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Oscillator representations and systems of ordinary differential equations

Edited by Charles L. Fefferman, Princeton University, Princeton, NJ, and approved October 18, 2000 (received for review August 17, 2000)
Abstract
Using representationtheoretic methods, we determine the spectrum of the 2 × 2 system with A, B ∈ Mat_{2}(ℝ) constant matrices such that A = ^{t}A > 0 (or <0), B = −^{t}B ≠ 0, and the Hermitian matrix A + iB positive (or negative) definite. We also give results that generalize (in a possible direction) the main construction.
1. Introduction
It is a natural and important problem to find an efficient and invariant way of studying the spectrum of systems of differential equations. We deal first with the spectral problem of particular systems that are the Weyl quantization of noncommutative quadratic form (that we shall call noncommutative harmonic oscillators) of the kind where (x, ξ) ∈ ℝ^{2} ≃ T*ℝ, A, B ∈ Mat_{2}(ℝ) are constant 2 × 2matrices with A = ^{t}A definite, either positive or negative, B = −^{t}B ≠ 0, and A + iB > 0 (or < 0). The Weyl quantization of the above noncommutative quadratic form is Q(x, D_{x}): 𝒮(ℝ; ℂ^{2}) → 𝒮(ℝ; ℂ^{2}), Q(x, D_{x}): 𝒮′(ℝ; ℂ^{2}) → 𝒮′(ℝ; ℂ^{2}) continuously, and it is important to remark that in the above hypotheses the system Q(x, D_{x}), as an unbounded operator in L^{2}(ℝ; ℂ^{2}) (with maximal domain B^{2}(ℝ; ℂ^{2}) := {u ∈ L^{2}(ℝ; ℂ^{2}); x^{α}∂_{x}^{β} u ∈ L^{2}(ℝ; ℂ^{2}), ∀α, β, 0 ≤ α + β ≤ 2}) is selfadjoint with discrete spectrum made of eigenvalues μ_{k} ∈ ℝ (with finite multiplicities) such that μ_{k} → +∞ as k → +∞. The main problem in analyzing the spectrum of such systems comes from two sources: the noncommutativity of matrices and the noncommutativity of the quantized variables x and ξ.
Our aim is in the first place to find a systematic way of solving algebraically in L^{2}(ℝ; ℂ^{2}) (that is, taking into account L^{2}convergence or, at most, 𝒮′convergence) the above kind of systems. The operator Q(x, D_{x}) naturally possesses an sl_{2}(ℝ)action (and more generally a metaplectic group action), due to the Weyl quantization. Hence, the algebraic viewpoint consists of using this sl_{2}(ℝ)symmetry and the symmetries carried by A and B to develop a method that determines, as a first step, a candidate for an eigenvalue along with the formal sequence of coefficients of a potential eigenfunction belonging to that eigenvalue (in the sense that the eigenfunction has that sequence as coefficients with respect to some Schwartz L^{2}basis obtained by means of the oscillator representation of sl_{2}(ℝ); this may be thought of as a necessary condition). Then the second step consists of determining which candidate eigenvalues can be actual ones, by studying the convergence properties (in L^{2} or 𝒮′) of the related formal sequences of coefficients (see ref. 1). In Section 2, we shall give a survey of the results of refs. 2 and 3 described above. As it will be seen below, the key step is to obtain a threeterm recurrence system that can be “diagonalized.” In Section 3, we shall next consider a different system of ordinary differential operators and treat the relative spectral problem by employing the tensorproduct of the oscillator representation and the vector (standard) representation of sl_{2}(ℝ). That system (which is also interesting, for the most general noncommutative harmonic oscillator is given by A(−∂_{x}^{2}/2) + B(x∂_{x} + 1/2) + C(x^{2}/2) + D, with A = ^{t}A, B = −^{t}B, C = ^{t}C and D = ^{t}D) might look more difficult at first glance than the ones considered in Section 2, because of the presence of a zerothorder nonconstant matrixvalued term. However, the difficulty is only “virtual,” for the use of the aforementioned tensor product representation of sl_{2}(ℝ) allows one to treat the eigenvalue problem in a completely straightforward way: the above system is just a pair of “harmonic oscillators” in disguise. We shall also exhibit a family of systems parametrized by ɛ ∈ ℝ, which “interpolates” the pair of “harmonic oscillators” and a system of the above type. This family possesses the remarkable property that all the eigenvalues have multiplicity 1, provided ɛ ∉ (1/2)ℤ. This approach will be generalized in ref. 4 to treat more general 2 × 2systems and higherrank cases in a unified way.
