the original routines so that we can use them later on
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% right30
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b(in)g(Kato's)f(Theory)0 1425 y(Kato's)j(idea)h(w)o(as)f(to)g(in)o(tro)
q(duce)i(a)e(geometric)g(ev)o(olution)f(whic)o(h)i(satis\014es)g(the)g
(adiabatic)0 1474 y(theorem)d(without)f(an)h(error.)g(That)g(is,)f(a)h
(unitary)f Fj(U)853 1480 y Fe(a)874 1474 y Fk(\()p Fj(s)p
Fk(\),)h(so)g(that:)513 1557 y Fj(U)541 1563 y Fe(a)561
1557 y Fk(\()p Fj(s)p Fk(\))7 b Fj(P)f Fk(\(0\))12 b(=)g
Fj(P)6 b Fk(\()p Fj(s)p Fk(\))h Fj(U)880 1563 y Fe(a)900
1557 y Fk(\()p Fj(s)p Fk(\))p Fj(;)460 b Fk(\(9\))0 1640
y(where)17 b Fj(P)6 b Fk(\()p Fj(s)p Fk(\))16 b(is)g(a)f(sp)q(ectral)i
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Fj(U)952 1646 y Fe(a)973 1640 y Fk(\(0\))f(=)h(1.)h(The)g(second)h
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Fj(U)5 b Fk(\()p Fj(s)p Fk(\),)14 b(generated)h(b)o(y)435
1773 y Fj(i)468 1763 y Fk(_)456 1773 y Fj(U)5 b Fk(\()p
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b(=)g(1)p Fj(:)361 b Fk(\(10\))0 1856 y(with)14 b Fj(U)123
1862 y Fe(a)157 1856 y Fk(and)g(sho)o(w)f(that)h(the)h(t)o(w)o(o)e(are)
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/MyAppDict 100 dict dup begin def
/$F2psDict 200 dict def
$F2psDict begin
$F2psDict /mtrx matrix put
/col-1 {0 setgray} bind def
/col0 {0.000 0.000 0.000 srgb} bind def
/col1 {0.000 0.000 1.000 srgb} bind def
/col2 {0.000 1.000 0.000 srgb} bind def
/col3 {0.000 1.000 1.000 srgb} bind def
/col4 {1.000 0.000 0.000 srgb} bind def
/col5 {1.000 0.000 1.000 srgb} bind def
/col6 {1.000 1.000 0.000 srgb} bind def
/col7 {1.000 1.000 1.000 srgb} bind def
/col8 {0.000 0.000 0.560 srgb} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
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/col26 {0.750 0.380 0.000 srgb} bind def
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save
-44.0 -103.0 translate
 90 rotate
1 -1 scale
.9 .9 scale % to make patterns same scale as in xfig
90 rotate
-50.0 -475.0 translate

