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The purpose of this paper is to discuss how we can construct the heat kernel for (sub)-Laplacian in an explicit (integral) form in terms of a certain class of special functions. Of course, such cases will be highly limited. Here we only treat a typical operator, called Grushin operator. So, first we explain two methods to construct the heat kernel of a “step 2” Grushin operator. One is the eigenfunction expansion which leads to an integral form for the heat kernel, then we treat the formula by a method called, complex Hamilton–Jacobi method invented by Beals–Gaveau–Greiner. One of the main result in this paper is to construct an action function for a higher order oscillator. Until now, no explicit expression of the heat kernel for higher order cases have been given in an explicit form and we show a phenomenon that our action function will play a role toward the construction of the heat kernel of higher step Grushin operators.

We analyze the spectral zeta function for sub-Laplace operators on product manifolds M × N . Starting from suitable conditions on the zeta functions on each factor, the existence of a meromorphic extension to the complex plane and real analyticity in a zero neighbourhood is proved. In the special case of N = S 1 and using the Poisson summation formula, we obtain expressions for the zeta-regularized determinant. Moreover, we can calculate limit cases of such determinants by inserting a parameter into our formulas. This is a generalization of results in Furutani and de Gosson (2003) [1] and in particular it applies to an intrinsic sub-Laplacian on U ( 2 ) ≅ S 3 × S 1 induced by a sum of squares of canonical vector fields on S 3 ; cf. Bauer and Furutani (2008) [2] . Finally, the spectral zeta function of a sub-Laplace operator on Heisenberg manifolds is calculated by using an explicit expression of the heat kernel for the corresponding sub-Laplace operator on the Heisenberg group; cf. Beals et al. (2000) [18] and Hulanicki (1976) [19] .

The purpose of this paper is to study the spectral properties of a sub-Laplacian on S 3 , i.e., we discuss the analytic continuation of its spectral zeta function, give explicit expressions of the residues and especially, we provide an expression of the zeta-regularized determinant of the sub-Laplacian on S 3 . Also, we describe sub-Riemannian curves on S 3 based on the Hopf bundle structure, together with a proof of Chow’s theorem for this case in a strong sense (= connecting property by globally smooth curves). A characterization of sub-Riemannian geodesics on S 3 via an isoperimetric problem through the Hopf bundle is explained. Incidentally, we introduce a hypo-elliptic operator on P 1 C descended from the sub-Laplacian on S 3 , which we call a spherical Grushin operator. We determine the subspace where it degenerates and give an expression of the trace of its heat kernel by making use of the trace of the heat kernel of the sub-Laplacian. In case of S 7 , we limit ourselves to present the spectral zeta function of a sub-Laplacian.

We explain the explicit integral form of the heat kernel for the sub-Laplacian on two step nilpotent Lie groups G based on the work of Beals, Gaveau and Greiner. Using such an integral form we study the heat trace of the sub-Laplacian on nilmanifolds L\G where L is a lattice. As an application a common property of the spectral zeta function for the sub-Laplacian on L\G is observed. In particular, we introduce a special class of nilpotent Lie groups, called pseudo H-type groups which are generalizations of groups previously considered by Kaplan. As is known such groups always admit lattices. Here we aim to explicitly calculate the heat trace and the spectrum of the (sub)-Laplacian on various low dimensional compact nilmanifolds including several pseudo H-type nilmanifolds L\G, i.e. where G is a pseudo H-type group. In an appendix we discuss a zeta function which typically appears as the Mellin transform for these heat traces.

Let L be a lattice in R{sup n}, then the Jacobi identity is written as (1.1) {summation}{sub {gamma}}{epsilon}{sub L} e{sup -4}{pi}{sup 2}{parallel}{sup 2}t = Vol(R{sup n}/L)/(4{pi}t){sup n/2} {summation}{sub {gamma}}{epsilon}{sub L} e{sup -{parallel}{sup 2}/4t}. As is well-known, the left side of (1.1) is the trace of the heat kernel on the flat torus R{sup n}/L and the right side reveals the lengths of closed geodesics on it corresponding to each element in L.

In this article, we discuss three topics in the area of sub-Riemannian geometry and analysis.

We have seen in the previous chapters how an elliptic operator can be associated in a natural way with a geometric Riemannian structure. In a similar way sub-elliptic operators arise from similar structures, called sub-Riemannian structures, which will be discussed next. References for sub-Riemannian manifolds are [27] and [92].

The Lagrangian and Hamiltonian formalisms will be useful in the following chapters when the heat kernel will be computed using the path integral and geometric variational methods. In the following we shall present a brief overview of the variational theory needed in the sequel.