Applications of the study of the spectrum of N × N noncommutative harmonic oscillators are in the field of lower bounds and hypoellipticity of systems of pseudodifferential operators (see refs. 5–10).
We finally remark that the spectral problem for Q(x, D_{x}) can be translated into a family of Fuchsian type thirdorder equations with four regular singularities, in the complex unit disk (see ref. 11). We believe that the results on the multiplicity of the spectrum of Q(x, D_{x}) may provide a crucial information to determine the monodromy of the ordinary differential equation discussed in ref. 11.
2. Study of the Spectrum of Q
The condition that the Hermitian matrix A + iB be definite, either positive or negative, is equivalent to requiring that det A > (pf(B))^{2}, and also equivalent to requiring that the operator Q(x, D_{x}) be elliptic, i.e., det Q(x, ξ) ≠ 0 for (x, ξ) ≠ (0, 0), det Q(x, ξ) being positively homogeneous of degree 4 in (x, ξ). Hence, Q(x, D_{x}), as an unbounded operator with domain (maximal domain) for any given ϕ ∈ 𝒮(ℝ; ℂ^{2}), has a discrete spectrum made of real eigenvalues with finite multiplicities, diverging (in absolute value) to +∞.
Define Since B = ±pf(B)J, and by the commutativity of J with SO(2), one can easily reduce the study of Q to that of where, after possibly conjugating by K, we may assume α, β > 0. We may thus suppose A = I(α, β), B = J. Set ℓ = > 0, and define 1 Since [Ψ(ℓ), Ψ^{†}(ℓ)] = I, putting yields that H, X^{+}, X^{−} satisfy the commutationrelations of sl_{2}(ℝ) 2 This gives in fact the tensor product of the oscillator representation and the twodimensional trivial representation of sl_{2}(ℝ). For v ∈ ℂ^{2}∖{0}, set Since X^{−} annihilates ξ_{0}(v) [ξ_{0}(v) is a lowestweight vector], ξ_{N}(v) is a weight vector of H with weight N + 1/2, and it also follows that whenever v and w are nonzero vectors such that 〈v, w〉_{ℂ2} = 0, {ξ_{N}(v), ξ_{N}(w)}_{N∈ℤ+} is an orthogonal basis of L^{2}(ℝ; ℂ^{2}). The operator Q_{(α,β)} is unitarily equivalent to (ℓ/)Q̃, where Q̃ := A^{1/2}HA^{1/2}. The problem is therefore to understand the structure of the spectrum of Q̃. There are now (at least) two ways of studying the spectral problem for Q̃: (1) studying the equivalent problem of finding λ and η such that (H − λA^{−1})η = 0 (the twisted eigenvalue problem); and (2) studying directly the eigenvalue problem (Q̃ − λI)η = 0. In both cases, the key point is to get an appropriate recurrence formula that allows one to control the coefficients of the eigenfunctions. (One can easily see that an expansion in terms of the usual Hermite functions is not “convenient” for this purpose.) As we shall explain below, we get a system of recurrence equations that, by suitable rotations, can be diagonalized into a scalar threeterm recurrence equation (this is highly nontrivial, for we do not know a priori that that is possible). We remark that the crucial point is the choice of the basis to be used in the discussion. Since A^{−1} = σ_{+}I + σ_{−}KJ, A^{1/2} = μ_{+}I + μ_{−}KJ, in both problems 1 and 2 the action of the operator K appears. Since K does not commute with H (whereas J does), what is missing here is the explicit formula for the action of K on the ξ_{N}(v) in terms of the ξ. Set ξ_{N}^{j} = ξ_{N}(e_{j}), j = 1, 2, where {e_{1}, e_{2}} is the canonical basis of ℂ^{2}, and, for v_{+} := [_{−i}^{ 1}], define ξ_{N}^{+} := ξ_{N}(v_{+}).