% This junk string is used by the show operators
/PATsstr 1 string def
/PATawidthshow { 	% cx cy cchar rx ry string
  % Loop over each character in the string
  {  % cx cy cchar rx ry char
    % Show the character
    dup				% cx cy cchar rx ry char char
    PATsstr dup 0 4 -1 roll put	% cx cy cchar rx ry char (char)
    false charpath		% cx cy cchar rx ry char
    /clip load PATdraw
    % Move past the character (charpath modified the
    % current point)
    currentpoint			% cx cy cchar rx ry char x y
    newpath
    moveto			% cx cy cchar rx ry char
    % Reposition by cx,cy if the character in the string is cchar
    3 index eq {			% cx cy cchar rx ry
      4 index 4 index rmoveto
    } if
    % Reposition all characters by rx ry
    2 copy rmoveto		% cx cy cchar rx ry
  } forall
  pop pop pop pop pop		% -
  currentpoint
  newpath
  moveto
} bind def
/PATcg {
  7 dict dup begin
    /lw currentlinewidth def
    /lc currentlinecap def
    /lj currentlinejoin def
    /ml currentmiterlimit def
    /ds [ currentdash ] def
    /cc [ currentrgbcolor ] def
    /cm matrix currentmatrix def
  end
} bind def
% PATdraw - calculates the boundaries of the object and
% fills it with the current pattern
/PATdraw {			% proc
  save exch
    PATpcalc			% proc nw nh px py
    5 -1 roll exec		% nw nh px py
    newpath
    PATfill			% -
  restore
} bind def
% PATfill - performs the tiling for the shape
/PATfill { % nw nh px py PATfill -
  PATDict /CurrentPattern get dup begin
    setfont
    % Set the coordinate system to Pattern Space
    PatternGState PATsg
    % Set the color for uncolored pattezns
    PaintType 2 eq { PATDict /PColor get PATsc } if
    % Create the string for showing
    3 index string		% nw nh px py str
    % Loop for each of the pattern sources
    0 1 Multi 1 sub {		% nw nh px py str source
	% Move to the starting location
	3 index 3 index		% nw nh px py str source px py
	moveto			% nw nh px py str source
	% For multiple sources, set the appropriate color
	Multi 1 ne { dup PC exch get PATsc } if
	% Set the appropriate string for the source
	0 1 7 index 1 sub { 2 index exch 2 index put } for pop
	% Loop over the number of vertical cells
	3 index 		% nw nh px py str nh
	{			% nw nh px py str
	  currentpoint		% nw nh px py str cx cy
	  2 index show		% nw nh px py str cx cy
	  YStep add moveto	% nw nh px py str
	} repeat		% nw nh px py str
    } for
    5 { pop } repeat
  end
} bind def

% PATkshow - kshow with the current pattezn
/PATkshow {			% proc string
  exch bind			% string proc
  1 index 0 get			% string proc char
  % Loop over all but the last character in the string
  0 1 4 index length 2 sub {
				% string proc char idx
    % Find the n+1th character in the string
    3 index exch 1 add get	% string proe char char+1
    exch 2 copy			% strinq proc char+1 char char+1 char
    % Now show the nth character
    PATsstr dup 0 4 -1 roll put	% string proc chr+1 chr chr+1 (chr)
    false charpath		% string proc char+1 char char+1
    /clip load PATdraw
    % Move past the character (charpath modified the current point)
    currentpoint newpath moveto
    % Execute the user proc (should consume char and char+1)
    mark 3 1 roll		% string proc char+1 mark char char+1
    4 index exec		% string proc char+1 mark...
    cleartomark			% string proc char+1
  } for
  % Now display the last character
  PATsstr dup 0 4 -1 roll put	% string proc (char+1)
  false charpath		% string proc
  /clip load PATdraw
  neewath
  pop pop			% -
} bind def
% PATmp - the makepattern equivalent
/PATmp {			% patdict patmtx PATmp patinstance
  exch dup length 7 add		% We will add 6 new entries plus 1 FID
  dict copy			% Create a new dictionary
  begin
    % Matrix to install when painting the pattern
    TilingType PATtcalc
    /PatternGState PATcg def
    PatternGState /cm 3 -1 roll put
    % Check for multi pattern sources (Level 1 fast color patterns)
    currentdict /Multi known not { /Multi 1 def } if
    % Font dictionary definitions
    /FontType 3 def
    % Create a dummy encoding vector
    /Encoding 256 array def
    3 string 0 1 255 {
      Encoding exch dup 3 index cvs cvn put } for pop
    /FontMatrix matrix def
    /FontBBox BBox def
    /BuildChar {
	mark 3 1 roll		% mark dict char
	exch begin
	Multi 1 ne {PaintData exch get}{pop} ifelse  % mark [paintdata]
	  PaintType 2 eq Multi 1 ne or
	  { XStep 0 FontBBox aload pop setcachedevice }
	  { XStep 0 setcharwidth } ifelse
	  currentdict		% mark [paintdata] dict
	  /PaintProc load	% mark [paintdata] dict paintproc
	end
	gsave
	  false PATredef exec true PATredef
	grestore
	cleartomark		% -
    } bind def
    currentdict
  end				% newdict
  /foo exch			% /foo newlict
  definefont			% newfont
} bind def
% PATpcalc - calculates the starting point and width/height
% of the tile fill for the shape
/PATpcalc {	% - PATpcalc nw nh px py
  PATDict /CurrentPattern get begin
    gsave
	% Set up the coordinate system to Pattern Space
	% and lock down pattern
	PatternGState /cm get setmatrix
	BBox aload pop pop pop translate
	% Determine the bounding box of the shape
	pathbbox			% llx lly urx ury
    grestore
    % Determine (nw, nh) the # of cells to paint width and height
    PatHeight div ceiling		% llx lly urx qh
    4 1 roll				% qh llx lly urx
    PatWidth div ceiling		% qh llx lly qw
    4 1 roll				% qw qh llx lly
    PatHeight div floor			% qw qh llx ph
    4 1 roll				% ph qw qh llx
    PatWidth div floor			% ph qw qh pw
    4 1 roll				% pw ph qw qh
    2 index sub cvi abs			% pw ph qs qh-ph
    exch 3 index sub cvi abs exch	% pw ph nw=qw-pw nh=qh-ph
    % Determine the starting point of the pattern fill
    %(px, py)
    4 2 roll				% nw nh pw ph
    PatHeight mul			% nw nh pw py
    exch				% nw nh py pw
    PatWidth mul exch			% nw nh px py
  end
} bind def

