QUANTUM EQUIVALENT MAGNETIC FIELDS THAT ARE NOT CLASSICALLY

EQUIVALENT

CHAMPS MAGNÉTIQUES QUANTIQUEMENT ÉQUIVALENTS MAIS CLASSIQUEMENT NON-ÉQUIVALENTS

CAROLYN GORDON, WILLIAM KIRWIN, DOROTHEE SCHUETH, AND DAVID WEBB

Dedicated to Pierre Bérard and Sylvestre Gallot on the occasion of their sixtieth birthdays.

Abstract. We construct pairs of compact Kähler-Einstein manifolds (M_{i}, g_{i}, ω_{i}) (i = 1, 2) of complex dimension n with the following properties: The canonical line bundle L_{i} = ⋀^{n} T^{∗}M_{i} has Chern class [ω_{i}/2π], and for each positive integer k the tensor powers L_{1}^{⊗k} and L_{2}^{⊗k} are isospectral for the bundle Laplacian associated with the canonical connection, while M_{1} and M_{2} – and hence T^{∗}M_{1} and T^{∗}M_{2} – are not homeomorphic. In the context of geometric quantization, we interpret these examples as magnetic fields which are quantum equivalent but not classically equivalent. Moreover, we construct many examples of line bundles L, pairs of potentials Q_{1}, Q_{2} on the base manifold, and pairs of connections ∇_{1}, ∇_{2} on L such that for each positive integer k the associated Schrödinger operators on L^{⊗k} are isospectral.

Résumé: On construit des couples de variétés de Kähler-Einstein compactes (M_{i}, g_{i}, ω_{i}) (i = 1, 2) de dimension complexe n avec les propriétés suivantes: La première classe de Chern associée au fibré en droites canonique L_{i} = ⋀^{n} T^{∗}M_{i} est ω_{i}/2π, et pour tout entier positif k, les puissances tensorielles L_{1}^{⊗k} et L_{2}^{⊗k} sont isospectrales pour le Laplacien associé à la connexion canonique, mais M_{1} et M_{2} – et, en conséquence, T^{∗}M_{1} et T^{∗}M_{2} – ne sont pas homéomorphes. Dans le contexte de la quantification géométrique, nous interprétons ces examples comme des champs magnétiques qui sont équivalents au sens quantique mais pas au sens classique. En plus, on construit beaucoup d’exemples de fibrés en droites L, de couples de potentiels Q_{1}, Q_{2} sur la variété de base et de couples de connexions ∇_{1}, ∇_{2} telles que pour tout entier positif k les opérateurs de Schrödinger associés sur L^{⊗k} soient isospectraux.

Contents

1. Introduction 2 2. Geometric quantization 3 2.1. Hamiltonian system associated with a magnetic field 3 2.2. Quantization of the Hamiltonian system 4 3. You can’t hear a magnetic field 8 3.1. The Sunada technique 8

2000 Mathematics Subject Classification. Primary 58J53; Secondary 53C20. The first author and last author were supported in part by NSF Grants DMS 0605247 and DMS 0906169. The

third author was partially supported by DFG Sonderforschungsbereich 647.

3.2. Construction of examples 10 4. Isospectral connections and potentials on a line bundle and its tensor powers 11 References 13

1. Introduction

Let L be a Hermitian line bundle over a closed Riemannian manifold (M, g). The Riemannian metric g on M and the connection ∇ on L together give rise to a Laplace operator Δ acting on the space C^{∞}(M, L) of smooth sections of L by

(1.1)  Δ = −trace(∇^{2}),

where

C^{∞}(M, L) ⟶^{∇} C^{∞}(T^{∗}M ⊗ L) ⟶^{∇} C^{∞}(T^{∗}M ⊗ T^{∗}M ⊗ L)

are the connections on L and on T^{∗}M ⊗ L (the latter is obtained from the Levi-Civita connection on T^{∗}M and the given connection ∇ on L; we denote it by ∇ as well) and the trace is with respect to the Riemannian metric g. The connection ∇ gives rise to a connection, and thus also a Laplacian, on the kth tensor power L^{⊗k} of L over M for each positive integer k. We will denote its spectrum, which is necessarily discrete, by Spec(L, ∇, k).

How much information is encoded in these spectra? For example, do they determine the connection? The curvature of the connection? The Chern class of the bundle? The geometry of the base manifold? We will primarily focus on a variant of the second question.

A closed 2-form ω on a Riemannian manifold (M, g) is sometimes viewed as a magnetic field. The classical Hamiltonian system for a charged particle moving in the magnetic field is given by (T^{∗}M, Ω, H). Here Ω is the symplectic structure on the phase space T^{∗}M given by Ω := ω_{0} + π^{∗}ω, where ω_{0} is the Liouville form and π : T^{∗}M → M is the projection, and the Hamiltonian H is given by H(q, ξ) = \frac{1}{2}g_{q}(ξ, ξ). If \frac{1}{2π}ω represents an integer cohomology class, then there exists a complex line bundle L with Chern class [\frac{1}{2π}ω]. Endow L with a Hermitian structure and a Hermitian connection with curvature −iω. Through the procedure of geometric quantization, the space of square integrable sections of L^{⊗k} is viewed as the “quantum Hilbert space,” and the quantum Hamiltonian is the operator Ĥ_{k} = −\frac{ℏ^{2}}{2} (−Δ − \frac{1}{6}R) with ℏ = \frac{1}{k}, where R is the scalar curvature of M. Thus we ask:

• Does the collection of all Spec(L, ∇, k), k = 1, 2, 3, …, determine the symplectic structure Ω on T^{∗}M? That is, does “quantum equivalence” of two magnetic fields imply their “classical equivalence”?

