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In the present paper, we determine the group of automorphisms of pseudo $H$-type Lie algebras, which are two-step nilpotent Lie algebras closely related to the Clifford algebras $\Cl(\mathbb R^{r,s})$., 38 pages, several tables. arXiv admin note: text overlap with arXiv:1703.04948

We construct a codimension 3 completely non-holonomic subbundle on the Gromoll–Meyer exotic 7-sphere based on its realization as a base space of a Sp ( 2 ) -principal bundle with the structure group Sp ( 1 ) . The same method can be applied to construct a codimension 3 completely non-holonomic subbundle on the standard 7-sphere (or more general on a ( 4 n + 3 ) -dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides a new and simple proof for the standard sphere S 7 .

In this note, we explain a relation between the Popp measures of sub-Riemannian structures on the total space of a principal bundle and the base manifold. Then we determine several concrete cases explicitly.

Let \({\mathscr {N}}\) be a 2-step nilpotent Lie algebra endowed with a non-degenerate scalar product \(\langle .\,,.\rangle \), and let \({\mathscr {N}}=V\oplus _{\perp }Z\), where Z is the centre of the Lie algebra and V its orthogonal complement. We study classification of the Lie algebras for which the space V arises as a representation space of the Clifford algebra \({{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\), and the representation map \(J:{{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(V)\) is related to the Lie algebra structure by \(\langle J_zv,w\rangle =\langle z,[v,w]\rangle \) for all \(z\in {\mathbb {R}}^{r,s}\) and \(v,w\in V\). The classification depends on parameters r and s and is completed for the Clifford modules V having minimal possible dimension, that are not necessary irreducible. We find necessary conditions for the existence of a Lie algebra isomorphism according to the range of the integer parameters \(0\le r,s

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-homeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-homeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.

By employing a new reduction procedure we derive explicit expressions for the fundamental solutions of a family P k , λ of degenerate second order differential operators on R N + l . Here λ is a complex parameter located in the strip | Re ( λ ) | N + k − 1 . As is pointed out in [2] P k , 0 has a geometric background and arises as a Grushin-type operator induced by a sub-Riemannian structure on a k + 1 -step nilpotent Lie group. Our method leads to new formulas for the inverse of the Kohn-Laplacian Δ λ which has been widely studied before in the framework of pseudo-convex domains and CR geometry. As an application we show that in all cases the fundamental solutions have a meromorphic extension in the parameter λ to C ∖ Q . All poles are simple and Q ⊂ R is an explicitly given discrete set. We recover the invertibility of Δ 1 modulo the classical Szego projection. This phenomenon had been observed before in [11] .

We examine a class of Grushin type operators P k where k ∈ N 0 defined in (1.1). The operators P k are non-elliptic and degenerate on a sub-manifold of R N + l . Geometrically they arise via a submersion from a sub-Laplace operator on a nilpotent Lie group of step k + 1 . We explain the geometric framework and prove some analytic properties such as essential self-adjointness. The main purpose of the paper is to give an explicit expression of the fundamental solution of P k . Our methods rely on an appropriate change of coordinates and involve the theory of Bessel and modified Bessel functions together with Weber's second exponential integral.

We classify the trivializable sub-Riemannian structures on odd-dimensional spheres S N that are induced by a Clifford module structure of R N + 1 . The underlying bracket generating distribution is of step two and spanned by a set of global linear vector fields X 1 , … , X m . As a result we show that such structures only exist in the cases where N = 3 , 7 , 15 . The corresponding hypo-elliptic sub-Laplacians Δ sub are defined as the (negative) sum of squares of the vector fields X j . In the case of a trivializable rank four distribution on S 7 and a trivializable rank eight distribution on S 15 we obtain a part of the spectrum of Δ sub . We also remark that in both cases there is a relation between the eigenfunctions and Jacobi polynomials.

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.