% Save the original routines so that we can use them later on
/oldfill	/fill load def
/oldeofill	/eofill load def
/oldstroke	/stroke load def
/oldshow	/show load def
/oldashow	/ashow load def
/oldwidthshow	/widthshow load def
/oldawidthshow	/awidthshow load def
/oldkshow	/kshow load def

% These defs are necessary so that subsequent procs don't bind in
% the originals
/fill	   { oldfill } bind def
/eofill	   { oldeofill } bind def
/stroke	   { oldstroke } bind def
/show	   { oldshow } bind def
/ashow	   { oldashow } bind def
/widthshow { oldwidthshow } bind def
/awidthshow { oldawidthshow } bind def
/kshow 	   { oldkshow } bind def
/PATredef {
  MyAppDict begin
    {
    /fill { /clip load PATdraw newpath } bind def
    /eofill { /eoclip load PATdraw newpath } bind def
    /stroke { PATstroke } bind def
    /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
    /ashow { 0 0 null 6 3 roll PATawidthshow }
    bind def
    /widthshow { 0 0 3 -1 roll PATawidthshow }
    bind def
    /awidthshow { PATawidthshow } bind def
    /kshow { PATkshow } bind def
  } {
    /fill   { oldfill } bind def
    /eofill { oldeofill } bind def
    /stroke { oldstroke } bind def
    /show   { oldshow } bind def
    /ashow  { oldashow } bind def
    /widthshow { oldwidthshow } bind def
    /awidthshow { oldawidthshow } bind def
    /kshow  { oldkshow } bind def
    } ifelse
  end
} bind def
false PATredef
% Conditionally define setcmykcolor if not available
/setcmykcolor where { pop } {
  /setcmykcolor {
    1 sub 4 1 roll
    3 {
	3 index add neg dup 0 lt { pop 0 } if 3 1 roll
    } repeat
    setrgbcolor - pop
  } bind def
} ifelse
/PATsc {		% colorarray
  aload length		% c1 ... cn length
    dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
  } ifelse } ifelse
} bind def
/PATsg {		% dict
  begin
    lw setlinewidth
    lc setlinecap
    lj setlinejoin
    ml setmiterlimit
    ds aload pop setdash
    cc aload pop setrgbcolor
    cm setmatrix
  end
} bind def