We answer this question negatively by example. We consider the case in which (M, g, ω) is a Kähler manifold; in fact, we focus on Hermitian locally symmetric spaces of noncompact type, normalized such that the Einstein constant is −1. For such spaces, the line bundle with Chern class [ω/2π] is the canonical line bundle of (M, g, ω). We will show that for every normalized, simply-connected irreducible Hermitian symmetric space X of noncompact type of real dimension at least four, there exist arbitrarily large finite families of Hermitian locally symmetric spaces (M_{i}, g_{i}, ω_{i}) covered by X such that Spec(L_{i}, ∇_{i}, k) = Spec(L_{j}, ∇_{j}, k) for all k and all i, j (where ∇_{i} is the canonical connection on the canonical line bundle) but such that the cotangent bundles of the various M_{i} are mutually non-homeomorphic. Hence, the phase spaces (T^{∗}M_{j}, Ω_{j}) for the magnetic flows of the various (M_{j}, ω_{j}) are not symplectomorphic, and yet the measurable quantum energy spectra are the same. Our method is based on Sunada’s isospectrality technique along with D. B. McReynolds’s recent construction of arbitrarily large finite families of mutually isospectral locally symmetric spaces.

In the example outlined above, the classical phase spaces of the “quantum equivalent” systems fail not only to be symplectomorphic, but even to be homeomorphic. In a companion article, we will construct by a different method an example of quantum equivalent magnetic fields on a fixed manifold M (a torus) for which the associated symplectic structures on T^{∗}M are not symplectomorphic.

Our technique is similar to that of R. Kuwabara [11], who constructed pairs of connections on a fixed line bundle L over, for example, a Riemann surface M such that Spec(L, ∇_{1}, k) = Spec(L, ∇_{2}, k) for all k. In the final section of this paper, we review and slightly extend his construction.

The paper is organized as follows: In Section 2, we describe some of the relevant framework of geometric quantization, which will allow for a physical interpretation of the isospectrality results. This material is of course well-known to experts in geometric quantization, but we include it here in the hopes that it may be of interest to a wider audience. In Section 3, we describe Sunada’s technique in our context and show how it leads to the examples described above of Hermitian locally symmetric spaces (of real dimension four and higher) that are quantum equivalent but not classically equivalent. We also address the case of Riemann surfaces. Finally, in Section 4, we consider isospectral connections and potentials on a fixed line bundle.

This article, like many others of the authors, was influenced by Pierre Bérard’s work. We are pleased to celebrate many years of friendship on the occasion of his birthday.

2. Geometric quantization

2.1. Hamiltonian system associated with a magnetic field.

On ℝ^{3}, a magnetic field may be viewed as an exact 2-form ω, identified with the curl of the magnetic potential field A. The 1-form α = A^{♭} defines a connection ∇ := d − iα on the (trivial) Hermitian line bundle ℝ^{3} × ℂ with curvature −iω = −i dα.^{1}

In analogy with the situation in ℝ^{3}, a closed 2-form ω on a Riemannian manifold (M, g) can be interpreted as a magnetic field. The Hamiltonian system for a charged particle moving in the magnetic field has phase space (T^{∗}M, Ω) with Ω := ω_{0} +π^{∗}ω, where ω_{0} is the Liouville form on T^{∗}M (that is, ω_{0} = −dλ, where λ is the canonical 1-form on the cotangent bundle), and π : T^{∗}M → M is the projection; see [10], for example. The classical trajectories of the particle are given by the Hamiltonian flow of the (kinetic energy) Hamiltonian H(q, ξ) := \frac{1}{2}g_{q}(ξ, ξ). When ω = 0, so that Ω = ω_{0}, this flow is just the usual geodesic flow describing a free particle moving on M.

1The appearance of i = \sqrt{−1} here is a matter of convention. We choose the convention which is common in mathematics, specifically in geometric quantization.

Notation 2.1. We will say that (M_{1}, g_{1}, ω_{1}) and (M_{2}, g_{2}, ω_{2}) are classically equivalent if the associated Hamiltonian systems (T^{∗}M_{1}, Ω_{1}, H_{1}) and (T^{∗}M_{2}, Ω_{2}, H_{2}) are equivalent; that is, if there exists a symplectomorphism φ : (T^{∗}M_{1}, Ω_{1}) → (T^{∗}M_{2}, Ω_{2}) such that φ^{∗}H_{2} = H_{1}.

Notation and Remarks 2.2. In case [ω/2π] is an integral cohomology class, let L be the line bundle over M with Chern class [ω/2π]. Endow L with a Hermitian structure and let P be the associated principal circle bundle. Let ∇^{L} be a Hermitian connection on L with curvature −iω. The connection ∇ and the Riemannian metric on M give rise to a Riemannian metric g̃ on P, sometimes called a Kaluza-Klein metric. Consider the associated geodesic flow on T^{∗}P. The circle action on the principal bundle P gives rise to a Hamiltonian action of the circle S^{1} on T^{∗}P. The symplectic reduction of the geodesic flow on P by the S^{1} action yields the Hamiltonian system of the magnetic flow on T^{∗}M described above. In this brief description we have followed Kuwabara; see [12] for more information.

In preparation for Subsection 2.2, we note that the connection ∇^{L} and the Riemannian metric g on M give rise to a Laplace operator Δ on the space C^{∞}(M, L) of smooth sections of L given by (1.1). By the usual construction, ∇^{L} induces a Laplace operator, also denoted Δ, on the space C^{∞}(M, L^{⊗k}), where L^{⊗k} is the kth tensor power of L for k = 1, 2, 3, . . .. The space C^{∞}(M, L^{⊗k}) may be identified with the space C_{k}^{∞}(P) of smooth complex-valued functions f on P satisfying the equivariance condition f(α.x) = α^{−k}f(x) for α ∈ S^{1} and x ∈ P. The Laplace operator on C^{∞}(M, L^{⊗k}) is unitarily equivalent to the restriction of Δ_{P} − 4k^{2}π^{2} to C_{k}^{∞}(P), where Δ_{P} is the Laplace-Beltrami operator of (P, g̃).

