"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Wang, Benjamin G."@en . "2009-11-27T18:53:42Z"@en . "2004"@en . "Master of Science - MSc"@en . "University of British Columbia"@en . "We consider Schr\u00F6dinger Operators with periodic electric and magnetic field\r\nwith zero flux through a fundamental cell of the periodic lattice with dimension\r\nd. We show that, for a generic small electric/magnetic field and a generic small\r\nFermi energy, the corresponding Fermi surface is at most dimension d-2, convex\r\nand not invariant under inversion at any point."@en . "https://circle.library.ubc.ca/rest/handle/2429/15869?expand=metadata"@en . "1164151 bytes"@en . "application/pdf"@en . "Asymmetric Fermi Surfaces for Periodic Schrodinger Operators by Ben jamin G . W a n g B . S c , T h e Univers i ty of B r i t i s h C o l u m b i a , 2 0 0 2 < A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E i n T h e Facu l ty of Gradua te Studies (Depar tment of Mathemat ics ) We accept this thesis as conforming to the required s tandard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A October 6, 2004 \u00C2\u00A9 Ben jamin G . W a n g , 2004 UBCl m THE UNIVERSITYOF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES Library Authorization In p resen t ing this thes is in part ia l fu l f i l lment o f the requ i remen ts for an a d v a n c e d d e g r e e at the Univers i ty o f Bri t ish C o l u m b i a , I ag ree tha t the L ibrary shal l m a k e it f ree ly ava i lab le fo r re fe rence a n d s tudy. 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N a m e of Au thor (please print) OH / |0 / T-OOLr Date ( d d / m m / y y y y ) Ti t le o f Thes is : Asy P M ^ C few Surfaces -for PenocJic- G c l v r o c W / ' OperAt^rs ' J D e g r e e : D e p a r t m e n t of Master of- sdence Year : 7PO<4-T h e Univers i ty o f Brit ish C o l u m b i a V a n c o u v e r , BC C a n a d a g r a d . u b c . c a / f o r m s / ? f o r m l D = T H S p a g e 1 o f 1 last updated: 7-Oct-04 11 A b s t r a c t W e consider Schrodinger Operators w i t h per iodic electric and magnetic field w i t h zero flux through a fundamental cel l of the periodic lat t ice w i t h d imens ion d. W e show that , for a generic smal l e lectr ic /magnet ic field and a generic smal l F e r m i energy, the corresponding F e r m i surface is at most d imension d-2, convex and not invariant under inversion at any point . iii C o n t e n t s Abstract ii Contents iii Acknowledgements iv I Thesis 1 1 Preliminaries . . . 2 2 The Main Result 7 3 Analyticity of the Fermi surfaces 9 4 Proof of the Main Theorem 16 Bibliography 23 A c k n o w l e d g e m e n t s T h a n k s to anyone who's taught me m a t h before, g iv ing me the knowledge to be able to do this thesis. Specia l thanks to m y supervisor D r . Fe ldman , who had an enormous amount of patience w i t h me. P a r t I T h e s i s 2 Chapter 1 P r e l i m i n a r i e s Def in i t ion : L e t f(x) be a function i n R d . T h e n the vector 7 is cal led a pe r iod of / i f f(x + 7 ) = /(ar) Vx G Rd. If 7 1 , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , 7^ G R d are independent vectors then r = { n i 7 i + 71272 + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 -ndjd I nj \u00E2\u0082\u00AC Z } is called a non-degenerate latt ice. L e t T be such a latt ice. Suppose there is a crys ta l lat t ice w i t h ions at T that generate electric and magnetic potentials V ( x ) and A ( x ) per iodic w i t h respect to r. T h e n the H a m i l t o n i a n for a single electron mov ing i n the crysta l is H=^-{iV + A{x))2 + V(x) T h i s H a m i l t o n i a n commutes w i t h a l l of the t ransla t ion operators (T1){x) = {x + 1) 7 G T Suppose for now H and T 7 were