/PATDict 3 dict def
/PATsp {
  true PATredef
  PATDict begin
    /CurrentPattern exch def
    % If it's an uncolored pattern, save the color
    CurrentPattern /PaintType get 2 eq {
      /PColor exch def
    } if
    /CColor [ currentrgbcolor ] def
  end
} bind def
% PATstroke - stroke with the current pattern
/PATstroke {
  countdictstack
  save
  mark
  {
    currentpoint strokepath moveto
    PATpcalc				% proc nw nh px py
    clip newpath PATfill
    } stopped {
	(*** PATstroke Warning: Path is too complex, stroking
	  with gray) =
    cleartomark
    restore
    countdictstack exch sub dup 0 gt
	{ { end } repeat } { pop } ifelse
    gsave 0.5 setgray oldstroke grestore
  } { pop restore pop } ifelse
  newpath
} bind def
/PATtcalc {		% modmtx tilingtype PATtcalc tilematrix
  % Note: tiling types 2 and 3 are not supported
  gsave
    exch concat					% tilingtype
    matrix currentmatrix exch			% cmtx tilingtype
    % Tiling type 1 and 3: constant spacing
    2 ne {
	% Distort the pattern so that it occupies
	% an integral number of device pixels
	dup 4 get exch dup 5 get exch		% tx ty cmtx
	XStep 0 dtransform
	round exch round exch			% tx ty cmtx dx.x dx.y
	XStep div exch XStep div exch		% tx ty cmtx a b
	0 YStep dtransform
	round exch round exch			% tx ty cmtx a b dy.x dy.y
	YStep div exch YStep div exch		% tx ty cmtx a b c d
	7 -3 roll astore			% { a b c d tx ty }
    } if
  grestore
} bind def
/PATusp {
  false PATredef
  PATDict begin
    CColor PATsc
  end
} bind def

% this is the pattern fill program from the Second edition Reference Manual
% with changes to call the above pattern fill
% left30
11 dict begin
/PaintType 1 def
/PatternType 1 def
/TilingType 1 def
/BBox [0 0 1 1] def
/XStep 1 def
/YStep 1 def
/PatWidth 1 def
/PatHeight 1 def
/Multi 2 def
/PaintData [
  { clippath } bind
  { 32 16 true [ 32 0 0 -16 0 16 ]
	{<c000c000300030000c000c000300030000c000c000300030
	000c000c00030003c000c000300030000c000c0003000300
	00c000c000300030000c000c00030003>}
     imagemask } bind
] def
/PaintProc {
	pop
	exec fill
} def
currentdict
end
/P1 exch def
1.1111 1.1111 scale %restore scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
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/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
%%EndProlog

$F2psBegin
10 setmiterlimit
n 0 612 m 0 0 l 792 0 l 792 612 l cp clip
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45.000 slw
% Polyline
n 3600 4275 m 8100 4275 l gs col-1 s gr 
0.000 slw
% Polyline
n 3600 1650 m 8100 1650 l 8100 4275 l 3600 4275 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
15.00 15.00 sc P1 [16 0 0 -8 240.00 110.00] PATmp PATsp ef gr PATusp 
7.500 slw
% Polyline
gs  clippath
7584 4710 m 7944 4800 l 7584 4890 l 8040 4890 l 8040 4710 l  cp clip
3435 1341 m 3525 981 l 3615 1341 l 3615 885 l 3435 885 l  cp clip
n 3525 900 m 3525 4800 l 8025 4800 l gs col-1 s gr gr

% arrowhead
30.000 slw
n 3435 1341 m 3525 981 l 3615 1341 l  col-1 s
% arrowhead
n 7584 4710 m 7944 4800 l 7584 4890 l  col-1 s
/Times-Roman ff 420.00 scf sf
1725 1050 m
gs 1 -1 sc (spec\(H\)) col-1 sh gr
/Times-Roman ff 420.00 scf sf
7650 5250 m
gs 1 -1 sc (s) col-1 sh gr
$F2psEnd
rs
end