The space of smooth sections C^{∞}(M, L^{⊗k}) is endowed with the standard L^{2} inner product given for smooth sections s and t by

⟨s, t⟩ := (\frac{k}{2π} )^{n} ∫ s.t \frac{ω^{n}}{n!} ,

where s.t denotes the pointwise Hermitian inner product on each fibre. This inner product defines a Hilbert space consisting of square-integrable sections of L^{⊗k}, of which the space C^{∞}(M, L^{⊗k}) of smooth sections is a dense subspace. The operator Δ is an unbounded operator on this Hilbert space with dense domain C^{∞}(M, L^{⊗k}). The theory of unbounded operators on Hilbert spaces is well-developed (see, for example, the classic texts [16, Vol II, Chap. 8] and [4]), and we mention here only that Δ admits a self-adjoint extension, still denoted by Δ, with dense domain D containing the space of smooth sections of L^{⊗k}. In the following, when we say that Δ is an operator on the Hilbert space of L^{2}-sections, it is to be understood in this usual sense of an unbounded operator with dense domain.

2.2. Quantization of the Hamiltonian system.

Using geometric quantization, one associates to a classical mechanical system (satisfying suitable requirements) a quantum mechanical system, consisting of a Hilbert space H_{k} and a quantum Hamiltonian operator Ĥ_{k} : H_{k} → H_{k} for each integer k > 0. (Here Planck’s constant is given by ℏ = 1/k.) For the Hamiltonian system (T^{∗}M, Ω, H) in Subsection 2.1, the quantization may be carried out provided that ω/2π represents an integral cohomology class of M. In this case, one obtains Theorem 2.3 below. Following the statement of the theorem and related remarks, we will briefly outline the procedure of geometric quantization. For a complete presentation, see the classic references [21] and [17]; see also [9] and [2].

Theorem 2.3. [21, p. 204] We use Notation 2.2 and assume that [ω/2π] is an integral cohomology class. The quantum Hilbert space associated to the classical Hamiltonian system (T^{∗}M, Ω, H) of Subsection 2.1 is given for each integer k > 0 by H_{k} = L^{2}(M, L^{⊗k}) (the space of square-integrable sections of L^{⊗k}) and the quantization of the Hamiltonian H is the (unbounded) operator

(2.1)  Ĥ_{k} = −\frac{ℏ^{2}}{2} (−Δ − \frac{1}{6}R)

on H_{k}, where R is the scalar curvature of the metric g. (Here ℏ = \frac{1}{k}.)

The allowed energy values of the charged particle in the magnetic field, which are what one would see if one “measured” the energy of the quantum particle, are the eigenvalues of Ĥ_{k}.

Remark 2.4. The definition of the Laplacian Δ on L^{⊗k}, and thus of the operators Ĥ_{k}, depends on a choice of connection on L with the specified curvature −iω. However, in the examples that we will give in Subsection 3.2, there will be a natural choice of connection with that curvature.

Notation 2.5. Let (M_{i}, g_{i}), i = 1, 2, be a compact Riemannian manifold and let ω_{i} be a closed 2-form on M_{i} such that [ω_{i}/2π] is an integral cohomology class. For each integer k > 0, let Ĥ_{k}^{i} : L^{2}(M, L_{i}^{⊗k}) → L^{2}(M, L_{i}^{⊗k}) be the associated quantum Hamiltonian as given in Theorem 2.3. We will say that (M_{1}, ω_{1}) and (M_{2}, ω_{2}) are quantum equivalent (with respect to the connections used to define the line bundle Laplacians) if for every k, the operators Ĥ_{k}^{1} and Ĥ_{k}^{2} have the same spectrum.

We now outline the quantization procedure. Consider the classical Hamiltonian system (T^{∗}M, Ω, H) given in Subsection 2.1. Recall that Ω = ω_{0} + π^{∗}ω. Let π^{∗}L be the pullback of L to a bundle over T^{∗}M and π^{∗}∇^{L} the pullback of the connection. Since the Liouville form ω_{0} on T^{∗}M is exact, the Hermitian line bundle L_{Ω} with Chern class [Ω/2π] may be identified with π^{∗}L. Writing ω_{0} = dΘ, the Hermitian connection ∇ := π^{∗}∇^{L} − iΘ on L_{Ω} has curvature −iΩ.

The prequantization of the Hamiltonian system (T^{∗}M, Ω) is the space of square-integrable sections of L_{Ω}^{⊗k} with respect to the standard inner product

⟨s, t⟩ := (\frac{k}{2π} )^{n} ∫_{T^{∗}M} s.t \frac{Ω^{n}}{n!} ,

where s.t denotes the (pointwise) Hermitian product on L_{Ω}^{⊗k}. For a smooth function f on T^{∗}M (we are interested in particular in the Hamiltonian H above), one associates a prequantum Hamiltonian operator f̂^{preQ}, given by the Kostant-Souriau construction:

(2.2)  f̂^{preQ} := \frac{i}{k} ∇_{X_{f}}^{L_{Ω}^{⊗k}} + f,

where X_{f} is the Hamiltonian vector field associated to f, defined by Ω(X_{f}, ·) = df(·). The Kostant-Souriau prequantization (2.2) satisfies Dirac’s quantization conditions:

(1) the map f ↦ f̂^{preQ}is linear,

(2) the quantization ˆ1^{preQ} of the constant map 1 is the identity operator, and

(3) [f̂^{preQ}, ĝ^{preQ}] = −iℏ{ \atop \{f, g\}}^{preQ}, where \{·, ·\} is the Poisson bracket, [·, ·] is the operator commutator, and ℏ = 1/k.