matrices, then we could find an o r thonormal basis of simultaneous eigenvectors for bo th H and T and these eigenvectors obey T~,a = Aa]77 V7 \u00C2\u00A3 r A s T 7 is unitary, a l l its eigenvalues must be complex numbers of modulus one. So there must exist real numbers /3 Q ) 7 such that A \u00E2\u0080\u009E i 7 = e1^\"^. N o w because have wh ich gives Pa,-y + Pa,y = Pa,i+~i' m o d 2w V7, 7 ' G T T h e n given any d numbers (3\, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ,/3d the system of linear equations (wi th un-knowns kd) H \u00E2\u0080\u00A2 k = ft 1 < % < d that is J^7t,jfcj = Pi 1 < i < d j = i (where 7 ^ is the j t h component of 7*) has a unique solut ion because the linear independence of 7 i , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 7d implies that the m a t r i x [lij}in,k = e.n{k)4>n,k T7n,k \u00E2\u0080\u0094 elfe''Yci>n,fc for a l l 7 G T\" means that n,k{X + \"f) = e '^1'4>n,k{x) for a l l xG R d and 7 G T . If the e l f e ' 7 were not there, this would mean that (j>n,k is per iodic w i t h respect to T. W e can make a simple change of variables to el iminate the elk'^. Define T h e n subbing i t into ( P . l ) gives ^-(tV + A - fc)2Vyfe + V1>n,k = ^-(iS7 + A - kf e~ik-x ^ + Ve-ik-xn,k 2m 2m = -J-(iV + A - k)(e~ik-x(iV + A)4>n,k) + e-ik-xVct>n,k 2m = e-ik-x^-(iV + 4) Vn,fc + e-ik-xVn,k = en(k, A, V)^k 2m Denote by Nfc the set of values of n that appear i n pairs a \u00E2\u0080\u0094 (k, n) and define Hk = s p a n { 0 \u00E2\u0080\u009E ) f c I n G N f c } T h e n , formally, ignoring that k runs over an uncountable set, L2(Rd) = span{c i n , f e | k G Rd/T*,n\u00E2\u0082\u00AC Nfc} = \u00C2\u00AEk&*/r#Hk Set W/c = span{^\u00E2\u0080\u009E,fc I n G Nfc} A s mu l t i p l i c a t i on by elk'x is a uni ta ry operator, 7ik is un i t a r i ly equivalent to i i k and L 2 ( R d ) is un i t a r i ly equivalent to \u00C2\u00AEhmd/r#'ri-k- T h e restr ic t ion of the Schrodinger operator H to r\k is ( i V + A - k)2 + V appl ied to functions that Chapter 1. Preliminaries 4 are per iodic w i t h respect to V. Therefore, at least formally, we know that i n order to find the spec t rum of H = ^(iV + A)2 + V act ing on L2(Rd), i t suffices to f ind, for each k G Rd/T* the spect rum of Hk =.^{iV + A - k)2 + V. T o make this rigorous, we shal l make L2(Rd) un i t a r i ly equivalent to \u00C2\u00AEK\u00C2\u00A3Rd/r#'rtk by cons t ruc t ing a un i ta ry operator U from the space of L2 functions f(x),x G R d to the space of L2 functions ip(k,x),k G Rd/T#,x G Rd/T w i t h the proper ty that (UHU*il>){k,x) = Hki/>{k,x) N o w define S(Rd/T* x Rd/T) = {il>\u00E2\u0082\u00AC C \u00C2\u00B0 \u00C2\u00B0 ( R d x Rd) I ip(k,x + j) = tp{k, x) V 7 G V eib-xip{k + b,x) = il>(k,x) V6 G T*} and S(Rd) = {/ G C \u00C2\u00B0 \u00C2\u00B0 ( R d ) | sup (l+x2\u00E2\u0084\u00A2)(TT ^-r~f{x)) < c o Vn,h,--- ,id\u00C2\u00A3N} .i=i dxi W i t h the inner product on S(Rd/T* x Rd/T) given by (tp,4>)r = -W4T / d k I dxi>{k,x)(k,x) I1 *H Jw/r# Jw/r S(Rd/T* x Rd/T) is almost a Hi lbe r t space. T h e only miss ing a x i o m is com-pleteness. C a l l the comple t ion L2(Rd/T* x Rd/T) N o w set {wl>)(x) = TJL [ ddkeik*i>(k, . . x) R r f / r # ( u / ) ( f c , i ) = X ! e - < M x + 7 ) / ( a : + 7) T h e n notice u : S{Rd/T# x Rd/T) -> {k,x)ddk = J(-i)M-^(eik-x)1>(k,x)d*k = J eik-xi^(-^)a^(k,x)ddk Chapter 1. Preliminaxies 5 A n d also because ip(k,x) and a l l its derivatives are bounded, xa f eik-xi;(k,x)ddk]\= f d%(eikx iW^-i){k,x))ddk Jw/r* ' JR S(Rd/V# x Rd/T), Proof: F i x / \u00C2\u00A3 S{Rd) and set i>(k,x) = ^ e - i f c ( x + 7 ) / ( ^ + 7 ) A s f(x) and a l l of its derivatives are bounded by the series ^ r t = i d x e d k e converges absolutely and uni formly i n k and x (on any