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1570 y Fh(\001)230 1564 y Fk(\()p Fj(s)p Fk(\))f(=)362
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1507 y Fi(Z)445 1602 y Fh(\000)474 1564 y Fj(dz)c Fk(\(1)g
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Fj(s)p Fk(\)\))e Fj(R)p Fk(\()p Fj(z)r(;)g(s)p Fk(\))886
1553 y(_)871 1564 y Fj(P)g Fk(\()p Fj(s)p Fk(\))p Fj(R)p
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1570 y Fh(\001)1230 1564 y Fk(\()p Fj(s)p Fk(\)\))p Fj(:)94
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Fj(C)676 973 y Fd(\003)710 988 y Fk(algebras)h(and)g(mo)q(dels)e(of)h
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n(ts)0 1266 y Fk(This)e(w)o(ork)g(w)o(as)g(partially)f(supp)q(orted)j
(b)o(y)e(a)g(gran)o(t)g(from)e(the)j(Israel)g(Academ)o(y)e(of)h
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(h)0 1365 y(at)j(the)g(T)m(ec)o(hnion.)0 1502 y Fl(References)92
1593 y Fk([1])20 b(V.)29 b(Arnold,)g Fg(Mathematic)n(al)g(Metho)n(ds)h
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1643 y(\(1980\).)92 1726 y([2])20 b(J.)26 b(E.)g(Avron)g(and)g(A.)g
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1776 y(ph/9805022)92 1858 y([3])20 b(J.)15 b(E.)g(Avron,)g(J.)g(S.)g
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(for)g(dense)157 1908 y(p)n(oint)f(sp)n(e)n(ctr)n(a)p
Fk(,)e Fg(Comm.)h(Math.)h(Phys.)30 b Fa(128)14 b Fk(\(1990\),)e
(497{507.)92 1991 y([4])20 b(J.)10 b(E.)g(Avron,)f(R.)g(Seiler)i(and)f
(L.)f(G.)g(Y)m(a\013e,)h Fg(A)n(diab)n(atic)h(the)n(or)n(ems)g(and)h
(applic)n(ations)157 2040 y(to)20 b(the)g(quantum)i(Hal)r(l)d(e\013e)n
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b(Math.)j(Ph)o(ys.)28 b Fa(153)13 b Fk(\(1993\),)g(649-650\).)92
2173 y([5])20 b(M.V.)d(Berry)m(,)i(Pro)q(c.)f(Ro)o(y)m(.)e(So)q(c.)i
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157 2223 y(phase:)c(Fiv)o(e)g(y)o(ears)h(after,)e(in)h
Fg(Ge)n(ometric)g(phases)i(in)f(physics)f Fk(\(A.)g(Shap)q(ere)h(and)
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h(Lond.)g(A)157 2405 y Fa(429)p Fk(,)13 b(61-72,)f(\(1990\).)728
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%%Page: 10 10
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Fa(51)13 b Fk(\(1928\),)157 494 y(165{169.)92 577 y([9])20
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Fa(19)p Fk(,)f(511,)g(\(1978\).)72 710 y([10])19 b(P)m(.)c(Ehrenfest,)i
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y([19])19 b(H.)14 b(Narnhofer)i(and)e(W.)g(Thirring,)g
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2055 y(me)n(chanics)p Fk(,)d(Ph)o(ys.)g(Rev.)f(A)28 b
Fa(26)p Fk(,)13 b(3646,)f(\(1982\).)72 2138 y([20])19
b(G.)25 b(Nenciu,)h Fg(Line)n(ar)g(A)n(diab)n(atic)g(The)n(ory:)f(Exp)n
(onential)i(Estimates)p Fk(,)e(Comm.)157 2188 y(Math.)13
b(Ph)o(ys.)h Fa(152)p Fk(,)f(479-496,)f(\(1993\))72 2271
y([21])19 b(E.)f(Ott,)g Fg(Go)n(o)n(dness)i(of)f(er)n(go)n(dic)g(adiab)
n(atic)g(invariants)p Fk(,)36 b(Ph)o(ys.)18 b(Rev.)g(Lett.)h
Fa(42)157 2320 y Fk(\(1979\),)13 b(1628-1631.)717 2530
y(10)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF

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