Indeed, the prequantization (2.2) is derived precisely so that it satisfies (1) – (3) above. (See [21, Chap. 8]). Unfortunately, the pair (L^{2}(T^{∗}M, L_{Ω}^{⊗k}), f ↦ f̂^{preQ}) does not define a “good” quantization, essentially because L^{2}(T^{∗}M, L_{Ω}^{⊗k}) is too big. For example, in the case ω = 0 and M = ℝ^{n}, which corresponds to a free particle moving in Euclidean space, the line bundle L_{Ω}^{⊗k} is trivial and the prequantum Hilbert space is then L^{2}(T^{∗}ℝ^{n} = ℝ^{n} × ℝ^{n}). The variables in the first ℝ^{n} factor give the position of the particle, and the variables in the second ℝ^{n} factor describe the momentum. But one knows from quantum mechanics that a wave function cannot be simultaneously a function of both position and momentum.

In order to obtain a Hilbert space of the “correct” size, one first chooses a polarization of (T^{∗}M, Ω). A polarization of a symplectic manifold is an integrable (real or complex) Lagrangian distribution. If, as is the case here, the phase space is a cotangent bundle, one may take the vertical polarization, i.e., the distribution given by the tangent spaces to the fibers of T^{∗}M. (Note that this distribution is indeed Lagrangian with respect to Ω as well as ω_{0}.) This means that we are considering wave functions which depend only on position, not on momentum. Thus, the vertical polarization corresponds to the “position-space representation” in quantum mechanics.

Once a choice of polarization P is made, one would ideally like to define the quantum Hilbert space to be the subspace L_{P}^{2}(T^{∗}M, L_{Ω}^{⊗k}) of the prequantum Hilbert space L^{2}(T^{∗}M, L_{Ω}^{⊗k}) consisting of those sections that are covariantly constant in the P directions and then restrict the Kostant-Souriau prequantum Hamiltonians f̂^{preQ} to this subspace. However, there are two problems here. First, L_{P}^{2} = {0}! Secondly, even for a polarization P that yields a nontrivial quantum Hilbert space,^{2} the Kostant-Souriau quantization f̂^{preQ} does not in general preserve the quantum Hilbert space. Indeed, f̂^{preQ} will only preserve H_{k} if the Hamiltonian flow of f preserves the polarization P. One can show that the Ω-Hamiltonian flow of H does not preserve any polarization.

Fortunately, there is only one more piece of the puzzle which will remedy both of the remaining problems at once: the so-called half-form correction. The half-form correction boils down to tensoring L^{⊗k} with a square root of the canonical bundle associated to the polarization. Sections of such a bundle are called half-forms.

2A typical example of such a polarization is available whenever T^{∗}M admits a Kähler structure, for example when M is a compact Lie group. In this case, one can take P to be the holomorphic tangent bundle.

The half-form correction, due essentially to Blattner, Kostant and Sternberg, allows one to quantize a larger set of functions than just those whose flows preserve the polarization, and in particular one can quantize the standard Hamiltonians which appear in wave mechanics, of which our H = \frac{1}{2}∥ξ∥^{2} is an example. Moreover, the quantum Hilbert space associated to the vertical polarization, in the presence of the half-form correction, will turn out to be just L^{2}(M, L^{⊗k}), which is exactly what one would naively expect for the position-space representation.^{3}

The BKS construction in our setting is as follows. (We refer the interested reader to [21], Chap. 9, for more details and proofs; see also [9].) Choose a line bundle δ such that δ ⊗ δ = ⋀^{n} T M (this is possible because ⋀^{n} T M is trivializable), and let ν be a section of δ with ν^{2} = vol_{g}(M), where vol_{g}(M) is the Riemannian volume form on M. Sections of δ are called half-forms (associated to the vertical polarization), and the half-form corrected quantum Hilbert space is defined to be

Ĥ_{k} := L_{P}^{2}(T^{∗}M, L_{Ω}^{⊗k} ⊗ π^{∗}δ)

where the inner product is defined by the canonical pairing of half-forms. In particular, a section of L_{Ω} → T^{∗}M which is vertically covariantly constant is uniquely determined by its value on the zero-section M, and the inner product of two such sections is therefore given by^{4}

(2.3)  ⟨sν, tν⟩ = (\frac{k}{2π} )^{n} ∫_{M} s.t vol_{g}(M).

Hence, we see that the quantum Hilbert space associated to the vertical polarization can be identified with L^{2}(M, L^{⊗k}).

Now that we have the correct quantum Hilbert space, we need to quantize the Hamiltonian flow of the kinetic energy H. Let ρ_{t} denote the Hamiltonian flow of H on T^{∗}M. In order to define the quantization of the Hamiltonian H, we evolve ψν for a short time (that is, apply exp(−iktĤ^{preQ}) to the first factor, and the pull-back ρ_{t}^{∗} to the second factor), and then project the result back into Ĥ_{k}.

