2 J.H. SCHENKER AND M. AIENMAN Figure. A schematic depiction of the spectral relation (H) =? ((H o )). of functions in `2(G) which vanish at O G. Thus

Transcription

1 THE CREATION OF SPECTRAL GAPS BY GRAPH DECORATION JEFFREY H. SCHENKER (a) AND MICHAEL AIENMAN (a;b) Abstract. We present a mechanism for the creation of gaps in the spectra of self-adjoint operators dened over a Hilbert space of functions on a graph, which is based on the process of graph decoration. The resulting Hamiltonians can be viewed as associated with discrete models exhibiting a repeated local structure and a certain bottleneck in the hopping amplitudes.. Introduction Energy spectra characterized by the presence of bands and gaps are familiar from the Bloch theory of periodic systems. In this note, we present another mechanism for the creation of spectral gaps which does not rely on translation invariance. The band-gap spectral structure plays an important role in the theory of the solid state [], as well as in the properties of dialectric and acustic media [2]. Of particular interest are also situations in which localized states are injected into existing gaps (see ref. [3] for a mathematical discussion with further references). These applied models are mentioned only as distant analogies; the topic we discuss pertains to spectral properties of Hamiltonians of discrete models, whose \hopping terms" can be viewed as associated with graphs exhibiting a repeated local structure and a certain bottleneck in hopping amplitudes. To present the principle described herein it is convenient to introduce the notion of \graph decoration". Given two graphs? and G, we may \decorate"? with G by \gluing" a copy of G to each vertex v of? in such a way that v is identied with the appropriate copy of some distinguished vertex O G 2 G (see x2 for a formal denition, and gures 2 and 3 for typical examples). Given self-adjoint operators H o on `2(?) and A on `2(G) there is a natural way to dene an operator extension H of H o and A to `2(? / G) where? / G denotes the decorated graph just described. In the absence of certain degeneracy, there is a simple relation between the spectra of H and H o, denoted here by (H ::: ), which allows us to conclude that intervals around certain energies " j are excluded from the spectrum of H. Specically, there is a function such that and is of the form (H) =? ((H o )) ; (.) (E) = E + c + X j w j " j? E ; (.2) where w j > 0 and " j ; c 2 R (see gure ). In fact, we shall see that " j are exactly the eigenvalues of the operator b P A b P where b P is the projection onto the subspace Date: Aug. 4, 2000.

2 2 J.H. SCHENKER AND M. AIENMAN Figure. A schematic depiction of the spectral relation (H) =? ((H o )). of functions in `2(G) which vanish at O G. Thus, we have the appealing picture in which the eigenenergies of the decorated graph are repelled by resonances with the \inner spectrum" of the decoration. In x2 we dene graph decoration and describe the operator extension mentioned above. In x3 we present our main result (Prop. 3.) which describes the spectral relationship presented above in full detail. Finally, in x4 we provide several examples and applications of Prop Graph decoration In this section, we suppose that we are given a graph? and a self adjoint operator H o on `2(?), the space of square summable functions on the vertices of? (see below). Our goal is to describe a certain class of graph extensions of? and a corresponding class of operator extensions of H o. Recall that a graph G is described by specifying two sets: () V (G) whose elements are called vertices, and (2) E(G) a set of (unordered) pairs of vertices called edges. The edges play a secondary role in our discussion, for we are mainly concerned with the Hilbert space of square summable functions mapping V (G)! C, which we denote `2(?). The situation of interest is when E(G) is dened so that a given operator A on `2(G) is compatible with G, by which we mean that the o diagonal matrix elements hxjajyi vanish whenever fx; yg 62 E(G). Correspondingly, our notation generally identies a graph G with its vertex set: by x 2 G we indicate x 2 V (G) and we shall say that a graph is countable (nite) if V (G) is countable (nite). The graph extensions of? shall be obtained by \gluing" copies of a second graph G to each vertex of?. The extended graph may be visualized as a eld in which We use the Dirac bra-ket notation for matrix elements in the standard basis jxi = x, with x(y) the Kronecker function.

