Morse inequalities for ordered eigenvalues of generic self-adjoint families

In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet–Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians.

1. The particulars of the operator depend on what is being conducted: electrons, light, sound, etc.

2. Throughout the paper, we use integer coefficient homology, unless specified otherwise.

3. Note that this equivalence is not immediate and is established in the beginning of the proof of Theorem 3.2 below.

4. The similarity is natural since our Theorem 1.12 reproduces the classical Morse inequalities if one sets \(n=1\).

5. This is usually a stepping stone to defining the Clarke subdifferential, but we will limit ourselves to Clarke directional derivative which is both simpler and sufficient for our needs.

6. Note that there is a misprint in the direction of the inequality in [42, Sect. 6].

7. The reason for this codimension to be equal to \(\dim \mathrm{Sym}_{\nu}(\mathbb{F}) - 1\) is as follows: Symmetric matrices with non-repeated eigenvalues can be encoded by their eigenvalues and unit eigenvectors. When an eigenvalue is repeated \(\nu \) times, there is a loss of \(\nu -1\) parameters from the eigenvalues plus an extra freedom of choice of an orthonormal basis in the corresponding eigenspace. Thus the desired codimension is equal to \(\nu -1\) plus the dimension of the space of orthonormal bases of \(\mathbb{F}^{\nu}\), adding up to \(\dim \mathrm{Sym}_{\nu}(\mathbb{F}) - 1\).

8. The definition of a nondepraved point in [35, Sec. I.2.3] contains three conditions. Conditions (c) and (a) of [35, Sec. I.2.3] correspond to parts (1) and (2) of Proposition 4.7, respectively. The third condition — condition (b) of [35, Sec. I.2.3] — holds automatically in our case because \(x\) is non-degenerate as a smooth critical point of \(\lambda _{k}\big|_{S}\), by condition (S) assumed in Theorem 1.12). Thus we omit here the general description of condition (b), which is rather technical.

9. We also mention that [35] uses the term “critical” for the points \(y\) that are critical when the function in question is restricted to their respective stratum of constant multiplicity.

10. From properties of isometries and the inclusion \(\mathbf{E}^{\mathcal{R}}_{k} \subset \mathbf{E}_{k}\) it can be seen that \((\mathcal{U}^{*}\mathcal{U}_{\mathcal{R}})^{*} \mathcal{U}^{*} \mathcal{U}_{\mathcal{R}} = I_{\nu _{\mathcal{R}}}\).

11. Note that in equation (4.10) the same notation ⊥ is used for two different operations: on one hand, for the operation of taking orthogonal complement for subspaces \(\mathbb{F}^{\nu}\) and, on the other hand, for the operation of taking orthogonal complement for subspaces of \(\mathrm{Sym}_{\nu}\).

12. In every particular case of ℱ, one still needs to establish non-degeneracy of the critical point “by hand”. In some well-studied cases, such as discrete magnetic Schrödinger operators [3, 31], degenerate critical points are endemic.

13. And in fact only for them [61].

14. The word “weak” here is used to distinguish it from the jet version of the Thom transversality theorem which is usually called strong [7].

15. For bundles whose fibers are stratified spaces, smoothness is defined in the usual way — as smoothness of trivializing maps. Smooth maps between stratified submanifolds are maps which are restrictions of smooth maps on the corresponding ambient manifolds, see [35, p. 13].

16. Here we use that the codimension of the set of \(n_{1}\times n_{2}\) matrices of rank \(r\) is equal to \((n_{1}-r)(n_{2}-r)\).

17. The constant is independent of \(F\) but may depend on the norm used for \(F\); in case of the operator norm, Weyl inequality yields \(C=2\).

18. Namely, \(\Phi (R,0)=0\) for all \(R\).

19. A particularly convenient form for this task can be found in [12, Thm 4.3].

20. An analogous result for cohomologies can be found in [58].

21. See [34, Prop 15.2.10] or [22, Chap. 5]. It is also sometimes known as Poincaré–Verdier duality, see [45, VI.3].

We are grateful to numerous colleagues who aided us with helpful advice and friendly encouragement. Among them are Andrei Agrachev, Lior Alon, Ram Band, Mark Goresky, Yuji Kodama, Khazhgali Kozhasov, Peter Kuchment, Sergei Kuksin, Sergei Lanzat, Antonio Lerario, Jacob Shapiro, Stephen Shipman, Frank Sottile, Bena Tshishiku, and Carlos Valero. We also thank anonymous reviewers for many insightful comments that improved our paper.

GB was partially supported by NSF grants DMS-1815075 and DMS-2247473. IZ was partially supported by NSF grant DMS-2105528 and Simons Foundation Collaboration Grant for Mathematicians 524213.

Authors and Affiliations

1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

   Gregory Berkolaiko & Igor Zelenko

Corresponding author

Correspondence to Gregory Berkolaiko.

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Appendix A: Hellmann–Feynman Theorem

In this section we review the mathematical formulation of the formula that is known in physics as Hellmann–Feynman Theorem or first-order perturbation theory. We base our formulation on [48, Thm II.5.4] (see also [37]).