For a series of weighted Bergman spaces over bounded symmetric domains in ℂn, it has been shown by Axler and Zheng [1]; Englis [10] that the compactness of Toeplitz operators with bounded symbols can be characterized via the boundary behavior of its Berezin transform B a . In case of the pluriharmonic Bergman space, the pluriharmonic Berezin transform B ph fails to be one-to-one in general and even has non-compact operators in its kernel. From this point of view, perhaps surprisingly we show that via B ph the same characterization of compactness holds for Toeplitz operators on the pluriharmonic Fock space.

We derive the spectral zeta function in terms of certain Dirichlet series for a variety of spherical space forms M G . Extending results in [C. Nash, D. O’Connor, Determinants of Laplacians on lens spaces, J. Math. Phys. 36 (3) (1995) 1462–1505] the zeta-regularized determinant of the Laplacian on M G is obtained explicitly from these formulas. In particular, our method applies to manifolds of dimension higher than 3 and it includes the case where G arises from the dihedral group of order 2 m . As a crucial ingredient in our analysis we determine the dimension of eigenspaces of the Laplacian in form of some combinatorial quantities for various infinite classes of manifolds from the explicit form of the generating function in [A. Ikeda, On the spectrum of a Riemannian manifold of positive constant curvature, Osaka J. Math. 17 (1980) 75–93].

The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that any 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with Lie brackets. This choice of the standard pseudo-metric form allows to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo $H$-type algebras have bases with rational structural constants, which implies that the corresponding pseudo $H$-type groups admit lattices., 50 pages

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.

We describe a relation between Atiyah–Patodi–Singer boundary condition and a global elliptic boundary condition, which naturally appears in formulating a splitting formula for a spectral flow, when we decompose the manifold into two components. Then we give a variant of the splitting formula with the Hormander index as a correction term.

We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line bundles and trivial on some subspaces. Also we remark a relation of the {\it Quillen determinant line bundle} and the {\it Maslov line bundle}., Comment: 11 pages, to appear in the Journal of Geometry and Physics

We study a problem of the geometric quantization for the quaternionprojective space. First we explain a Kahler structure on the punctured cotangent bundleof the quaternion projective space, whose Kahler form coincides withthe natural symplectic form on the cotangent bundle and show thatthe canonical line bundle of this complex structure is holomorphicallytrivial by explicitly constructing a nowhere vanishing holomorphicglobal section. Then we construct a Hilbert space consisting of acertain class of holomorphic functions on the punctured cotangentbundle by the method ofpairing polarization and incidentally we construct an operatorfrom this Hilbert space to the L 2 space of the quaternionprojective space. Also we construct a similar operator between thesetwo Hilbert spaces through the Hopf fiberation.We prove that these operators quantizethe geodesic flow of the quaternion projective space tothe one parameter group of the unitary Fourier integral operatorsgenerated by the square root of the Laplacian plus suitable constant.Finally we remark that the Hilbert space above has the reproducing kernel.

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz-Radon orthogonal matrices.

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.

The Fourier transform has been known as one of the most powerful and useful methods of finding fundamental solutions for operators with constant coefficients. Sometimes the application of a partial Fourier transform might be more useful than the full Fourier transform. In this chapter, by the application of the partial Fourier transform, we shall reduce the problem of finding the heat kernel of a complicated operator to a simpler problem involving an operator with fewer variables. After solving the problem for this simple operator, the inverse Fourier transform provides the heat kernel for the initial operator represented under an integral form. In general, this integral cannot be computed explicitly, but in certain particular cases it actually can be worked out. We shall also apply this method to some degenerate operators.

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].

In 1948 Feynman [42] provided an informal expression for the propagators of the famous Schrodinger equation, involving an integration over all the continuous paths with respect to a nonexistent infinite-dimensional Lebesgue measure. In fact, this is not really an integral, since there is no measure to give the integral. Since then, a large number of papers have tried to explain the precise mathematical meaning of the Feynman integral.

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 describe a method that was first studied by Beals [12] and Aarao [2, 1] to construct heat kernels for a large class of operators that may or may not be group invariant, including certain degenerate elliptic and kinetic operators. Once again, we shall start with the Heisenberg group. Consider first the n-dimensional Hermite operator $$\Delta - {\lambda }^{2}\vert x{\vert }^{2} ={ \sum \nolimits }_{k=1}^{n} \frac{{\partial }^{2}} {\partial {x}_{k}^{2}} - {\lambda }^{2}{ \sum \nolimits }_{k=1}^{n}{x}_{ k}^{2}.$$ (13.1.1)

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.