3 SPECTRAL GAPS AND GRAPH DECORATION 3 are tethered many identical kites (see gures 2 and 3). Formally given any graph G with a distinguished vertex O G we dene the decoration of? by (G; O G ), denoted? / G, to be the following graph:. V (? / G) = V (?) V (G). 2. E (? / G) = E eld [ E kite, where: (a) E eld = f f(x; O G ); (y; O G )g j fx; yg 2 E(?) g. (b) E kite = f f(x; h); (x; g)g j x 2 V (?) and fh; gg 2 E(G) g. We think of the space `2(? / G) as the tensor product `2(?) `2(G), which is natural since the vertex set of? / G is V (?) V (G). The subspace of functions which are supported on? o = f(x; O g ) : x 2?g is naturally identied with `2(?). We denote by P the orthogonal projection onto this space. Let A be a self adjoint operator on `2(G). A natural extension of H o to `2(?) `2(G), incorporating A, is H := P H o P + A : (2.) The above operator is appropriate to the geometry of graph decoration, for if H o and A are compatible with? and G respectively, then H is compatible with? / G. An example of an operator of the form described in eq. (2.) is provided by the discrete Laplacian. On any graph H the discrete Laplacian, H, is dened by [ H ](x) := X y: fx;yg2e(h) (y)? (x) : (2.2) For decorated graphs, if we take H o =?? and A =? G then the operator dened by (2.) is H =??/G. 3. A resolvent evaluation principle We now focus on the case jgj < and present our main result. 2 Proposition 3.. Let H be a bounded 3 self adjoint operator of the form described in eq. (2.) with G a nite graph. If jo G i is a cyclic vector for A, then where is a function of the form with c; " j 2 R, and w j > 0. (H) =? ((H o )) ; (3.) (E) = E + c + nx j= w j " j? E ; (3.2) Furthermore, whether or not jo G i is cyclic, there is a function of the form (3.2) such that for each x 2? and z 2 C n R x; OG j(h? z)? jx; O G = xj(ho? (z))? jx ; (3.3) and the spectral measure, e x, for H associated to jx; O G i is related to the spectral measure, x, for H o associated to jxi by e x (de) = 0 (E) x(d(e)) : (3.4) 2 This result may be easily extended to the case jgj = provided the spectrum of A is discrete. 3 More generally, H may be unbounded provided the set of functions with nite support forms a core for H.

4 4 J.H. SCHENKER AND M. AIENMAN Thus? ((H o )) (H) : (3.5) Remarks: Recall that given a self-adjoint operator K on a Hilbert space K, the spectral measure associated to a vector v 2 K is dened via the functional calculus and the Riesz-Markov theoerem as the unique regular Borel measure,, such that f(e)(de) = hv; f(k)vi ; (3.6) for each f 2 C o (R), the family of continuous functions on R which vanish at innity. Eq. (3.4) is a formal expression which indicates the following identity for the expectations of a function f 2 C o (R): ex (de) f(e) = x (d") X E2? (") 0 f(e) : (3.7) (E) Proof of Proposition 3.: The heart of Prop. 3. is the relation (3.3), so let us begin with a derivation of this equation. Fix x 2? and z 2 C n R. Recall that the Green function, G(y; u) := x; O G j(h? z)? jy; u ; (y; u) 2? / G ; (3.8) is the unique square summable solution to the equation A natural guess is that the solution factors: With this ansatz, eq. (3.9) yields for g and h: and (H? z)g = jx; O G i : (3.9) G(y; u) = g(y) h(u) : (3.0) g(y) [(A? z)h](u) = 0 ; u 6= O G ; (3.a) [H o g](y) h(o g ) + g(y) [(A? z)h](o g ) = jxi : (3.b) It is now an easy exercise to solve these equations using the Green functions for H o and A: eq. (3.a) gives g as a function of h, g(y) = h(o G ) * xj H o + [(A? z)h](o g) h(o g )? jy+ while eq. (3.b) shows that h is a multiple of the Green function for A, ; (3.2) h(u) = C O G j(a? z)? ju ; (3.3) with C an arbitrary factor which drops out of the resulting solution: G(y; u) = * xj H o + ho G j(a? z)? jo G i? jy+ OG j(a? z)? ju ho G j(a? z)? jo G i : (3.4)