compact set) for a l l ii,-\" ,id,ji,\"-jd- Consequently ip(k,x) exists and is C \u00C2\u00B0 \u00C2\u00B0 . A s for the per iod-ic i ty condit ions, if 7 G V, ip(k, x + f)=J2 e-^+^+^fix + 7 + V) 7 ' e r = e - i f e - ( x + 7 \" ) / ( a ; - r - 7 \" ) ' where 7\" = 7 + 7' 7\"er = ip{k,x) and i f b G T# ei(k+b)-x^j, + b*tX) = ^ et(fe+6)-xe-i(fc+6)-(x+7)/(a; + 7 ) = \u00C2\u00A3 e ^ + ^ f i x + 7) 7\u00E2\u0082\u00ACr 7er = J2 e-ik-y(x + 7) = eik'x Y, e-ik{k,x) So by proofs s imi lar to the lecture notes [FN] we have proposit ions that state: Proposition P . l Let A and V be C \u00C2\u00B0 \u00C2\u00B0 functions that are per iodic w i t h respect to T and set H = (iV + A(x))2 + V(x) Hk = (tV + A(x) - kf + V(x) Chapter 1. Preliminaries 6 with domains S and S(Rd/T# x Rd/T) respectively. Then, (uHutp)(k,x) = (Hkip)(k,x) for all ip G S(Rd/T* x Rd/T) Proposition P.2 i) The operators u and u have unique bounded extensions U : L2(Rd/T# x Rd/T) -* L2(Rd) and U : L2(Rd) -* L2(Rd/T# x Rd/T) and UU = HL2(Rrf/r#xHrf/r) UU = tL2(^,i) U = U* U = U* Proof : w i t h s imi lar treatment i n the s tudy of Fourier series, w i t h the per iod ic i ty condi t ion , we get uuip = ip for a l l V G S(Rd/T* x Rd/T) (P.2) uuf = f for a l l / G S(Rd) (P.3) (uf, ug) = (/, g) for a l l f,g\u00C2\u00A3 S(Rd) (PA) Set f = utp and g = u, so that by (P.4) (utp, u)T (P.5) N e x t set g = u(j>. T h e n by (P.2), ug = so that by (P.4) 2 and let r > d. Define A = {A = (AU---,Ad) G (Lr^R/T))d\ [ A(x)dx = 0} JwL/v V = {V G L^ / 2 (R/r)| / V(x)dx = 0} For (A, V) G A x V set #fc(i4, V) = (iV + A{x) - k)2 + V{x) W h e n d = 2,3, this operator Hk{A, V) describes an electron i n Rd w i t h quasimo-m e n t u m k moving under the influence of the magnetic field w i t h per iodic vector potent ia l A(x) = (Ai(x),--- ,Ad(x)) and electric field w i t h per iodic potent ia l V ( x ) . L a t e r we shal l show that ei{k,A,V) < e2{k,A,V) < ... are the eigenvalues of the operator Hk{A, V) on Lr(Rd/T). T h e res t r ic t ion of en(k, A, V) to the first B r i l l o u i n zone B of T is called the n- th band function of A . Observe that iffc(0,0) = ( i V \u00E2\u0080\u0094 k)2 and so the eigenvalues are (b \u00E2\u0080\u0094 k)2, w i t h the corresponding eigenvectors e~lb'x,b G T# . In par t icular , ei(fc ,0,0) = |fc| 2 T h e F e r m i surface of ( A , V ) w i t h energy A is defined as F\(A, V) = {k G B | en{k, A, V) = A for some n} Because H has real eigenvalues, let Hk{A, V)4>n = en(k, A, V)n = Hk(A,V)4>n = ( ( - tV + A(x) - k)2 + V(x))4>n = ((tV - A(x) + k)2 + V{x)Wn = H _ f c ( - A , V)K Therefore en(-k,-A,V) = en(k,A,V) for a l l n > 1. I n par t icular , when A = 0, en(-k,Q,V) =en(k,0,V) Chapter 2. The Main Result 8 for a l l n > 1, so that Fx{0,V) = -Fx(0,V) for a l l A and V . For a l l (A,V) G A x V , A G R and p G R d , define p - F A (A F) = {p - k | k G F A (A V)} T h e m a i n result of the paper is: T h e o r e m There is a neighbourhood Ao x Vo of the origin in A x V and XQ > 0 ('ij / o r a// (X,A,V) G (\u00E2\u0080\u0094 oo,Arj) x x Vo, F\(A,V) is either a strictly convex (d-1)-dimensional real anaylytic submanifold of B, or consists of one point, or is empty (ii)there is an open dense subset S of Ao x Vo of the origin in A x V such that for all (A, A, V) G S and all p e l 1 ' Fx(A,V)n(p-Fx(A,V)) has dimension at most d-2. Furthermore 0 such that i) The map DxU\u00E2\u0080\u0094>R, (k,A,V) i \u00E2\u0080\u0094 \u00C2\u00BB ei(k,A,V) is real analytic ii) For all (A, V) G U and all k G D ei(k,A,V) e\(k, A, V) is posi-tive definite. Furthermore infkeD^i{k, A,V) < Ao iv) For each (A,V) \u00E2\u0082\u00AC U and each A < A 0 the Fermi surface F\(A,V) is ei-ther empty, or consists of one point only, or is a real analytic