The projection is achieved by a generalization of the half-form pairing (2.3). One can show that the pushforward of the vertical polarization by ρ_{t} is an integrable Lagrangian distribution which is (at least for small t) transverse to the vertical polarization. Hence, there exists some function f_{t} ∈ C^{∞}(T^{∗}M) such that ρ_{t}^{∗}(vol_{g}(M)) ∧ vol_{g}(M) = f_{t}Ω^{n}/n!. The generalized (BKS) half-form pairing is then defined to be

3There are several further advantages, from both the mathematical and physical viewpoints, though they are not relevant to our current purposes. One, which is easy to describe, is that when using the BKS construction to quantize the simple harmonic oscillator (a well-known example from physics), a shift is introduced which results in the physically correct energy spectrum. Specifically, without the BKS construction, one obtains an energy spectrum consisting of integer multiples of ℏ. The physically correct spectrum, which is obtained using the BKS construction, is \{(n + \frac{1}{2})ℏ : n ∈ ℕ_{0}\}.

4We will abuse notation slightly and not distinguish between ν (or vol_{g}(M)) and its pullback π^{∗}ν (resp.

π^{∗} vol_{g}(M)).

(ρ_{t}^{∗}ν) .ν := \sqrt{f_{t}}.

This pairing can be shown to be nondegenerate (at least for small t), and therefore defines a bijection between (exp(−ikt Ĥ^{preQ}) ⊗ ρ_{t}^{∗})Ĥ_{k} and Ĥ_{k}. The quantum Hamiltonian Q̂_{k}(H) is obtained by computing the derivative with respect to t, evaluated at t = 0, of the operator on Ĥ_{k} given by first applying (exp(−iktĤ^{preQ}) ⊗ ρ_{t}^{∗}) and then projecting the result back into Ĥ_{k} using the BKS pairing. At the end of the day, we are really interested only in sections of L^{⊗k}; thus, we can take a section ψ in H_{k}, multiply it by ν, apply the BKS construction, and write the result in the form ψ′ν. The quantization of the Hamiltonian H is then defined to be Ĥ_{k}ψ := ψ′. In our case, this yields the expression in Theorem 2.3.

3. You can’t hear a magnetic field

3.1. The Sunada technique. We will use a variant of Sunada’s technique [18].

Definition 3.1. Let G be a finite group and let Γ_{1} and Γ_{2} be subgroups of G. We will say that Γ_{1} is almost conjugate to Γ_{2} in G if there is a bijection Γ_{1} → Γ_{2} carrying each element of Γ_{1} to a conjugate element in Γ_{2}; equivalently, each G-conjugacy class [g]_{G} intersects Γ_{1} and Γ_{2} in the same number of elements.

Sunada’s Theorem states that if a finite group G acts by isometries on a compact Riemannian manifold M and if Γ_{1} and Γ_{2} are almost conjugate subgroups of G acting freely on M, then Γ_{1}\M and Γ_{2}\M are isospectral.

Remarks 3.2.

(1) The almost conjugacy condition is equivalent to a representation theoretic condition as follows. The right multiplication of G on the cosets in Γ_{i}\G gives rise to a natural action of G on the finite-dimensional vector space R[Γ_{i}\G]. The subgroups Γ_{1} and Γ_{2} of G are almost conjugate if and only if there exists an isomorphism

τ : R[Γ_{1}\G] → R[Γ_{2}\G]

intertwining the actions of G.

(2) Assume that Γ_{1} and Γ_{2} are almost conjugate in G and let τ be the intertwining map in (1). Let W be any vector space on which G acts on the right. For i = 1, 2, let W^{Γ_{i}} be the subspace of vectors fixed by all elements of Γ_{i}. Then τ gives rise to a linear isomorphism, called “transplantation”

T : W^{Γ_{2}} → W^{Γ_{1}}.

Transplantation was first introduced in an example in [7] and systematized in [3] to give a new proof of Sunada’s Theorem; see also [22]. We are following the presentation in [8].

(3) Transplantation is functorial: if V and W are right G-spaces and ψ : V → W is a G-equivariant map, then the following diagram commutes:

W^{Γ_{2}} ⟶^{T_{W}} W^{Γ_{1}}
ψ↓  ↓^{ψ}
V^{Γ_{2}} ⟶^{T_{V}} V^{Γ_{1}}

Moreover, if W is an inner product space and if the G action is unitary, then the transplantation map is unitary.

Notation 3.3. Given a Hermitian line bundle L over a closed Riemannian manifold (M, g) and a Hermitian connection ∇ on L we denote by Spec(L, ∇, k) the spectrum of the associated Laplace operator Δ on C^{∞}(M, L^{⊗k}) (recall Notation and Remarks 2.2). For a potential Q ∈ C^{∞}(M), we denote by Spec(Q; L, ∇, k) the spectrum of Δ + Q on C^{∞}(M, L^{⊗k}).

Proposition 3.4. Let (M, g) be a compact Riemannian manifold, let L be a Hermitian line bundle over M, and let ∇ be a Hermitian connection on L. Let G be a finite group that acts on L carrying fibers to fibers, preserving ∇, and such that the induced action on M is by isometries. For i = 1, 2, suppose that Γ_{i} is a subgroup of G whose action on M is free. Thus L_{i} := Γ_{i}\L, i = 1, 2 is a Hermitian line bundle over M_{i} := Γ_{i}\M, and ∇ induces a connection ∇_{i} on L_{i}. If Γ_{1} and Γ_{2} are almost conjugate in G, then:

(i)

Spec(L_{1}, ∇_{1}, k) = Spec(L_{2}, ∇_{2}, k)

for all positive integers k. (ii) If, moreover, Q ∈ C^{∞}(M) is a G-invariant function, then

Spec(Q; L_{1}, ∇_{1}, k) = Spec(Q; L_{2}, ∇_{2}, k)

for all positive integers k, where we use the same notation Q for the smooth potentials on M_{1} and M_{2} induced by the potential Q on M.

This variant of Sunada’s Theorem is essentially contained in R. Kuwabara [11], although his interest was in pairs of connections on the same underlying bundle and in the case Q = 0.

