Spherical Functions and Spectrum of the Laplacian on Semi-homogeneous Trees

Abstract We determine the $$\ell ^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -spectrum of the isotropic nearest-neighbor stochastic transition operator $$\mu _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> acting on functions on the set V of vertices of a semi-homogeneous tree; in the much simpler setting of homogeneous trees, the spectrum has been known for a long time. The spectrum is given by the eigenvalues of spherical functions, normalized at a reference vertex $$v_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> . We first show that spherical functions are boundary integrals of generalized Poisson kernels that, unlike the homogeneous setting, are not complex powers of the usual Poisson kernel. Then we compute these generalized Poisson kernels via Markov chains and their generating functions, whence we work out explicit expressions for spherical functions, that turn out to have an $$\ell ^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> behavior different from the homogeneous setting; indeed, one of them, for an appropriate choice of $$v_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , belongs to $$\ell ^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> for some $$p&lt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Up to normalization, on each of the two homogeneity classes, that is, on each orbit $$V_+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> , $$V_-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mo>-</mml:mo> </mml:msub> </mml:math> of the Markov chain induced by $$\mu _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , the operator $$\mu _1^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>μ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> differs from the step-2 isotropic operator $$\mu _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> only by a shift. On the other hand, the recurrence relation associated to the semi-homogeneous $$\mu _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> is that of a polygonal graph, akin to that of $$\mu _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> on a homogeneous tree. By this token, we compute the spectra of $$\mu _1^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>μ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> on $$\ell ^p(V_+)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup>