Spectral Analysis Beyond $\ell ^2$ on Sierpinski Lattices

Abstract

We study the spectrum of the Laplacian on the Sierpinski lattices. First, we show that the spectrum of the Laplacian, as a subset of ${\mathbb {C}}$, remains the same for any $\ell ^p$ spaces. Second, we characterize all the spectral points for the lattices with a boundary point.

Appendix

In this appendix, we construct the 4-eigenfunctions on ${\widetilde{\mathcal {SG}}}$. Note that this induces a class of eigenvalues $\Sigma _4=\bigcup _{m=0}^\infty R^{\circ -m}\{4\}$.

We introduce the following orthogonal matrices

$$\begin{aligned} A_1= \begin{pmatrix} 1&{}0&{}0\\ 0&{}0&{}-1\\ 0&{}-1&{}0 \end{pmatrix},\quad A_2= \begin{pmatrix} 0&{}0&{}-1\\ 0&{}1&{}0\\ -1&{}0&{}0 \end{pmatrix}, \quad A_3= \begin{pmatrix} 0&{}-1&{}0\\ -1&{}0&{}0\\ 0&{}0&{}1 \end{pmatrix}. \end{aligned}$$

(4.1)

Recall that if we fix an infinite word $\omega =\omega _1\omega _2\ldots$, then there is a Sierpinski lattice defined by ${\widetilde{\mathcal {SG}}}=\bigcup _{m=0}^\infty F_{\omega _1}^{-1}F^{-1}_{\omega _2}\ldots F^{-1}_{\omega _m} V_m$. For convenience, we write

$$\begin{aligned} q_i^{(-m)}=F_{\omega _1}^{-1}F^{-1}_{\omega _2}\ldots F^{-1}_{\omega _m}\left( q_i\right) ,\quad i=0,1,2, \end{aligned}$$

and write

$$\begin{aligned} q_{li}^{(-m)}=F_{\omega _1}^{-1}F^{-1}_{\omega _2}\ldots F^{-1}_{\omega _m}\left( F_lq_i\right) ,\quad l\in W_m=\{0,1,2\}^m. \end{aligned}$$

Clearly $q_i=q_{\omega _m\omega _{m-1}\ldots \omega _1 i}^{(-m)}$.

Proposition 4.1

• (a). If ${\widetilde{\mathcal {SG}}}$ has no boundary, then there is a three dimensional $\ell ^\infty$-eigenspace of $\Delta$ corresponding to 4.

• (b). If ${\widetilde{\mathcal {SG}}}$ has a boundary point, then there is a two dimensional $\ell ^\infty$-eigenspace of $\Delta$ corresponding to 4.

Fig. 8

An illustration for extending f to be a 4-eigenfunction.(We take $\omega _1=2,\omega _2=1$ as shown in the left picture.)

Proof

(a). Let $f(q_0)=a,f(q_1)=b,f(q_2)=c$, where a, b, c are arbitrary real number. Define

$$\begin{aligned} \begin{pmatrix} f\left( q^{(-m)}_0\right) \\ f\left( q^{(-m)}_1\right) \\ f\left( q^{(-m)}_2\right) \end{pmatrix}= A^{-1}_{\omega _{m}}\ldots A^{-1}_{\omega _{2}}A^{-1}_{\omega _1}\begin{pmatrix} f\left( q_0\right) \\ f\left( q_1\right) \\ f\left( q_2\right) \end{pmatrix}, \end{aligned}$$

(4.2)

and

$$\begin{aligned} \begin{pmatrix} f\left( q^{(-m)}_{l0}\right) \\ f\left( q^{(-m)}_{l1}\right) \\ f\left( q^{(-m)}_{l2}\right) \end{pmatrix}= A_{l_1}A_{l_2}\ldots A_{l_m}\begin{pmatrix} f\left( q^{(-m)}_0\right) \\ f\left( q^{(-m)}_1\right) \\ f\left( q^{(-m)}_2\right) \end{pmatrix},\quad \forall l\in W_m. \end{aligned}$$

(4.3)

See Fig. 8 for an example of the extension of f. One can easily check that f is a 4-eigenfunction of $\Delta$ on ${\widetilde{\mathcal {SG}}}$ and f is bounded. By the above construction, we get a three dimensional eigenspace to 4.

On the other hand, noticing that 4 is not a forbidden eigenvalue, a 4-eigenfunction f is uniquely determined by $f|_{V_0}$.

(b). The proof of (b) is essentially the same. The eigenspace is 2 dimensional as a consequence of the eigenvalue equation at the boundary point. $\square$