For a proof by transplantation, observe that G acts on the right on the space C^{∞}(M, L^{⊗k}) of smooth sections of L^{⊗k} by (f.g)(x) = g^{−1}.f(g.x) for f ∈ C^{∞}(M, L^{⊗k}), g ∈ G, and x ∈ M. The space C^{∞}(M_{i}, L_{i}^{⊗k}) of smooth sections of L_{i}^{⊗k} may be identified with the space C^{∞}(M, L^{⊗k})^{Γ_{i}} of Γ_{i}-invariant elements of C^{∞}(M, L^{⊗k}). Thus by Remark 3.2, we obtain a transplantation map T : C^{∞}(M_{2}, L_{2}^{⊗k}) → C^{∞}(M_{1}, L_{1}^{⊗k}). Moreover, with this identification, the Schrödinger operator Δ_{i} + Q on C^{∞}(M_{i}, L_{i}^{⊗k}) (associated with the Riemannian metric on M_{i}, the connection ∇_{i}, and the potential Q) is the restriction to C^{∞}(M, L^{⊗k})^{Γ_{i}} of the Schrödinger operator Δ + Q of L^{⊗k}. Since Δ commutes with the action of G and since Q is G-invariant, we may let Δ + Q play the role of ψ in Remark 3.2. It follows that T intertwines the Schrödinger operators Δ_{1} + Q and Δ_{2} + Q on L_{1}^{⊗k} and L_{2}^{⊗k}, thus proving the theorem.

Theorem 3.5. We use the notation and hypotheses of Proposition 3.4, part (i). Let −iω_{j} be the curvature of the connection ∇_{j} on L_{j}, j = 1, 2. Then in the language of Notation 2.5 and Remark 2.4, (M_{1}, ω_{1}) and (M_{2}, ω_{2}) are quantum equivalent with respect to the connections ∇_{1} and ∇_{2}.

Proof. We apply part (ii) of Proposition 3.4 with the scalar curvature R of M in the role of Q. Note that R is necessarily G-invariant since G acts by isometries on M. □

3.2. Construction of examples.

Let (M, g, ω) be a Kähler manifold of complex dimension n. (Here ω is the Kähler form.) The canonical line bundle L_{M} over M is defined to be the nth exterior power of the holomorphic cotangent bundle. Since M is Kähler, the Levi-Civita connection on T M commutes with the complex structure and thus defines a holomorphic connection on the holomorphic tangent bundle. This connection gives rise to a holomorphic connection on L_{M} that we will call the canonical connection.

If X is a simply-connected Hermitian symmetric space of non-compact type and M is a compact locally symmetric space with universal covering X, we will call M an X-space. Every X-space M is a Hodge manifold, i.e., M is a Kähler manifold and a suitable real multiple of the Kähler form ω of M represents an integer cohomology class. More precisely, if the metric is rescaled such that X (and hence each X-space M) has Einstein constant −1 then the Chern class of the canonical bundle L_{M} is [ω/2π] (see [1], formulas (4.68) and (4.59); compare also [20], p. 219.) As in Remark 2.4, the notion of “quantum equivalence” of (M_{1}, ω_{1}) and (M_{2}, ω_{2}), where the M_{i} are X-spaces and ω_{i} their Kähler forms, will mean with respect to the canonical connections on the canonical bundles L_{M_{i}}.

Theorem 3.6. Let X be a simply-connected Hermitian symmetric space of noncompact type of real dimension at least four. Then there exist arbitrarily large families of non-isometric X-spaces M_{i} such that the (M_{i}, ω_{i}) are all mutually quantum equivalent but not classically equivalent. In fact the phase spaces (T^{∗}M_{i}, Ω_{i}) of the classical Hamiltonian systems are not symplectomorphic (or even homeomorphic).

Proof. D.B. McReynolds [13] showed, using Sunada’s Theorem, that for every simply-con-nected symmetric space X of non-compact type, there exist arbitrarily large collections of non-isometric X-spaces M_{i} whose Laplace-Beltrami operators are mutually isospectral. For each such collection, there exists an X-space M and a finite group G of isometries of M such that M_{i} = Γ_{i}\M, where the Γ_{i} are almost conjugate subgroups of G. In the setting that X is Hermitian symmetric, the isometries are holomorphic. Since all holomorphic isometries of M preserve both the canonical bundle and the canonical connection, we can now apply Theorem 3.5 to see that (M_{i}, ω_{i}) and (M_{j}, ω_{j}) are quantum equivalent for all i, j.

Mostow Strong Rigidity tells us that the various M_{i} have non-isomorphic fundamental groups, i.e., M_{i} = Λ_{i}\X with Λ_{i} and Λ_{j} non-isomorphic discrete uniform subgroups of the group of isometries of X when i ≠ j. The cotangent bundle T^{∗}M_{i} is the quotient of the (trivial) bundle T^{∗}X by the action of Λ_{i} and thus the various T^{∗}M_{i} are also nonhomeomorphic. □

The assumption on the dimension of X in Theorem 3.6, equivalently the exclusion of the case that X is the real hyperbolic plane, was needed only so that the phase spaces for the classical Hamiltonian systems would not be homeomorphic. In fact, we have the following:

Proposition 3.7. Let (M_{1}, g_{1}) and (M_{2}, g_{2}) be any pair of hyperbolic Riemann surfaces which are isospectral with respect to the Laplace-Beltrami operator acting on functions. Then (M_{1}, ω_{1}) and (M_{2}, ω_{2}) are quantum equivalent, where ω_{i} is the Kähler form of (M_{i}, g_{i}).

We emphasize that, in contrast to the Hermitian locally symmetric spaces in Theorem 3.6, the Riemann surfaces are not required to satisfy the conditions of Sunada’s Theorem.

Proof. H. Pesce [14] proved that every pair of isospectral compact Riemann surfaces (M_{1}, g_{1}) and (M_{2}, g_{2}) is strongly isospectral in the following sense: Let G = P SL(2, ℝ). A Hermitian vector bundle E over the hyperbolic plane X is said to be homogeneous if G acts on E, carrying fibers to fibers, such that the induced action on X is the standard action by isometries. The actions of G on X and E give rise to an action of G on the space of smooth sections of E. A self-adjoint elliptic differential operator D on E (i.e., on smooth sections of E) is said to be natural if it commutes with the G-action. In that case, if Γ is a discrete subgroup of G acting freely and properly discontinuously on X, then D induces a self-adjoint elliptic differential operator on the bundle Γ\E over the Riemann surface Γ\X. Compact Riemann surfaces M_{1} = Γ_{1}\X and M_{2} = Γ_{2}\X are said to be strongly isospectral if for each homogeneous Hermitian vector bundle E over X and each natural self-adjoint elliptic operator D on E, the induced operators on the bundles Γ_{1}\E over M_{1} and Γ_{2}\E over M_{2} are isospectral. (Aside: The key point in proving that isospectral compact Riemann surfaces M_{1} = Γ_{1}\X and M_{2} = Γ_{2}\X are always strongly isospectral is that isospectrality of the Riemann surfaces implies that the representations of G induced by the trivial representations of Γ_{1} and Γ_{2} are equivalent. This condition is considerably weaker than the Sunada condition, which requires that Γ_{1} and Γ_{2} be subgroups of some finite subgroup Γ of G and that the trivial representations of Γ_{1} and Γ_{2} induce equivalent representations of Γ.)

The proposition follows from the fact that the canonical Hermitian line bundle over X and all its tensor powers are homogeneous, and the Laplacian associated with the canonical connection is natural. □

Remark 3.8. Using Sunada’s technique, R. Brooks, R. Gornet, and W. Gustafson [6] constructed arbitrarily large finite families of mutually isospectral, non-isometric Riemann surfaces. (Their work motivated that of D.B. McReynolds cited above.) While the vast majority of known isospectral Riemann surfaces were constructed by Sunada’s technique, M.- F. Vignéras’s examples [19] and recent examples of C.S. Rajan [15] do not satisfy the Sunada condition.

4. Isospectral connections and potentials on a line bundle and its tensor powers

Using a trick introduced by R. Brooks [5], we can use Proposition 3.4 to obtain isospectral connections and potentials on a single line bundle and its tensor powers.

Corollary 4.1. In addition to the hypotheses of Proposition 3.4, assume that there exists a bundle map σ of L, projecting to an isometry (also to be denoted σ) of M, such that σ normalizes G and such that σΓ_{1}σ^{−1} = Γ_{2}. Continue to denote by σ the induced bundle map from L_{1}^{⊗k} to L_{2}^{⊗k}. Then

Spec(Q; L_{1}, ∇_{1}, k) = Spec(σ^{∗}Q; L_{1}, σ^{∗}∇_{2}, k)

for all positive integers k.

This corollary is contained in Kuwabara [11] for the case Q = 0.

Remark 4.2. One may choose ∇ to be σ-invariant as well as G-invariant, in which case ∇_{1} = σ^{∗}∇_{2}. We can then conclude that Q_{1} and σ^{∗}Q_{2} are isospectral potentials for the Schrödinger operator −Δ_{1}+potential.

We now explain how to use the corollary to obtain examples in which the base manifolds are Riemann surfaces. Brooks [5] gave explicit examples of finite groups G and Riemann surfaces (M, g) (with a hyperbolic Riemannian metric g) such that the following conditions are satisfied:

(i) The group G acts freely by orientation preserving isometries on the oriented Riemann surface (M, g).

(ii) There exists a pair of almost conjugate, nonconjugate subgroups Γ_{1}, Γ_{2} of G.

(iii) There exists an outer automorphism τ of G such that Γ_{2} = τΓ_{1}τ^{−1} and such that the action of G extends to a free action of the semi-direct product Ĝ of G and ⟨τ⟩ on (M, g) by orientation-preserving isometries.

Using these objects we obtain the following class of examples.

Example 4.3. We choose (M, g), G, Γ_{1}, Γ_{2}, τ, Ĝ as above and consider the Hermitian line bundle L_{N} over N := Ĝ\M. Denote its pullback to M by L. The group Ĝ acts on L by vector bundle isomorphisms. We choose a Ĝ-invariant Hermitian connection ˆ∇ on L by pulling back a Hermitian connection from L_{N}, and we choose a function f ∈ C^{∞}(M) which is G-invariant but not τ-invariant. Denoting the Riemannian volume form on M by ω, we let ∇ := ˆ∇ + i d^{∗}(fω). Note that ∇ is G-invariant, but not τ-invariant. Moreover, we choose any G-invariant potential Q ∈ C^{∞}(M). Finally, we let σ denote the vector bundle isomorphism of L induced by τ. Applying Proposition 3.4 together with Corollary 4.1 we obtain, for the vector bundle L_{1} := Γ_{1}\L over M_{1} := Γ_{1}\M and the induced connections ∇_{1} on L_{1}, resp. ∇_{2} on L_{2} := Γ_{2}\L:

Spec(Q; L_{1}, ∇_{1}, k) = Spec(σ^{∗}Q; L_{1}, σ^{∗}∇_{2}, k)

for each positive integer k.

Remark 4.4. (i) The choice of ∇ in the previous example guarantees that the resulting pairs of isospectral connections ∇_{1} and σ^{∗}∇_{2} have different curvature. In fact, the pullbacks to L of the connections ∇_{1} and σ^{∗}∇_{2} on L_{1} are ˆ∇ + i d^{∗}(fω) and ˆ∇ + i d^{∗}((τ^{∗}f)ω), respectively. The pullback to M of the difference of the corresponding curvature forms on M_{1} is given by i dd^{∗}((f − τ^{∗}f)ω). The 2-form (f − τ^{∗}f)ω has integral zero over M and is thus exact by Poincaré duality. On the other hand, this form is nonzero by our choice of f, and hence nonharmonic. This immediately implies that dd^{∗}((f − τ^{∗}f)ω) ≠ 0, as claimed.

(ii) Let τ̃ denote some lift of τ to the hyperbolic plane H_{2}, and let G̃, Γ̃_{i} denote the groups of all lifts of elements of G, resp. Γ_{i}, to H_{2}. Let N(Γ̃_{1}) denote the normalizer of Γ̃_{1} within Isom(H_{2}). Then τ̃ ∉ G̃φ̃ for any φ̃ ∈ N(Γ̃_{1}) because, otherwise, the relation τ̃Γ̃_{1}τ̃^{−1} = Γ̃_{2} would imply that Γ_{1} and Γ_{2} were conjugate in G. Note that N(Γ̃_{1}) consists precisely of the lifts of isometries of M_{1}. Therefore the fact that τ̃ ∉ G̃φ̃ for all φ̃ ∈ N(Γ̃_{1}) implies that it is possible to choose the G-invariant function f subject to the slightly stronger property that the functions f_{1} and f_{1}^{τ} which are induced by f and τ^{∗}f on M_{1}, respectively, do not differ by any isometry of M_{1}. Then, for any isometry φ of M_{1} we can apply the argument of (i) to the lift of f_{1} − φ^{∗}f_{1}^{τ} to M and conclude that now the curvature forms associated with ∇_{1} and σ^{∗}∇_{2} are not related by pullback by any isometry of M_{1}.

References

[1] W. Ballmann. Lectures on Kähler Manifolds. EMS, Zürich, 2006.

[2] S. Bates and A. Weinstein. Lectures on the Geometry of Quantization. Berkeley Mathematics Lecture Notes, 8. American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1997.

[3] P. Bérard. Transplantation et isospectralité I. Math. Ann. 292 (1992), no. 3, 547–559.

[4] F. Berezin and M. Shubin. The Schrödinger Equation. Mathematics and its Applications (Soviet Series), 66. Kluwer Academic Publishers Group, Dordrecht, 1991.

[5] R. Brooks. On manifolds of negative curvature with isospectral potentials. Topology 26 (1987), 63–66.

[6] R. Brooks, R. Gornet, and W. Gustafson. Mutually isospectral Riemann surfaces. Adv. Math. 138 (1998), 306–322.

[7] P. Buser. Isospectral Riemann surfaces. Ann. Inst. Fourier, 36 (1986), no. 2, 167–192.

[8] C. Gordon, E. Makover, and D. Webb. Transplantation and Jacobians of Sunada isospectral Riemann surfaces. Adv. Math. 197 (2005), 86–119.

[9] V. Guillemin and S. Sternberg. Geometric Asymptotics. Mathematical Surveys, No. 14. American Mathematical Society, Providence, R.I., 1977.

[10] V. Guillemin and S. Sternberg. Symplectic Techniques in Physics. Cambridge University Press, Cambridge, 1990.

[11] R. Kuwabara. Isospectral connections on line bundles. Math. Z. 204 (1990), 465–473.

[12] R. Kuwabara. Spectral geometry for Schrödinger operators in a magnetic field focusing on geometrical and dynamical structures on manifolds. In: Selected papers on analysis and differential equations, Amer. Math. Soc. Transl. Ser. 2 211 (2003), 25–46.

[13] D.B. MacReynolds. Isospectral locally symmetric manifolds. Preprint, arXiv:math/0606540v2.

[14] H. Pesce. Variétés hyperboliques et elliptiques fortement isospectrales. J. Funct. Anal. 134 (1995), no. 2, 363–391.

[15] C. S. Rajan. On isospectral arithmetic spaces. Amer. J. Math. 129 (2007), 791–806.

[16] M. Reed and B. Simon. Methods of Modern Mathematical Physics, vols. I-IV. Second edition. Academic Press, Inc., New York, 1980.

[17] J. Śniatycki. Geometric Quantization and Quantum Mechanics. Springer, New York, 1980.

[18] T. Sunada. Riemannian coverings and isospectral manifolds. Ann. Math. 121, 169—186.

[19] M.-F. Vignéras. Variétés Riemanniennes isospectrales et non isométriques. Ann. Math. 121 (1980), 21–32.

[20] R. Wells. Differential Analysis on Complex Manifolds. Springer, New York, 1980.

[21] N.M.J. Woodhouse. Geometric Quantization, second Edition. Oxford University Press, Inc., New York, 1991.

[22] S. Zelditch, Isospectrality in the FIO category. J. Differential Geom. 35 (1992), no. 3, 689–710.

Department of Mathematics, Dartmouth College, Hanover, NH 03755

E-mail address: csgordon@dartmouth.edu

CAMGSD, Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal *Current Address: Mathematisches Institut, University of Cologne, Weyertal 86 - 90, D-50931 Cologne, Germany

E-mail address: will.kirwin@gmail.com

Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany

E-mail address: schueth@math.hu-berlin.de

Department of Mathematics, Dartmouth College, Hanover, NH 03755

E-mail address: david.l.webb@dartmouth.edu