Let $H$ be a reproducing kernel Hilbert space contained in a wider space $L^2(X,\mu)$. We study the Hilbert-Schmidt property of Hankel operators $H_g$ on $H$ with bounded symbol $g$ by analyzing the behavior of the iterated Berezin transform. We determine symbol classes $\mathcal{S}$ such that for $g\in \mathcal{S}$ the Hilbert-Schmidt property of $H_g$ implies that $H_{\bar{g}}$ is a Hilbert-Schmidt operator as well and there is a norm estimate of the form $\|H_{\bar{g}}\|_{\text{HS}}\leq C\cdot \| H_g\|_{\text{HS}}$. Finally, applications to the case of Bergman spaces over strictly pseudo convex domains in $\mathbb{C}^n$, the Fock space, the pluri-harmonic Fock space and spaces of holomorphic functions on a quadric are given.

We show that an operator defined on the quaternion projective space is a zeroth order selfadjoint pseudo-differential operator of Hormander class L 1,0 0 . This operator arises when we compare two quantization operators of the geodesic flow on the quaternion projective space. Such quantization operators are defined on a Hilbert space consisting of holomorphic functions and the Hilbert space has reproducing kernel. We describe the reproducing kernels in the cases of sphere and quaternion projective space in terms of hypergeometric functions, and discuss their relation through fiber integration with respect to the complexified Hopf fibration.

We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the "radius" of the fiber goes to zero. We explain the lines of the calculations precisely for three dimensional cases and state the corresponding results for five dimensional Heisenberg manifolds. We see that the values themselves are of the product form with a factor which is that of the flat torus. So in the last half of this paper we derive general formulas of the zeta-regularized determinant for product type manifolds of two Riemannian manifolds, discuss the formulas for flat tori and explain a relation of the formula for the two dimensional flat torus and Kronecker's second limit formula., 42 pages, no figures

We construct a Kahler structure on the punctured cotangent bundle of the Cayley projective plane whose Kahler form coincides with the natural symplectic form on the cotangent bundle and we show that the geodesic flow action is holomorphic and is expressed in a quite explicit form. We also give an embedding of the punctured cotangent bundle of the Cayley projective plane into the space of 8 × 8 complex matrices.

First we summarize two different concepts of Cauchy data (‘Hardy’) spaces of elliptic differential operators of first order on smooth compact manifolds with boundary: the L2-definition by the range of the pseudo-differential Calderon-Szego projection and the ‘natural’ definition by projecting the kernel into the (distributional) quotient of the maximal and the minimal domain. We explain the interrelation between the two definitions. Second we give various applications for the study of topological, differential, and spectral invariants of Dirac operators and families of Dirac operators on partitioned manifolds.

We consider direct sum decompositions $\beta=\beta_{-}+\beta_{+}$ and $L=L_{-}+L_{+}$ of two symplectic Hilbert spaces by Lagrangian subspaces with dense embeddings $\beta_{-}\hookrightarrow L-$ and $L_{+}\hookrightarrow\beta_{+}$. We show that such criss-cross embeddings induce a continuous mapping between the Fredholm Lagrangian Grassmannians $\mathcal{F}\mathcal{L}_{\beta_{-}}(\beta)$ and $\mathcal{F}\mathcal{L}_{L_{-}}(L)$ which preserves the Maslov index for curves. This gives a slight generalization and a new proof of the Yoshida-Nicolaescu Spectral Flow Formula for families of Dirac operators over partitioned manifolds.

We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators. We discuss continuous variations of the form $A_D+C_t$, where $A_D$ is a fixed self-adjoint unbounded Fredholm operator and $\{C_t\}$ a family of bounded self-adjoint operators. We extend the definition of the spectral flow to such families of unbounded operators in a purely functional analytical way. We then prove that the spectral flow is equal to the Maslov index of the corresponding family of abstract Cauchy data spaces.