Spectral decimation & its applications to spectral analysis on infinite fractal lattices Joe P. Chen Department of Mathematics Colgate University QMath13: Mathematical Results in Quantum Physics Special Session on Quantum Mechanics with Random Features Georgia Tech, Atlanta, GA October 8–11, 2016 Joint works with S. Molchanov and A. Teplyaev Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 1 / 19
Motivation: Analysis on nonsmooth domains Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 2 / 19
Some fractals are nicer than others t t ✲ ✛ t t t t ✲ ✛ ✲ ✛ ✲ ✛ t t t t t t t t t t ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ 1 q p p q q p q p p q p q q p p q 1 Each of these fractals is obtained from a nested sequence of graphs which has nice, symmetric replacement rules. Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 3 / 19
Spectral decimation (= spectral similarity) a 0 a 1 a 2 Rammal-Toulouse ‘84, Bellissard ‘88, Fukushima-Shima ‘92, Shima ‘96, etc. A recursive algorithm for identifying the Laplacian spectrum on highly symmetric, finitely ramified self-similar fractals. Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 4 / 19
Spectral decimation Definition (Malozemov-Teplyaev ’03) Let H and H 0 be Hilbert spaces. We say that an operator H on H is spectrally similar to H 0 on H 0 with functions ϕ 0 and ϕ 1 if there exists a partial isometry U : H 0 → H (that is, UU ∗ = I) such that 1 U ( H − z ) − 1 U ∗ = ( ϕ 0 ( z ) H 0 − ϕ 1 ( z )) − 1 =: ϕ 0 ( z ) ( H 0 − R ( z )) − 1 for any z ∈ C for which the two sides make sense. A common class of examples: H 0 subspace of H , U ∗ is an ortho. projection from H to H 0 . Write H − z in block matrix form w.r.t. H 0 ⊕ H ⊥ 0 : � � I 0 − z X H − z = . X Q − z Then U ( H − z ) − 1 U ∗ is the inverse of the Schur complement S ( z ) w.r.t. to the lower-right block of H − Z : S ( z ) = ( I 0 − z ) − X ( Q − z ) − 1 X . Issue: There may exist a set of z for which either Q − z is not invertible, or ϕ 0 ( z ) = 0. Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 5 / 19
Spectral decimation: the main theorem Spectrum σ (∆) = { z ∈ C : ∆ − z does not have a bounded inverse } . Definition The exceptional set for spectral decimation is E ( H , H 0 ) def = { z ∈ C : z ∈ σ ( Q ) or ϕ 0 ( z ) = 0 } . Theorem (Malozemov-Teplyaev ’03) ∈ E ( H , H 0 ) : Suppose H is spectrally similar to H 0 . Then for any z / R ( z ) ∈ σ ( H 0 ) ⇐ ⇒ z ∈ σ ( H ) . R ( z ) is an eigenvalue of H 0 iff z is an eigenvalue of H. Moreover there is a one-to-one map between the two eigenspaces. Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 6 / 19
Example: Z + t t t t t t t t t t t t t t t t t t t Let ∆ be the graph Laplacian on Z + (with Neumann boundary condition at 0), realized as the limit of graph Laplacians on [0 , 2 n ] ∩ Z + . If z � = 2 and R ( z ) = z (4 − z ), then R ( z ) ∈ σ ( − ∆) ⇐ ⇒ z ∈ σ ( − ∆). σ ( − ∆) = J R , where J R is the Julia set of R . J R is the full interval [0 , 4]. Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 7 / 19
The pq -model A one-parameter model of 1D fractals parametrized by p ∈ (0 , 1). Set q = 1 − p . A triadic interval construction, “next easiest” fractal beyond the dyadic interval. Earlier investigated by Kigami ’04 (heat kernel estimates) and Teplyaev ’05 (spectral decimation & spectral zeta function). Assign probability weights to the three segments: q p m 1 = m 3 = 1 + q , m 2 = 1 + q Then iterate. Let π be the resulting self-similar probability measure. t t ✲ ✛ 1 1 m 1 m 2 m 3 t t t t ✲ ✛ ✲ ✛ ✲ ✛ 1 q p p q 1 t t t t t t t t t t ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ 1 q p p q q p q p p q p q q p p q 1 Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 8 / 19
Spectral decimation for the pq -model The spectral decimation polynomial is R ( z ) = z ( z 2 − 3 z +(2+ pq )) . pq n − 1 � R − m { 1 ± q } σ ( − ∆ n ) = { 0 , 2 } ∪ m =0 ✻ (2 , 2) r r ✲ r r r r (0 , 0) max( p , q ) Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 9 / 19