Lemma 2.1. The set B_{mix} := {ξ_{N}^{+}, Kξ_{N}^{+}}_{N∈ℤ+}is an orthogonal basis of L^{2}(ℝ; ℂ^{2}).
As the system preserves parity, we shall consider hereafter only the even case.
Dealing with Problem 1.
Since the action of H on the Kξ_{N} involves both the ξ and Kξ, we use both bases B_{can}^{+} := {ξ_{2N}^{1}, ξ_{2N}^{2}}_{N∈ℤ+} and B_{can}^{+,K} := {Kξ_{2N}^{1}, Kξ_{2N}^{2}}_{N∈ℤ+} at the same time, and the fact that they are related by K. The latter means that if ({a_{N}}_{N∈ℤ+}, {b_{N}}_{N∈ℤ+}) are the coordinates of a solution η to the twisted eigenvalue problem with respect to the basis B_{can}^{+}, and ({c_{N}}_{N∈ℤ+}, {d_{N}}_{N∈ℤ+}) those of η with respect to the basis B_{can}^{+,K}, then there exists an involutive linear function Using K̃, we get a nonconstant matrixcoefficient system of recurrence equations in ℂ^{4} for the (a_{N}, b_{N}, c_{N}, d_{N}) that can be reduced to two recurrence systems for (a_{N}, b_{N}) and (c_{N}, d_{N}) plus a linear relation between (a_{N}, b_{N}) and (c_{N}, d_{N}). At this point there exists an explicitly known constant matrix M_{0}(ℓ) such that (1/) M_{0}(ℓ) ∈ SO(2), where we arrive at the recurrence equations where N ∈ ℤ_{+}, v_{−1} = w_{−1} = 0 and The key point now is the compatibility condition: Suppose λ gives rise to a solution ({a_{N}}_{N∈ℤ+}, {b_{N}}_{N∈ℤ+}) to (P1_{N}^{λ}) that defines a function u_{1} := ∑_{N=0}^{+∞} (a_{N}ξ_{2N}^{1} + b_{N}ξ_{2N}^{2}) ∈ L^{2}(ℝ; ℂ^{2}), then with ({a_{N}}_{N∈ℤ+}, {b_{N}}_{N∈ℤ+}) we associate, in an algebraic fashion through (P3_{N}^{λ}), a solution ({c_{N}}_{N∈ℤ+}, {d_{N}}_{N∈ℤ+}) to (P2_{N}^{λ}) that corresponds to a function u_{2} := ∑_{N=0}^{+∞} (c_{N}Kξ_{2N}^{1} + d_{N}Kξ_{2N}^{2}) ∈ L^{2}(ℝ; ℂ^{2}), for which u_{1} = u_{2}iff K̃(({a_{N}}_{N∈ℤ+}, {b_{N}}_{N∈ℤ+})) = ({c_{N}}_{N∈ℤ+}, {d_{N}}_{N∈ℤ+}). In this and only in this case, η = u_{1} = u_{2}is a solution to the twisted eigenvalue problem.
