Off-diagonal singularities of the spectral function of elliptic systems with multiple eigenvalues of the principal symbol

The spectral function $e(x,y,\lambda)$ of a self-adjoint elliptic operator is the matrix-valued kernel of its spectral projector, and its sharp asymptotics as $\lambda\to\infty$ are the content of the local Weyl law. Hörmander's 1968 study of the spectral function of an elliptic operator established the optimal pointwise remainder $O(\lambda^{(n-1)/m})$ and described the kernel on and near the diagonal for scalar operators and for systems whose principal symbol has distinct eigenvalues; for systems with multiple (variable-multiplicity) eigenvalues the Fourier-integral-parametrix method fails, and the paper left open two problems: the optimal remainder estimate, and a complete description of the off-diagonal singularities. The first problem was later resolved by Ivrii's microhyperbolicity method, an energy-estimate technique that yields the sharp on-diagonal asymptotics for matrix operators without any simplicity assumption. This record concerns the second problem, which remains open: for a self-adjoint elliptic system with positive Hermitian principal symbol whose eigenvalue branches cross, describe $e(x,y,\lambda)$ off the diagonal — its uniform blow-up profile in the near-diagonal region $|x-y|=O(\lambda^{-1/m})$, sharp off-diagonal bounds, and the structure of its singularities, where the expected singular set is formed by pairs connected by the Hamiltonian flows of the eigenvalue branches, possibly including composite trajectories that switch branches at crossings.

Background

The spectral function of a self-adjoint elliptic operator $A$ on a compact manifold is the matrix-valued kernel $e(x,y,\lambda)=\sum_{\lambda_k<\lambda}v_k(x)v_k(y)^*$ of its spectral projector, where $v_1,v_2,\dots$ is an orthonormal basis of eigenfunctions with eigenvalues $\lambda_1\le\lambda_2\le\dots\to\infty$. The Weyl law describes the growth of this kernel on the diagonal: for an operator of order $m$ in dimension $n$, $e(x,x,\lambda)$ grows like $\lambda^{n/m}$. Hörmander's 1968 Acta Mathematica paper 'The spectral function of an elliptic operator' proved the sharp pointwise (local) form $e(x,x,\lambda)=\kappa(x)\lambda^{n/m}+O(\lambda^{(n-1)/m})$ for scalar elliptic differential operators and for systems whose principal symbol has pairwise distinct eigenvalues, together with a description of the kernel on and near the diagonal. An elliptic system means that $A$ acts on vector-valued functions or, more generally, on sections of a vector bundle, so that its principal symbol $a_m(x,\xi)$ — the top-order part of the operator frozen in phase-space coordinates $(x,\xi)$ — is a Hermitian matrix; its eigenvalue branches $h^{(1)}(x,\xi),\dots,h^{(D)}(x,\xi)$ may cross as $(x,\xi)$ varies, and when they do the system is said to have multiple, or non-simple, characteristics (variable eigenvalue multiplicity). The 1968 proof reduces $A$ to a first-order operator by extracting an $m$-th root $A^{1/m}$ and builds a Fourier integral operator parametrix, that is, an approximate inverse whose singularities propagate along the Hamiltonian flow — the flow on phase space generated by each eigenvalue branch of the principal symbol. This reduction requires non-crossing, scalar-like branches; for systems with variable multiplicity the $m$-th root is not a classical pseudo-differential operator with scalar principal symbol, and Ivrii's later survey states explicitly that such a parametrix 'cannot be constructed … even inside for elliptic systems with the eigenvalues of the principal symbol having the variable multiplicity'. At the time, only weaker remainder estimates were available for such systems. The on-diagonal part of this gap was later closed by Ivrii's microhyperbolicity method: microhyperbolicity is a positivity condition on the matrix symbol in a phase-space direction, automatically satisfied by positive elliptic symbols, which supports energy (positive-commutator) estimates replacing the unavailable parametrix and yields the sharp pointwise on-diagonal remainder for matrix operators with arbitrary eigenvalue multiplicity. What remains from the 1968 question is the off-diagonal part: the behavior of $e(x,y,\lambda)$ for $x\ne y$, uniformly in the near-diagonal region $|x-y|=O(\lambda^{-1/m})$ where the kernel blows up, and away from it, where singularities are expected precisely along pairs $(x,y)$ connected by the Hamiltonian flows of the eigenvalue branches, including composite ('broken') trajectories that switch branches at crossings. This off-diagonal theory is well developed for scalar operators but is not available for systems with multiplicities: the existing treatments of systems either restrict to simple eigenvalues or address only the counting function and the propagator (the wave-group operator $e^{-itA}$).

Problem Statement

Let $M$ be a smooth closed manifold of dimension $n$ without boundary, let $E\to M$ be a Hermitian vector bundle of rank $D$, and let $A$ be an essentially self-adjoint elliptic differential operator of even order $m\ge 2$ acting on sections of $E$, bounded from below, whose principal symbol $a_m(x,\xi)$ is positive definite for $\xi\ne 0$. No assumption is made on the multiplicity of the eigenvalue branches of $a_m$: branches are allowed to cross, with variable multiplicity. Let $v_1,v_2,\dots$ be an orthonormal basis of eigenfunctions of $A$ with eigenvalues $\lambda_k\to\infty$ and let

$$e(x,y,\lambda)=\sum_{\lambda_k<\lambda}v_k(x)v_k(y)^*$$

denote the matrix-valued spectral function. On the diagonal, the sharp local Weyl law $e(x,x,\lambda)=\kappa(x)\lambda^{n/m}+O(\lambda^{(n-1)/m})$ is known without simplicity assumptions. The open question, raised by the source for systems with multiple principal-symbol eigenvalues, is to describe $e(x,y,\lambda)$ off the diagonal to the same standard as the classical theory for scalar operators and for systems with simple characteristics. Prove or disprove:

(i) there is a uniform asymptotic formula for $e(x,y,\lambda)$ in the near-diagonal scaling region $\lambda^{1/m}|x-y|=O(1)$, identifying the blow-up profile of the kernel as $y\to x$ and the modification of this profile at points where the eigenvalue branches of $a_m$ cross;

(ii) $e(x,y,\lambda)$ satisfies sharp bounds for $x\ne y$, uniformly on compact subsets of the complement of the diagonal, with the optimal exponent identified and the uniformity with respect to $(x,y)$ and $\lambda$ made explicit;

(iii) the singular set of $e(x,y,\lambda)$ off the diagonal can be identified, and the asymptotic structure and strength of the singularities of $e(x,y,\lambda)$ on it described, as $\lambda\to\infty$; in the variable-multiplicity setting the expected structure is the closure of the pairs $(x,y)$ connected by the Hamiltonian flows of the eigenvalue branches of $a_m$, possibly including composite ('broken') trajectories that switch branches at crossings, and this expectation is to be proved or disproved rather than assumed.

The three parts (i)–(iii) are posed jointly, and each may be answered positively or negatively. Across (i)–(iii), the description sought is of asymptotics with explicit remainder terms, consistent with the known on-diagonal asymptotics and reducing to the known scalar and simple-characteristics results when the branches decouple or never cross.

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