smooth strictly convex (d-1)-dimensional real analytic manifold that is completely contained in D. Proof of the Corollary : F i r s t we state the Impl ic i t Func t i on T h e o r e m be-low. [L] Chapter 3. Analyticity of the Fermi surfaces 10 The Implicit Function Theorem let U, V be open sets in complex Banach spaces E, F respectively and let f:UxV-+G be analytic, and let (a,c) \u00C2\u00A3 U x V. If the partial derivative of f at (a,c) with respect to c, dcf:F-*G is a linear isomorphism, then there exists an open neighbourhood A of a and a unique analytic map u : A \u00E2\u0080\u0094> V such that f(x, u(x)) = f(a, c), u(a) = c on A. Proof of the Corollary : i) T h i s is the direct appl ica t ion of the Impl ic i t Func-t i o n T h e o r e m w i t h a = ( k , A , V ) = (k,0,0) and c = A = k2. N o w to check l inear i somorphism, w i t h the F(k, A, A, V) and u(k, A) we construct later, we have d e t ( 1 + 7 r a u ( M ) 7 r a ) ! U = f c 2 d, , A ,2k^b+ k2 - A - 1 , \u00C2\u00BB=i ber# u i , 2fc-6 + fc2-A-l^, 11 ( J + TTJfl 1 + b2 6\u00E2\u0082\u00ACr# = J - ( J J (1 + 2k'h\kl~X~l) \u00E2\u0080\u00A2 e / ^ ' \u00C2\u00AB ) U = f c a 6er# _ d ( 2fc-b + fc2-A-K T T f i x M - b - l , yr ' f(d,k,k\ ~ J \ \ + I T P )\\=k',b=o 11 ( i + 1 + b2 Ml e + (1 + 2 k b + k , 2 A \u00E2\u0080\u0094 - ) U = f c 2 , f > = o ^ ( fi 5 ( r f , f c , A , 6 ) ) | A = f c 2 ) b = 0 6er#,b#o = - n ( 1 + ^ ^ ) n ^ f e 2 ' b ) ^ \u00C2\u00B0 6er#,6^o fe\u00E2\u0082\u00ACr# Because we are res t r ic t ing ourselves w i t h i n the first B r i l l o u i n zone, ii) T h e spec t rum of Hk(0,0) is{|fc + 6| 2 | b \u00C2\u00A3 Hence for k \u00C2\u00A3 D, ei(fe,0,0) = \k\2 and e 2 (fc ,0,0) > e i ( fc ,0 ,0) . Since (k,A,V) \u00E2\u0080\u0094> ei(k,A,V) is real analyt ic , we just have to choose U sufficiently smal l and by cont inui ty e\(k,A, V) < e2{k,A, V)V(A,V) \u00C2\u00A3 U b) Chapter 3. Analyticity of the Fermi surfaces 11 i i i ) Here we define the Hessian to be the m a t r i x of the 2nd derivatives of the the mapp ing k \u00E2\u0080\u0094> e\(k,A, V). For (A, V) = (0 ,0) , the Hess ian at any point is given by 2 \u00E2\u0080\u00A2 1 and therefore is posit ive definite. Here by continuity, for U sma l l enough, the Hessian is posit ive definite for a l l (A, V) G U. iv) Fo r U sufficiently smal l the Hessian is posit ive definite and therefore by [M], we may wri te e\(k,A,V) as c + ' \u00C2\u00A3 ^ = 1 p 2 ;so if A < c then Fx(A,V) is empty, i f A = c then Fx (A, V) is one point and i f A > c then Fx(A,V) is real ana ly t i c smooth s t r ic t ly convex (d- l ) -d imens ional real analyt ic manifold that is com-pletely contained i n D . B u t because the mapp ing is diffeomorphic, we have the desired results i n the or ig inal k co-ordinates. Before we proceed to prove the theorem, we present some lemmas first. L e m m a l . Let \\B\\r = [tr (B*B)r/2)l/r where tr denotes the trace defined on trace class operators on L2(M.d/T). If B is a compact operator on L2(M.d/F) and r > 2, then a) \\B\\ < \\B\\r b) \\B\\r = \\B% P r o o f o f t h e L e m m a 1: a) Since B is compact , we can wri te o o B = ^An(V>n,-)n 71=1 ,w i th {Xn}^=1 G R + , A n \u00E2\u0080\u0094> 0, {\u00E2\u0080\u00A2tpn}^L1 and {(j>n)'^Ll o r thonormal sets being the s ingular value decomposi t ion. T h e n B*B \u00E2\u0080\u0094 Y^=i ^Wi'Pn, -)4>n, {B*B)r/2 = E \u00C2\u00A3 L l A \u00C2\u00A3 ( 0 n , - ) < A n , SO OO < max{Xn \ n G N } < ( \u00C2\u00A3 Xrn)1/r = \\B\\r 71=1 b) F r o m the singular value decomposi t ion we see that BB* = Y^=i ^ n W ' i and so | | B 1 | r = ( E n = i ^ n ) 1 / r = l | 5 | | P L e m m a 2 . Write [iV + A(x) - k)2 + V{x) - A = 1 - A + u{k, A) + w{k, A, V) with u(k,X) = -2ik-V + k 2 - X - l w(k,A,V) = iV-A + iA-W-2k-A + A2 + V Then there is a constant construct such that < constr,rA(l + \k\) WAhr + \\A\\2Lr + \\V\\Lr/2) < constr^di1 + \ k \ 2 + lAD ,) . 1 w{k,A,V) , 1 b) u(k, A) Chapter 3. Analyticity of the Fermi surfaces 12 c) Let 0 < e < There is a constant constr,r,d,k,\,A,v such that \((u(k,\) + w(k,A,V)),il>)\ \u00C2\u00A3 L2(Rd/T) P r o o f o f t h e L e m m a 2: a) we repeatedly apply the result that , for any r > 2 and a n y / \u00C2\u00A3 f (r#) and g \u00C2\u00A3 Lr{Rd/T) \\f(iV)g(x)\\ < vol{Rd/T) l ' r | | / | | r ( r # ) \\g\\L (*)\u00E2\u0080\u00A2 T h i s is proven just as i n [S] Theorem 4.1, except w i t h the per iodic domain . One first proves that the Hi lbe r t -Schmid t no rm of f(iS7)g{x) is bounded by vol(Rd/T)1/2 | | / | | ^ ( r # ) | |ff|li,2( R d because by the integral test -ds her* (1 + s2Y'2 . /solid angle / Jsolic dQ < oo , so the desired result follows from (*) w i t h g = A\, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , Ad, VV b) W i t h the eigenfunctions eib'x, b \u00C2\u00A3 T#, the spec t rum of y-j-_.M(fc ; \)-^J== is 2fc \u00E2\u0080\u00A2 b + fe2 - A - 1 1 + 6 2 b\u00C2\u00A3T* Chapter 3. Analyticity of the Fermi surfaces 13 For any r >d, the f (r#)-norm of 2 f c , 6 ^ 2 i , I A ~ 1 , wh ich is also the | | - | | r n o r m of 7i=^ w (M)7I=A> i s b o u n d e d b y 2k \u00E2\u0080\u00A2 b + k2 - A - 1 1 + b2 < 2\k\ l + b2 (|fc|2 + |A | + l ) l + b2 < constr^d(l + \k\2 + |A | ) rsdds c) Denote D = y/TT^A. T h e condi t ion on e implies that r(l - e) > d +^ > d so that 1 ( l + h 2 ) ' * 1 \u00E2\u0080\u0094 ) / 2 1 is s t i l l summable. So as i n part a), 1 D 1 _ e D D < < 1 -A D1-< constr>r{l + \k\2 + |A|) < constrir \\A\\Lr < COnstr,r \\V\\L,./2 Consequently, \((i\7A),iP)\ = \((WA),D-lDi>)\ = \(D-\WA),Di>)\ < HD-^iV^II ||JDV|| = \\D-1{iS/A)D-1+W1-^ < | | D - 1 ( i V i 4 ) \u00C2\u00A3 > - 1 + \u00C2\u00A3 | | | | \u00C2\u00A3 1 _ \u00C2\u00A3 0 | | \\Dil>\\ < IJD-^ VH | |ylD- 1 + \u00C2\u00A3 | | IIZ)1-^ !! HDVII < 1 \u00E2\u0080\u00A2 constv,r \\A\\Lr | | \u00C2\u00A3 > 1 _ ^ | | \\D%p\\ | < ( A i V ) < ^ ) l = \({Ai^)D-lD4>,D-l^Dl-^)\ < l l D - ^ ^ A t V ) ! ? - 1 ! ! | |D^| | H^1\" !^! ^ c o n s t r . r P H i . l l ^ H l l ^ - ^ l l \{(2k \u00E2\u0080\u00A2 A),i>)\ < constT,r\k\ \\A\\Lr \\Dl-*4>\\ H\\ \((A \u00E2\u0080\u00A2 A)4>,^)\ < constr,r \\A\\2Lr {{D1-^ {{D^W \({V^)\ < constv,r \\V\\L,n \\Dl~^\\ \\Dtp\\ \({ucf>^)\ < constVtr(\ + \k\2 + |A | ) ll^ ll | |Z ty | | P u t t i n g a l l these together we get the desired expression for part c). Proof of the Theorem: Because Ls{Rd/T) D L3'(R/T) for a l l 1 < s < 5', we m a y assume wi thou t loss of generali ty that r < d + 1. T h e n the l e m m a implies that 1 . 1 F(k,X,A, V) = detd+i (1+ 1 y/I^A u(k, A) w(k,A,V) ) Chapter 3. Analyticity of the Fermi surfaces 14 is a well-defined analy t ic function on Cd x C x Ac x V c - Here, d e t d + i ( l + B) is the regularised determinant which , for matrices , is defined by d (\u00E2\u0080\u0094lY d e t a i l + B) = exp(YJ ^-^trB1) det(l + B) i = i 1 T h i s regularised determinant is defined for B w i t h finite. See [S], The -orem 9.2. It is analyt ic since one can take l imi t s of finite rank approximat ions of B . L e t V be the domain of %/H - A . A n d let q : V x V -> C , q{, 4>) = {(f), {H - \)) be the form, then by the lemma, ( t A + A{x) - If + V(x) - A = 1 - A + u(k, A) + w(k, A, V) gives a well-defined quadrat ic form on V x V. Fur thermore , for e V, by l e m m a c) ((11 - A + u(k, A) + w(k, A, V)), )) = ((B - \u00C2\u00A3), 4>) - ((u(k, A) + w(k, A, V)), )} > const (]1_ A ) ( i - 0 / 2 ^ V / F T A ^ , For any 5 > 0 there is a constant cs such that T h i s is so because ( I - A ) (H - A) ( 1 _ E ) / 2 < S y/T^&t +cs\\\\ 5:(i+^) ( i- e ) / ai^)i a ber# < < 5 2 ^ ( l + 6 2 ) | ^ ) | 2 + c ^ | ^ ) | 2 6er# t>er# = 52 IvT^V^ I + c\ Uf < (5 | | v T ^ V ^ I + c5 \\,4>)) > V I - -const c|| V l - A4> However notice ( a \u00C2\u00B1 (3)2 > 0, so \et(3\ < \(a2 + /? 