Spectral decimation for the pq -model The spectral decimation polynomial is R ( z ) = z ( z 2 − 3 z +(2+ pq )) . pq n − 1 � R − m { 1 ± q } σ ( − ∆ n ) = { 0 , 2 } ∪ m =0 λ 0 , 0 = 0 λ 0 , 1 = 2 λ 1 , 0 = 0 λ 1 , 2 λ 1 , 1 λ 1 , 3 = 2 λ 2 , 0 = 0 λ 2 , 9 = 2 λ 2 , 6 λ 2 , 2 λ 2 , 4 λ 2 , 8 λ 2 , 1 λ 2 , 5 λ 2 , 7 λ 2 , 3 Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 10 / 19
The pq -model on Z + t t t t t t t t t t ✲ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ 1 q p p q q p q p p q p q q p p q q p ∆ p is not self-adjoint w.r.t. ℓ 2 ( Z + ), but is self-adjoint w.r.t. the discretization of the aforementioned self-similar measure π . Let ∆ + p = D ∗ ∆ p D , where D : ℓ 2 ( Z + ) → ℓ 2 (3 Z + ) , ( Df )( x ) = f (3 x ) . Then ∆ p is spectrally similar to ∆ + p . Moreover, ∆ p and ∆ + p are isometrically equivalent (in L 2 ( Z + ) or in L 2 ( Z + , π )). Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 11 / 19
The pq -model on Z + t t t t t t t t t t ✲ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ ✛ ✲ 1 q p p q q p q p p q p q q p p q q p Spectrum σ ( H ) = { z ∈ C : H − z does not have a bounded inverse } . Facts from functional analysis: σ ( H ) is a nonempty compact subset of C . σ ( H ) equals the disjoint union σ pp ( H ) ∪ σ ac ( H ) ∪ σ sc ( H ). pure point spectrum ∪ absolutely continuous spectrum ∪ singularly continuous spectrum Theorem (C.-Teplyaev, J. Math. Phys. ’16) If p � = 1 2 , the Laplacian ∆ p , regarded as an operator on either ℓ 2 ( Z + ) or L 2 ( Z + , π ) , has purely singularly continuous spectrum. The spectrum is the Julia set of the polynomial R ( z ) = z ( z 2 − 3 z +(2+ pq )) , which is a topological Cantor set of Lebesgue measure zero. pq One of the simplest realizations of purely singularly continuous spectrum. The mechanism appears to be simpler than those of quasi-periodic or aperiodic Schrodinger operators. ( cf. Simon, Jitomirskaya, Avila, Damanik, Gorodetski, etc.) See also recent work of Grigorchuk-Lenz-Nagnibeda ‘14, ‘16 on spectra of Schreier graphs. Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 12 / 19
Proof of purely singularly continuous spectrum (when p � = 1 2 ) (2 , 2) ✻ q q ✲ q q q q (0 , 0) max( p , q ) Spectral decimation: ∆ p is spectrally similar to ∆ + p , and they are isometrically equivalent. 1 After taking into account the exceptional set, R ( z ) ∈ σ (∆ p ) ⇐ ⇒ z ∈ σ (∆ p ). Notably, the repelling fixed points of R , { 0 , 1 , 2 } , lie in σ (∆ p ). ∞ ∞ 1 , � R ◦− n (0) ⊂ σ (∆ p ). Meanwhile, since 0 ∈ J ( R ), � R ◦− n (0) = J ( R ). By 2 n =0 n =0 So J ( R ) ⊂ σ (∆ p ). 1 , R ◦ n ( z ) ∈ σ (∆ p ) for each n ∈ N . On the one hand, σ (∆ p ) is If z ∈ σ (∆ p ), then by 3 compact. On the other hand, the only attracting fixed point of R is ∞ , so F ( R ) (the Fatou set) contains the basin of attraction of ∞ , whence non-compact. Infer that ∈ F ( R ) = ( J ( R )) c . So σ (∆ p ) ⊂ J ( R ). z / Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 13 / 19
Proof of purely singularly continuous spectrum (when p � = 1 2 ) (2 , 2) ✻ q q ✲ q q q q (0 , 0) max( p , q ) Thus σ (∆ p ) = J ( R ). When p � = 1 2 , J ( R ) is a disconnected Cantor set. 4 So σ ac (∆ p ) = ∅ . Find the formal eigenfunctions corresponding to the fixed points of R , and show that none of 5 them are in ℓ 2 ( Z + ) and in L 2 ( Z + , π ). Thus none of the fixed points lie in σ pp (∆ p ). By spectral decimation, none of the pre-iterates of the fixed points under R are in σ pp (∆ p ). So σ pp (∆ p ) = ∅ . Conclude that σ (∆ p ) = σ sc (∆ p ). 6 Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 14 / 19
The Sierpinski gasket lattice ( SGL ) Let ∆ be the graph Laplacian on SGL . ∈ { 2 , 5 , 6 } and R ( z ) = z (5 − z ), If z / then R ( z ) ∈ σ ( − ∆) ⇐ ⇒ z ∈ σ ( − ∆). σ ( − ∆) = J R ∪ D , where J R is the Julia set of R ( z ) and �� ∞ m =0 R − m { 3 } � D := { 6 } ∪ . J R is a disconnected Cantor set. Thm. (Teplyaev ’98) On SGL , σ (∆) = σ pp (∆). Eigenfunctions with finite support are complete. → Localization due to geometry. Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 15 / 19
Localized eigenfunctions on SGL Joe P. Chen (Colgate) Spectral decimation QMath 13 Atlanta 10/2016 16 / 19
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