To study the recurrence (P1_{N}^{λ}), we first observe that by virtue of the initial condition v_{−1} = 0, it is actually a scalar recurrence, depending only on the initial condition v_{0} ∈ ℂ^{2}. At this point, all the solutions to (P1_{N}^{λ}) [and hence to (P2_{N}^{λ})] are of the form h(λ) ⊗ v_{0}, for some sequence h(λ) = {h_{N}(λ)}_{N∈ℤ+} ∈ ℂ^{ℤ+} depending only on λ. One next sees that it is possible to construct, for λ ∈ ℝ, only two kinds of solutions: the ones for which there exists N_{0} such that h_{N}(λ) = 0 for all N ≥ N_{0} + 1 [such solutions are called finitetype (λ, N_{0})solutions] that are hence obviously associated with Schwartz candidates as solutions to the twisted eigenvalue equation; the ones that are not of finite type, for which there exists N_{0} ≥ 0 such that h_{N}(λ) = 0 for every N ≤ N_{0} − 1 and h_{N}(λ) ≠ 0 for infinitely many N ≥ N_{0} [one sets h_{−1}(λ) ≡ 0]. The latter solutions are not in general associated with any reasonable distribution in 𝒮′(ℝ; ℂ^{2}) unless λ satisfies an equation of the kind q_{N0}(λ) = f_{N0}(λ), where q_{N0} is a suitably defined rational function and f_{N0} is a suitably defined continued fraction [both depending on Λ_{2N}(λ) and d_{2N}(λ)], case in which the associated function is actually a Schwartz function [such solutions are called infinitetype (λ, N_{0})solutions]. Notice that the algebraiccompanion solutions k(λ) ⊗ w_{0} to (P2_{N}^{λ}), algebraically obtained from a solution h(λ) ⊗ v_{0} of (P1_{N}^{λ}) (of finite or infinite type) through (P3_{N}^{λ}), are automatically associated with Schwartz functions. Hence, for a fixed λ that gives rise to finitetype or infinitetype solutions, we get at least two solutions, for we have the freedom of the choice of v_{0} ∈ ℂ^{2}. On the other hand, the aforementioned compatibility condition rules out that freedom: if one defines for h(λ)(≠ 0) of finite or infinite type then dim_{ℂ} W_{λ}(h) ≤ 1. The reason why W_{λ}(h) might be zerodimensional (corresponding to v_{0} = w_{0} = 0) comes from the fact that the operator K̃ acts in a highly unknown fashion, and we cannot a priori conclude that a solution h(λ) ⊗ v_{0} to (P1_{N}^{λ}) and the algebraic companion k(λ) ⊗ w_{0} relative to (P2_{N}^{λ}) are related (for the same λ) through the compatibility condition.
Let us define (and analogously the sets ∑_{0}^{−} and ∑_{∞}^{−} relative to the odd case). One has also the following description of ∑_{0}^{+}: upon defining polynomials j_{2N} ∈ ℝ[λ] inductively by the recurrence formula j_{−2}(λ) = 1, j_{0}(λ) = d_{0}(λ) [they appear as determinants of particular Jacobi matrices (see ref. 2)], then (and analogously in the odd case). Put ∑_{0} := ∑_{0}^{+} ∪ ∑_{0}^{−}, ∑_{∞} := ∑_{∞}^{+} ∪ ∑_{∞}^{−}, and for λ ∈ ∑_{0} ∪ ∑_{∞} define V_{λ}^{+} := {u ∈ L^{2}(ℝ; ℂ^{2}); Q̃(x, D_{x})u = λu and u is even} and V_{λ}^{−} := {u ∈ L^{2}(ℝ; ℂ^{2}); Q̃(x, D_{x})u = λu and u is odd}. It is important to notice that at this point we do not know as yet that the V_{λ}^{±} are eigenspaces belonging to λ (because we do not know as yet that any given λ in ∑_{0} ∪ ∑_{∞} is an eigenvalue). Thus, as proved in ref. 2, we have the following theorem (that might be thought of as the first half of Theorem 2.4 below).
Theorem 2.2. One hasand
Dealing with Problem 2.