2) and so constc<511| VT^A> < i c o n s t2 c ^ M 2 + i V f ^ A ^ Therefore ((1-A + u(k,X) + w(k,A,V))4>,4>)) > - - ( c o n s t c g \" ^ 2 Chapter 3. Analyticity of the Fermi surfaces 15 A n d the form is semibounded. A g a i n since \[2 const c$ 1 72 W e have 1 4 -const v / T^~A> < 2 c o n s t 2 c 2 ||||2 + J y/1 - Acf> < \{{l-A+u(k,\)+w(k,A,V)),))\ < const I n order to show that t - A + u(k, A) + A , V) is closed, we must show it is complete under the no rm \\\\2+1 = ((11 - A + u(k,X) + w(k,A,V))4>, < \\cp\\+1 < const2 V1-A& N o w if 4>n are Cauchy w i t h respect to ||||+1 then \\4>n n,m\u00E2\u0080\u0094*oo 0 : Vl^A^n - ) y/i - A((pn - 4>m] . 0 = J - | | ^ \u00E2\u0080\u009E - ^ | | + 1 Here we used the fact that \/t \u00E2\u0080\u0094 A is complete. Hence there is a unique asso-ciated self-adjoint semibounded operator Hk(A,V) T h e resolvent of (A \u00E2\u0080\u0094 A ) - 1 at i is given by s _ ^ + i , which is compact because the eigenvalues { \ b G r#} converges to zero. T h e n by the resolvent ident i ty 1 1 1 1 (u + w) Hk{A,V) - X + i 1 - A + i + u + k 1 1 - A + i 1 - A + i + u + w and l e m m a 2 a), b) , the resolvent of Hk{A,V) at i is also compact . Hence the spec t rum of Hk(A, V) is discrete. T h e n A e Spec(Hk{A, V)) i f and on ly i f there exists tj) e VHk(A,v) C V such that 11 - A + i (Hk(A,V)-X) VT^A VI^~AiJ} = 0 T h i s is the case if and only if -y==(Hk{A, V)-X)^=K has a non- t r iv ia l kernel. B y [S] T h e o r e m 9.2e), this is the case if and only if F(k, A, A, V) = 0 16 Chapter 4 P r o o f o f t h e Main T h e o r e m For s impl i c i ty we wri te e ( k , A , V ) = ex(k,A, V). B y part (iii) of the corol lary i n secion II , for each ( A , V ) G U the function e ( - , A , V ) has a unique ex t r emum fcmtn(A,V). T h i s ex t remum is a non-degenerate m i n i m u m . F r o m the imp l i c i t funct ion we know that fcm,\u00E2\u0080\u009E(A,V) depends ana ly t ica l ly on ( A , V ) . T h e same is true for the corresponding value A m i \u00E2\u0080\u009E ( A , V ) = e ( / c m i n ( A , V ) , A , V ) . F r o m part (iii) of the corol lary i n section II we have A m , \u00E2\u0080\u009E ( A , V ) < Ao- N o w let P = {(A, A , V ) G K x U | A m i n ( A , V ) < A < A 0 } . T h e n for each (A, A , V ) G P the F e r m i surface F\(A,V) is a smooth , real analyt ic , s t r ic t ly convex (d- l ) -d imens iona l mani fo ld w h i c h is not empty. For k G F\ ( A , V ) denote the outward uni t no rma l vector to F\(A,V) at k by n(k) . If ( A , A , V ) G P then for each \u00C2\u00A3 on the uni t sphere there is a unique point k\(\u00C2\u00A3,A,V) G Fx ( A , V ) s u c h that n(fc.\(f , A , V ) ) = A g a i n it follows from the impl i c i t funct ion theorem that 5 ^ - i x P \u00E2\u0080\u0094\u00E2\u0080\u00A2 D, (\u00C2\u00A3, A, A, V) H - > fcA(\u00C2\u00A3, A,V)' is a real analyt ic map . T o prove the theorem stated i n the In t roduct ion we have to show that for (A, A, V) i n an open dense subset of P and a l l p G M d the intersection Fx(A, V ) n (p\u00E2\u0080\u0094F\(A, V) has dimension at most d-2. Since for a l l (A, A,V) e P the mani fo ld is real ana ly t ic smooth and s t r ic t ly convex, one has either dim(Fx(A, V)n(p- Fx(A, V)) < d-2 or FX(A, V) = (p - FX(A, V) If the first case is true then we're done. If Fx(A, V) = (p - Fx{A, V)) then the inversion i n the point p/2 maps the point of Fx{A, V) w i t h n o r m l vector \u00C2\u00A3 to the point w i t h \u00E2\u0080\u0094\u00C2\u00A3. i.e. FX(A, V) = (p-Fx(A, V)) => kx(Z, A, V)+kx(^, A, V)+kx(-Z, A, V) = p V \u00C2\u00A3 G Sd Therefore the set of a l l (A, A,) G P for which there is a point p GRd such that dim(Fx{A, V)n(p- F\{A, V))) > d-2 is contained i n the set S' = { (A,A,V) G P | V\u00E2\u0082\u00AC(M\u00C2\u00A3, A,V) + kx{-Z,A,V)) = 0VZ\u00C2\u00A3 S^1} Observe that S' is the interect ion of the analyt ic hypersurfaces S'= p| {(X,A,V)eP\ Vt(kx(t,A,V) + kx(-t;,A,V))=0} Chapter 4. Proof of the Main Theorem 17 Hence to show that the complement of S' is open and dense it suffices to find one (A, A, V) G P that does not lie i n S'. W e can do this by choosing V = 0 and a par t icular vector potent ia l A which is on ly 2 dimensional , and showing that for sma l l t a n d some A the t r ip le (A, t \u00E2\u0080\u00A2 A, 0) does not i n S'. T h i s then also shows that the complement of S' D {(A, A, V) G P \ V = 0} is open and dense m{(\,A,V)eP\V = 0} Hence i n the fol lowing ca lcula t ion we w i l l on ly consider the points (A,r; \u00E2\u0080\u00A2 ^4,0) of P w i t h a su i tab ly choosen two dimensional vector potent ia l A . Therefore we restrict ourselves to the case d = 2 and delete the V-var iab le i n the nota t ion . W e begin by comput ing the first 3 derivatives of e(k, t \u00E2\u0080\u00A2 A) at the or ig in for a rb i t ra ry A . W e use nota t ion f(k) = \u00E2\u0080\u00A2^f(k,t)\t=o L e m m a . L e t A \u00C2\u00A3 A- Denote e(k,t) = e(k,t-A). T h e n there exists a constant C such that for a l l k G D e(k) =0 e(k)=C-2 \u00C2\u00A3 l j r T | ( 2 f c + 6).i(6)| faer#\{o} e(k) =12Re \u00C2\u00A3 ^ 1 [A(-c) \u00E2\u0080\u00A2 A(c- b)[(2k + b) \u00E2\u0080\u00A2 A(b)\ 6 ,cer#\{o} Here A{x) = \u00C2\u00A3 b e r # A(b)eibx w i t h A{b) = {Ax(b), A2(b)) being its Fourier co-efficients. A l s o , for each A G (0, Ao) and every \u00C2\u00A3 G S1 jtkx(U \u00E2\u0080\u00A2 A)\t=0 = 0 2%/A\u00C2\u00A3 \u00E2\u0080\u00A2 ^ pkx&t- A)\t=0 - ~e(V\0 2 v / A \u00C2\u00A3 - ^ f c A ( \u00C2\u00A3 , i - A ) | t = 0 = - e ( v / A \u00C2\u00A3 ) P r o o f : L e t ibk(t) be the eigenfunction w i t h eigenvalue e(k,t) for the operator Hk(t \u00E2\u0080\u00A2 A) normalised by Vfc(0) = and < Vfe(0), Mt) >= 1 Vvol , where vo l is the volume of R 2 / r . T h e constant function is an acceptable eigenfunction since at t = 0, i?fc(0 \u00E2\u0080\u00A2 ^ 4) = (zV \u00E2\u0080\u0094 k)2. T h e n for smal l t and k G D , tpk(t) is an analyt ic function of t and k, so we may differentiate Hip = e Chapter 4. Proof of the Main Theorem 18 and get Hip + Hip = eip + eip (4.1) Hip + 2Hip + Hip = lip + 2eip + eip (4.2) Hip + 3Hip + 3Hip + H'ip= eip + 3eip + 3eip + eip (4.3) N e x t we can take the inner product of these w i t h ip(0) to get i= (4.4) e =< Hip, ip>+2< Hip, iP > (4.5) e = +3 < Hip,tp > +3 < Hip\ip > (4.6) Since we have < ipk(0),ipk(t) >= 1, differentiate and we w i l l get ^ < ipk{t),ipk(0) > = < ipk{t),ipk(0) >= 0- So s imi la r ly < ^ f e ( t ) , Vfc(0) > = < # f c ( t ) ,^ f c (0) > = 0 and < Hip\{t),ipk(0) > = < Hipk(t),ipk(0) > = < H'ijk(t),ipk{0) > = 0. B u t now 4-HM \u00E2\u0080\u00A2 A) \u00E2\u0080\u0094 A - (i\7 + t- A \u00E2\u0080\u0094 k) + (iV + t- A \u00E2\u0080\u0094 k) \u00E2\u0080\u00A2 A dt = 2A-(W + t-A-k) + i ( ^ + ^ ) (4.7) So at t = 0, we have * ^ ^ 1 = ^ s(-24-4+!