5 SPECTRAL GAPS AND GRAPH DECORATION 5 Setting u = O G and y = x in this expression yields (3.3) with? (z) := ho G j(a? z)? jo G i : (3.5) Because dim `2(G) is nite, is a rational function with nitely many simple real poles (which occur at the zeros of O G j(a? E)? jo G ). Hence, the partial fraction expansion (alternatively, the represenation theory of Herglotz functions) shows that is of the form displayed in eq. (3.2): n? X (z) = z? c + j= w j " j? z ; (3.6) with c; " j 2 R and w j > 0. (The coecient of z is one, since z=(z)! as z!.) We now turn to the verication of the relation between the spectral measures expressed in (3.4). First consider the situation when? is nite. For a self-adjoint operator B on a nite dimensional vector space there is a useful formula for the spectral measure,, associated to a vector : (de) =? h ; (B? E)? i de ; (3.7) where is the Dirac-delta \function." (This formula oers a simple route to the \spectral averaging" principle discussed in ref. [4]; its derivation is an instructive exercise which we leave to the reader.) Coupled with eq. (3.3), (3.7) easily yields: e x (de) = =? hx; O G j(h? E)? jx; O G i? hxj(h o? (E))? jxi de 0 (E) d(e) = 0 (E) x(d(e)) : (3.8) When? is innite we must turn to a more abstract derivation of eq. (3.4). Writing each side of eq. (3.3) as a spectral integral we nd that E? z de x(e) = E? (z) d x(e) : (3.9) Expanding the right side of this equality with partial fractions yields E? z de x(e) = X 2? (E) 0 ()? z d x(e) : (3.20) This equation can be viewed as a special case of: X f(e) de x (E) = 0 () f() d x(e) ; (3.2) 2? (E) and indeed (3.20) implies (3.2), for all f 2 C o P (R), since by the Stone-Weierstrass Theorem the set of nite sums of the form j c j E?z j is dense in C o (R). As mentioned previously, (3.2) is the statement claimed in (3.4). Finally, the spectral inclusion? ((H o )) (H) follows since? ((H o )) = [ x2? supp.(e x ) ; (3.22) which may be veried using (3.4). If fjx; O G ig is a cyclic family for H then eq. (3.22) shows further that? ((H o )) = (H). In case jo G i is a cyclic vector for

6 6 J.H. SCHENKER AND M. AIENMAN A, this family is easily seen to be cyclic for H. This completes the proof of the proposition. We conclude this section with several remarks regarding proposition 3. and its proof:. Eigenfunctions and generalized eigenfunctions of H factor in the same way as the Green function (see eq. (3.0)). That is satises H = E provided and (u) = (y; u) = (y) (u) (3.23) OG j(a? E)? ju ho G j(a? E)? jo G i ; (3.24a) H o = (E) : (3.24b) 2. The relationship between the spectrum of H o and H is a stronger relationship than the simple inclusion? ((H o )) (H): the spectral type is preserved under the map?. So, a bound state for H o gives rise to n bound states for H. Similarly, a band of absolutely continuous spectrum for H o gives rise to n bands of absolutely continuous spectrum for H. If H o possesses singular continuous spectrum, then such spectrum also occurs in the spectrum of H. 3. Generically, jo G i is a cyclic vector for A, and (H) =? ((H o )). However, even when jo G i is not cyclic, we may still determine the spectrum of H. We need only decompose the space `2(G) as a direct sum V V? with each summand invariant under A and jo G i cyclic for Aj V. Then, (H) =? ((H o )) [ (Aj V?) ; (3.25) which may be veried by noting that H Ho + Aj V 0 = 0 Aj V? : (3.26) Note that the eigenvalues in (Aj V?) occur with multiplicity a multiple of j?j. 4. The poles " j of are eigenvalues of A b = P b AP b, where P b is the projection of `2(G) onto the space of functions which vanish at O G. To see this, recall that " j satisfy O G j(a? " j )? jo G = 0. Thus the Green functions j(y) = OG j(a? " j )? jy themselves are eigenfunctions: A b j = " j j. Conversely, if jo G i is cyclic for A then every eigenvalue of b A is a pole of. 5. The coecients c and w j in the partial fraction expansion of (eq. (3.2)) satisfy and c =? ho G j A j O G i ; (3.27a) nx w j = O G j A 2 j O G? hog j A j O G i 2 ; (3.27b) j= w j = lim E!" j (" j? E)(E) = ho G j(a? " j )?2 jo G i : (3.27c)

7 SPECTRAL GAPS AND GRAPH DECORATION 7 Figure 2. A portion of the graph discussed in x4. in the case d = 2. The rst two equalities may be veried by expanding in a Laurent series around Examples and Applications 4.. Splitting the spectrum of the Laplacian on d. For a simple example of the phenomenon, consider the operator H =? d /G where G is the graph consisting of two vertices V (G) = fo G ; G g and a single edge E(G) = ffo G ; G gg (see gure 2). As described at the end of x2, H is of the form (2.) with H o =? d and A =? G. The spectrum of H o =? is d (Ho) = [0; 4d] (4.) and all the associated spectral measures are purely absolutely continuous. In this case, the function is easy to calculate:?? E (E) =? (? E) 2 = E? +?? E ; (4.2) and the vector jo G i is cyclic. Thus (H) = = E j 0 E? + h0; + 2d? p + 4d 2 i [? E 4d h2; + 2d + p + 4d 2 i and the spectral measures are purely absolutely continuous. ; (4.3) 4.2. An example in which jo G i is not cyclic. Consider now the discrete Laplacian on the graph d / G where G is the fully connected graph with three vertices V (G) = fo G ; G ; 2 G g (see gure 3). The involution R on `2(G) obtained by interchanging j G i and j2 G i commutes with G. Hence G leaves invariant the subspaces V +(?) of functions which are symmetric (anti-symmetric) with respect to this involution. A non-normalized basis of simultaneous eigenfunctions for R and (? G ) consists of: jo G i+j G i+j2 G i 2 V + with eigenvalue 0, 2 jo G i?j G i?j2 G i 2 V + with eigenvalue 3, and j G i?j2 G i 2 V? with eigenvalue 3.

8 8 J.H. SCHENKER AND M. AIENMAN Figure 3. A situation in which jo G i is not cyclic (x4.2). Using this basis, it is easy to see that jo G i is a cyclic vector for the restriction of G to V +. Furthermore, we can calculate :? (E) = 2 = E? 2 + 2? E ; (4.4) 3 + 3?E 3 0?E and note that (? G j V? ) = f3g. Thus, (??/G ) = f3g [ where " are the solutions to = 0; "? [ 3; " + ; E j 0 E? 2 + 2? E 4d (4.5) E? = 4d ; (4.6)? E with "? < and " + >. The spectrum is purely absolutely continuous except for the presence of an innitely degenerate eigenvalue at E = Persistence of band edge localization. The operator H may include disorder, in the form of a random potential at the sites of? o. It is generally expected that in such a situation the spectrum of H o will exhibit Anderson localization (i.e., dense pure point spectrum) at all the spectral edges. (This has been rigorously shown to be true in various situations [2, 5, 6, 7, 8]). Let us note that the mechanism of gap creation via graph decorations preserves such band edge localization, even if the randomness is not introduced at all the sites of the decorated graph.

9 SPECTRAL GAPS AND GRAPH DECORATION 9 References [] C. Kittel, Quantum Theory of Solids. (Wiley: New York, London, Sydney, 963). [2] A. Figotin and A. Klein, \Midgap defect modes in dielectric and acoustic media," SIAM J. Appl. Math, 58, 748, (998). [3] P. A. Deift and R. Hempel, \On the existence of eigenvalues of the Schrodinger operator H? W in a gap of (H)," Comm. Math. Phys., 03, 46, (986). [4] B. Simon and T. Wol, \Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians," Comm. Pure Appl. Math., 39, 75, (986). [5] J. M.Barbaroux, J.-M. Combes, and P. D. Hislop, \Localization near band edges for random Schrodinger operators," Helv. Phys. Acta, 70, 6, (997). [6] W. Kirsch, P. Stollman, and G. Stolz, \Localization for random perturbations of periodic Schrodinger operators," Rand. Op. Stoch. Eq., 6, 24, (998). [7] F. Klopp, \Internal Lifshits tails for random perturbations of periodic Schrodinger operators.," Duke Math. J., 98, 335, (999). [8] M. Aizenman, J. H.Schenker, R. M. Friedrich, and D. Hundertmark, \Finite-volume fractional-moment criteria for Anderson localization," to appear in Comm. Math. Phys. Princeton University, Departments of Mathematics (a) and Physics (b), Princeton, NJ 08544, USA. address: schenker@princeton.edu, aizenman@princeton.edu