In this case, the key ingredients for dealing directly with problem 2 are the formula and the use of the basis B_{mix} (see Lemma 2.1). Again, we restrict to the eveneigenfunction case. Hence, consider the equation (Q̃ − λI)η = 0, with η of the form η = ∑_{N=0}^{+∞} (a_{N}ξ_{2N}^{+} + b_{N}Kξ_{2N}^{+}). Upon setting a_{−1} = b_{−1} = 0, z_{N} := ℓ^{−2N} (1 − iℓ)^{N} (μ_{−}a_{N} − iμ_{+}b_{N}), we get the recurrence equations i.e., exactly the recurrence equations (P2_{N}^{λ}). It is a key observation now, and the reason why the basis B_{mix} is of fundamental importance, that the compatibility condition is automatically satisfied. Hence, since λ ∈ ∑_{0} ∪ ∑_{∞} gives rise to Schwartz solutions to the eigenvalue equation, it follows from (P2_{N}^{λ}) = (P2_{N}^{λ})′ that λ ∈ Spec(Q̃). Thus Spec(Q̃(x, D_{x})) = ∑_{0} ∪ ∑_{∞}. Now, since any finitetype (λ, N_{0} + 1)solution of (P1_{N}^{λ}) corresponds to a finitetype (λ, N_{0})solution of (P2_{N}^{λ}), and hence of (P2_{N}^{λ})′, we have the following crucial fact.
Lemma 2.3. One has ∑_{0}^{+} ⊂ ∑_{∞}^{+}and ∑_{0}^{−} ⊂ ∑_{∞}^{−}.
As a consequence of this approach and of Theorem 2.2 above, we get, as proved in ref. 3, the following rather complete description.
Theorem 2.4. One hasandRemark 2.5: In general ∑_{0}^{±} ≠ ∅ (see ref. 2).
3. A New System
As an example of some possible directions of generalization of the system defined by the operator (see Eq. 1), we propose here the spectral problem relative to the operator i.e., the study of the equation Notice that, as an unbounded operator in L^{2}(ℝ; ℂ^{2}) with domain B^{2}(ℝ; ℂ^{2}), Q_{vect}(x, D_{x}) is selfadjoint with compact resolvent (this again by virtue of the global ellipticity of its principal part). It seems quite nontrivial to obtain the eigenvalues of Q_{vect}(x, D_{x}). We want to show that exploiting the oscillator representation allows one to solve the spectral problem. Notice that Q_{h}(x, D_{x}) is unitarily equivalent (through a symplectic scaling) to Q_{(,)}(x, D_{x}) (in the notations of Section 2). The main problem here is to treat the zerothorder part of Q_{vect}(x, D_{x}). That seems difficult, but that is not the case, by virtue of the tensor product representation of sl_{2}(ℝ) to be constructed below. Hence, let {H, X^{+}, X^{−}} be the basis of sl_{2}(ℝ), which satisfies Eq. 2. Put Then it is clear that [ψ, ψ^{†}] = 1, and hence the map ω: sl_{2}(ℝ) → End_{ℂ} (𝒮(ℝ)) gives the oscillator representation of sl_{2}(ℝ) on 𝒮(ℝ). Because the action of sl_{2}(ℝ) leaves the parity invariant, we have the irreducible decomposition of ω. Put ω^{±} := ω_{𝒮±}_{(ℝ)}. Then v_{0} = e^{−x2}^{/2} (resp. ψ^{†}v_{0}) gives the lowest weight vector of the irreducible representation of (ω^{+}, 𝒮_{+}(ℝ)) (resp. of (ω^{−}, 𝒮_{−}(ℝ))) (see ref. 12). Let (π, V) (V ≃ ℂ^{2}) be the vector representation of sl_{2}(ℝ): We have the following proposition.