This chapter deals with a construction of heat kernels from the geometric point of view. Each operator will be associated with a geometry. Investigating the geodesic flow in this geometry, one can describe the heat kernels for a large family of operators. The idea behind this method is that the heat flow propagates along the geodesics of the associated geometry. The “density” of the heat flow is described by a volume function that satisfies a transport equation which is an analog of the continuity equation from fluid dynamics. This corresponds to the density of paths given by the van Vleck determinant in the path integral approach. This method works for elliptic operators with or without potentials or linear terms. The method can be modified to work even in the case of sub-elliptic operators, as the reader will become familiar with in Chaps. 9 and 10. This method was initially applied for the Heisenberg Laplacian; see, for instance [28].

This section deals with the study of the heat kernel of a sub-Laplacian on the three-dimensional sphere, and a Grushin-type operator on S2, called the spherical Grushin operator. This is a hypoelliptic operator on S2 and is defined by the method explained in the previous chapter. The method of investigation is the explicit determination of eigenvalues and eigenfunctions of the Laplacian and sub-Laplacian in terms of harmonic polynomials (see [97, 7, 8, 9]).

This chapter deals with probabilistic methods of obtaining the heat kernel. The main idea of this subject is that the heat kernel can be represented as a transition density of an associated stochastic process, as pointed out by Kolmogorov [80] in the early 1930s. The probabilistic methods were also useful in obtaining the heat kernel of the Heisenberg Laplacian, as shown by Hulanicki [68] and Gaveau [49] in the late 1970s.

In this chapter, we are going to use a harmonic analysis method to construct the heat kernels and fundamental solutions of the sub-Laplacian on the Heisenberg group. This method relies on Laguerre calculus. We shall start with a beautiful idea of Mikhlin from his 1936 study of convolution operators on ℝ2. Let K be a principal value (P.V.) convolution operator on ℝ2: $$\mathbf{K}(f)(x) {=\lim }_{\epsilon \rightarrow 0}{ \int \nolimits \nolimits }_{\vert y\vert >\epsilon }K(y)f(x - y)dy,$$ where f ∈ C0 ∞ (ℝ2) and K ∈ C ∞ (ℝ2 ∖ { (0, 0)}) is homogeneous of degree − 2 with vanishing mean value; i.e., $${\int \nolimits \nolimits }_{\vert y\vert =1}K(y)dy = 0.$$ Thus we can write $$K(x) = \frac{f(\theta )} {{r}^{2}},\qquad x = {x}_{1} + i{x}_{2} = r{e}^{i\theta },$$ where $$f(\theta ) ={ \sum \nolimits }_{m\in \mathbb{Z},m\neq 0}{f}_{m}{e}^{im\theta }.$$ Suppose that g is another smooth function on [0, 2π] with $$g(\theta ) ={ \sum \nolimits }_{m\in \mathbb{Z},m\neq 0}{g}_{m}{e}^{im\theta }.$$ Then g induces a principal value convolution operator G on ℝ2 with kernel \(\frac{g(\theta )} {{r}^{2}}\). In [91], we found the following identity.

In this section we shall provide an explicit multiplication form of the Laplacian for a certain class of two-step nilpotent Lie groups including Heisenberg groups; see [46]. The result presented in this section is more precise than the explicit formula of the heat kernel given for the first time in the paper of Hulanicki [68], as explained in Remark 9.1.5.

We shall deal next with the nonsymmetric form of the Heisenberg group. The Heisenberg group considered in this section will be the set ℍ n = ℝ n ×ℝ n ×ℝ with the following group law: $$(x,y,t) {_\ast} ({x}^{{\prime}},{y}^{{\prime}},{t}^{{\prime}}) = (x + {x}^{{\prime}},y + {y}^{{\prime}},t + {t}^{{\prime}} + x \cdot {y}^{{\prime}}),$$ where (x, y, t), (x ′ , y ′ , t ′ ) ∈ ℝ n ×ℝ n ×ℝ and $$x \cdot {y}^{{\prime}} ={ \sum \nolimits }_{k=1}^{n}{x}_{ k}{y}_{k}^{{\prime}}.$$

Finding the heat kernel of an elliptic operator on a compact manifold using the eigenvalues method is a well-known method in mathematical physics and quantum mechanics. Roughly speaking, the eigenvalues and eigenfunctions of an operator determine its heat kernel. The formula is an infinite series that involves products of eigenfunctions; see Theorem 6.1.1. It is interesting that in several cases this series can be written as an elementary function by using the associated bilinear generating function.