(^+^)) <48) N o w form the inner product w i t h ip(0) = \u00E2\u0080\u00A2 1 we w i l l get e==^-<(-2k-A + i ( ^ + ^ ) ) , l > vol dxi ox2 B u t recal l i n the beginning of the paper we specified A = {A = (Ai, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2'\u00E2\u0080\u00A2 , Ad) 6 (Lr&(Rd/T))d I /R ( J / r A{x)dx = 0}, so < k-A, 1 > = < hAi(x), 1 > + < k2A2{x), 1 > = / kxAi(x)dx+ / k2A2(x)dx JWl/T JWLd/T N o w < ^^\")1 > = < 75J^)1 Jw1/r^X2^'X ~ A2{x)\evaluated at boundaries \u00E2\u0080\u0094 0 because A(x) is per iodic . Hence we have e = 0 F r o m (4) we have iP = -(H - e)-lHip (4.9) Chapter 4. Proof of the Main Theorem 19 Here we define (H \u00E2\u0080\u0094 e) 1 to be 0 on ^ (0 ) and the inverse of (H - e) on the or thogonal complement of ip(0). P lugg ing into (8) e = < Hip, i>>-2< H(H - e ) - 1 ^ , V> > F r o m (7) H = 2{A\ + A22) so that < Hip,ip > is a constant C = ^ \u00E2\u0080\u00A2 < (A\ + A2), 1 > independent of k. U s i n g (8) we get Since we can wri te A ( x ) as 2^6er# A(b)elbx, w e n a v e A l s o because Hk(0,0) = ( i V - k)2, Hk(0,0) 1 = e(0) 1, e(0) = k2. So (Hk(0,0) - e(Q))eibx = (b2 + 2k \u00E2\u0080\u00A2 b)eib-x W e finally get e \" = c - ^ \u00C2\u00A3 w T 2 k \u00E2\u0080\u0094 b { { 2 k + b ) \u00E2\u0080\u00A2 A { h ) e i b x ' \u00C2\u00A3 ( ( 2 f c + b ) \u00E2\u0080\u00A2 A { b ) ) e i b X ) 6er#\{o} 6er# = C ' 2 \u00C2\u00A3 ^ ^ 1 ( 2 * + 6) - ^ (6) |\u00C2\u00BB t e r # \ { 0 } N o w , from (2) we get iP^ ~(H-e)-\Hip-e^-2eip + 2Hip) = -{H - e^Hip + e ( # - c ) \" V(0) -2-0-ip-2(H- e^HiH - t)~lHip = ~(H - t)-lHiP + 2(H - e)-lH(H - e)~lHiP (4.10) N o w combine (6),(9),(10) and the fact that H \u00E2\u0080\u0094 0, we get e = -3< H(H -e)-xHib,i)>-3< H(H - e ) \" 1 ^ , V> > + & = -%Re <{H- e^Hip, #i/> > +6 < H(H - e ) \" 1 ^ ^ - e ) - 1 j f y , V > Chapter 4. Proof of the Main Theorem 20 Since (H - e)~lHiP = \u00E2\u0080\u0094^={H - e)\"1 \u00C2\u00A3 (2k + b) \u00E2\u0080\u00A2 A(b)eibx vol 6 g r # b-x y/vol ^ b2 + 2k-b v U U i ber#\{o} Hip = -?=(A\ + Al) = \u00C2\u00A3 E \u00E2\u0080\u00A2 A ( c ) e ^ x V v d ber* c e r # 2 Y, Mb~ c) \u00E2\u0080\u00A2 A(c)eibx v\u00C2\u00B0l u - r .* and ^ - < H e i b x , e i c x > = - ! - < f - 2 A \u00E2\u0080\u00A2 (6 + k) + t V \u00E2\u0080\u00A2 A ] e i b x , e i c : c > wo/ no/ = j [-2A -(b+k) + iV- A]ei{b-C)xdx vol J R d / r = \u00E2\u0080\u0094, i - 2 ( 6 + k)-A(c-b) + ii(c- b)A(c - b)dx vol J R d / r = -(2k + b + c)-A(c-b) we get 1 =12Re E b 2 + 1 2 f c , 6 [ ( 2 f c + 6) \u00E2\u0080\u00A2 - c) \u00E2\u0080\u00A2 i ( c ) ] 6,cer#\{0} (>,cer#\{0} T h e above completes the statement about the derivatives of e. W e now prove the statement about the derivatives of t \u00E2\u0080\u0094> k\(\u00C2\u00A3,t \u00E2\u0080\u00A2 A) for fixed A G (0, Ao) and \u00C2\u00A3 G S 1 . To simplify nota t ion put = 4) Different iat ing the ident i ty C(K(\u00C2\u00A3, t), t) = A we get V f c e ( K ( \u00C2\u00A3 , *),*) \u00E2\u0080\u00A2 ^ ( \u00C2\u00A3 , t) + *), t) = 0 (4.11) Since e = 0 and Vfce(/s(\u00C2\u00A3, 0), 0) = 2k = 2 \ /A\u00C2\u00A3 , sett ing t = 0 gives \u00C2\u00A3 \u00E2\u0080\u00A2 \u00C2\u00AB ( 0 = 0 (12.a) Chapter 4. Proof of the Main Theorem 21 L e t denote the vector (\u00E2\u0080\u0094&2,\u00C2\u00A3i) perpendicular to \u00C2\u00A3 = (\u00C2\u00A31,62), then by the defini t ion of k\ we have \u00E2\u0080\u00A2 \7ke(K(\u00C2\u00A3,t),t) = 0. Differentiat ing this ident i ty we get ^ \u00E2\u0080\u00A2 (Hessian(e) \u00E2\u0080\u00A2 + V f e e ) = 0 Since for t = 0, we have Hessian(e) = Hessian(2fc) = 2 x 1 and e = 0, we get \u00C2\u00A3 x - \u00C2\u00AB ( O = 0 (12.b) C o m b i n i n g (12a) and (12b) gives k = 0 (4.12) Differentiat ing (11) again and le t t ing t = 0 gives A(0 \u00E2\u0080\u00A2 Hessiank (e) \u00E2\u0080\u00A2 K(\u00C2\u00A3) + 2 V f c e ( * ( \u00C2\u00A3 , 0)) \u00E2\u0080\u00A2 \u00C2\u00AB (\u00C2\u00A3 ) + V f c e ( r < \u00C2\u00A3 , 0)) \u00E2\u0080\u00A2 K(\u00C2\u00A3) + "Thesis/Dissertation"@en . "2004-11"@en . "10.14288/1.0080065"@en . "eng"@en . "Mathematics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Asymmetric Fermi surfaces for periodic Schrodinger Operators"@en . "Text"@en . "http://hdl.handle.net/2429/15869"@en .