Proposition 3.1. Put ρ(H) := Q_{vect}(x, D_{x}), Then (ρ, 𝒮(ℝ; ℂ^{2})) defines a representation of sl_{2}(ℝ). Furthermore, ρ is equivalent to the tensor product representation (ω ⊗ π, 𝒮(ℝ) ⊗ ℂ^{2}). In fact, the operator exp(x^{2}J/2) defines the intertwining operator between these representations:where, recall, ω ⊗ π(Y) = ω(Y) ⊗ 1 + 1 ⊗ π(Y) [we have here identified 𝒮(ℝ) ⊗ ℂ^{2}with 𝒮(ℝ; ℂ^{2})]. In particular, the system defined by the operator Q_{vect}(x, D_{x}) is unitarily equivalent to the system defined by the operator ω ⊗ π(H).
Proof: All the statements follow from the fact Details are left to the reader.
Using the irreducible decomposition of the tensor product representation ω ⊗ π we have the following theorem.
Theorem 3.2. Put ξ_{0}^{±} := e^{x2}^{J/2} ϕ_{0}^{±}and ξ_{1}^{±} := e^{x2}^{J/2} ϕ_{1}^{±}, where Define, for N ≥ 0, ξ_{0,N}^{±} := ρ(X^{+})^{N} ξ_{0}^{±}, ξ_{1,N}^{±} := ρ(X^{+})^{N} ξ_{1}^{±}. Then Remark 3.3: As an sl_{2}(ℝ)module, the above theorem implies the irreducible decomposition of (ρ, 𝒮(ℝ; ℂ^{2})): where the closure refers to the 𝒮topology (the same decomposition holds for L^{2}(ℝ; ℂ^{2}) with closure in the L^{2}topology). In particular, ξ_{0}^{+}, ξ_{1}^{+} and ξ_{0}^{−}, ξ_{1}^{−} give the lowestweight vectors of the irreducible summands, respectively. The L^{2}structure of Spec(Q_{vect}(x, D_{x})) is given by Furthermore, the proof of the above theorem gives the following result (see ref. 4).
Theorem 3.4. Let The system Q_{ɛ}(x, D_{x}) interpolates systems Q_{h}(x, D_{x}) and Q_{vect}(x, D_{x}), it has spectrum given by the numbers 2N + 1/2 ± ɛ (with even relative eigenfunctions) and 2N + 3/2 ± ɛ (with odd relative eigenfunctions), where N ∈ ℤ_{+}, with multiplicity one for any N ≥ 0 when ɛ ∉ (1/2)ℤ.
Remark 3.5: We remark that the eigenfunctions of Q_{ɛ}(x, D_{x}) do not depend on ɛ (see ref. 4). This is the main reason why we think the eigenfunction basis of Q_{ɛ} may serve as a useful tool for studying more general systems.
We shall give further generalizations in ref. 4. One of our main motivations is to provide a class of examples whose spectral problems can be explicitly solved in a unified way. Indeed, if we take any unitary transformation U(x) in place of e^{x2}^{J/2} in Proposition 3.1, we may write down a number of examples that look more difficult, but that actually are obviously all unitarily equivalent to system Q_{vect}(x, D_{x}). Moreover, it is also quite interesting to consider the eigenvalue problem for even for the special case C = A, for one may write (A and C are real selfadjoint matrices, B real skewadjoint).
Acknowledgments
We thank the Department of Mathematics of University of Reims (Reims, France) for their kind hospitality. A.P. was in part supported by the Department of Mathematics of University of Reims, and M.W. by GrantinAid for Scientific Research (B) No. 11440010 from the Ministry of Education, Science, and Culture of Japan.
Footnotes

↵§ To whom reprint requests should be addressed. Email: parmeggi{at}dm.unibo.it.

This paper was submitted directly (Track II) to the PNAS office.

Article published online before print: Proc. Natl. Acad. Sci. USA, 10.1073/pnas.011393898.

Article and publication date are at www.pnas.org/cgi/doi/10.1073/pnas.011393898
 Received August 17, 2000.
 Copyright © 2001, The National Academy of Sciences
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