Theorem A.1

Let \(T: \mathbb{R}\to \mathrm{Sym}_{n}(\mathcal{F})\) be differentiable at \(x=0\). Let \(\lambda \) be an eigenvalue of \(T(0)\) of multiplicity \(\nu \), \(\mathbf{E}\subset \mathcal{F}^{n}\) be its eigenspace. Then, for small enough \(x\), there are exactly \(\nu \) eigenvalues of \(T(x)\) close to \(\lambda \) and they are given by

where \(\{\mu _{j}\}\) are the eigenvalues of the \(\nu \times \nu \) matrix \(\big(T'(0)\big)_{\mathbf{E}}\), see (1.4).

Appendix B: Twisted Thom space homologies from Poincaré–Lefschetz duality

The Poincaré–Lefschetz duality (see, e.g. [38, Theorem 3.43]) states that if \(Y\) is compact orientable \(n\)-dimensional manifold with boundary \(\partial Y\), then

There is also a twisted analogue^{Footnote 21} of Poincaré–Lefschetz duality for non-oriented manifolds: if \(Y\) is compact non-orientable \(n\)-dimensional manifold with boundary \(\partial Y\), then

Here, the twisted homology \(H_{*}(Y;\widetilde{\mathbb{Z}})\) was already introduced in Sect. 6. To define twisted cohomology groups, denote by \(\widetilde{Y}\) the orientation cover of \(Y\) and by \(\tau \) the corresponding orientation-reversing involution. \(H^{*}(Y;\widetilde{\mathbb{Z}})\) are the cohomologies of the cochain complex defined on the spaces of cochains \(c\) satisfying \(c\bigl(\tau (\alpha )\bigr)=-c(\alpha )\) for every chain \(\alpha \) in \(\widetilde{Y}\) (see [38, Se. 3H] for a more general point of view). Such cochains will be called skew-symmetric cochains. Note that the space of skew-symmetric cochains can be identified with the dual space to the space of skew-symmetric chains, as expected.

(a) Assume now that \(\nu \) is even. Then the base \(\operatorname{Gr}(i-1,\nu -1)\) of the vector bundle \(\widehat{E}_{i,\nu}\) is non-orientable and, since the vector bundle is also non-orientable, the total space \(\mathcal{B}(\widehat{E}_{i,\nu})\) is orientable. By the usual Poincaré–Lefschetz duality (B.1),

where \(\dim \widehat{E}_{i, \nu}\) is the dimension of the total space \(\mathcal{B}(\widehat{E}_{i,\nu})\). In the last identification we used that the base \(\mathrm{Gr}(i-1,\nu -1)\) is the deformation retract of the total space of the bundle.

Further, since \(\operatorname{Gr}(i-1,\nu -1)\) is non-orientable when \(\nu \) is even, we use the twisted analog of Poincaré duality for nonorientable manifolds (see [38, Theorem 3H.6] as well as (B.2) with \(\partial Y=\emptyset \)) to get

where we used

(b) Consider the case of odd \(\nu \). Then the base \(\operatorname{Gr}(i-1,\nu -1)\) is orientable, the bundle is non-orientable and therefore the total space \(\mathcal{B}(\widehat{E}_{i,\nu})\) is non-orientable. By the twisted Poincaré-Lefschetz duality (B.2),

The orientation double cover \(\widetilde{E}_{i,\nu}\) of \(\widehat{E}_{i,\nu}\) can be constructed from the tautological bundle of the oriented Grassmannian \(\widetilde{\operatorname{Gr}}_{\mathbb{R}}(i-1,\nu -1)\) in the same way as \(\widehat{E}_{i,\nu}\) was constructed from the tautological bundle of the Grassmannian \(\operatorname{Gr}_{\mathbb{R}}(i-1,\nu -1)\) by relations (6.2) and (6.5). In particular, \(\widetilde{E}_{i,\nu}\) is a bundle of rank \(s(i)\) over the oriented Grassmannian \(\widetilde{\operatorname{Gr}}_{\mathbb{R}}(i-1,\nu -1)\). Therefore, retracting the unit ball bundle \(\mathcal{B}(\widetilde{E}_{i,\nu})\) of \(\widetilde{E}_{i,\nu}\) to its base, we get that the integer cohomology groups of \(\mathcal{B}(\widetilde{E}_{i,\nu})\) are isomorphic to the integer cohomology groups of the oriented Grassmannian \(\widetilde{\operatorname{Gr}}_{\mathbb{R}}(i-1,\nu -1)\), i.e.

Moreover, the retraction can be made to preserve the spaces of skew-symmetric chains, which implies that

When \(\nu \) is odd, \(\operatorname{Gr}_{\mathbb{R}}(i-1,\nu -1)\) is orientable and so is \(\widetilde{\operatorname{Gr}}_{\mathbb{R}}(i-1,\nu -1)\). Moreover, the map from the usual Poincaré duality (see [38, Thm. 3.30] as well as (B.1) with \(\partial T = \emptyset \)) applied to \(\widetilde{\operatorname{Gr}}_{\mathbb{R}}(i-1,\nu -1)\) sends the equivalence classes of skew-symmetric cochains to the corresponding skew-symmetric chains. Thus, we arrive to

To summarize, we get the corresponding line in (1.13) whether \(\nu \) is even or odd.

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