Validated ordered eigenvalue enclosures for an index-one Hessian

Let $V$ and $U$ be as in Regular minimum-energy paths on a declared collision-free tube, let $x\in U$, and let $H={\mathrm{\nabla }}^{2}V(x)$ use the convention of The Hessian matrix and critical points of a scalar field. Since $V\in C^4(U)$, it is $C^2$; The Hessian of a $C^2$ scalar field is symmetric gives $H^T=H$. In the standard Euclidean inner product this makes the endomorphism represented by $H$ self-adjoint. Therefore the real spectral theorem Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis applies, and $H$ has ordered eigenvalues ${\lambda }_1(H)\le \cdots \le {\lambda }_d(H)$.

A validated ordered eigenvalue enclosure is a list of intervals $I_j=[{\mathrm{\ell }}_j,u_j]$ together with an independently checked certificate proving ${\lambda }_j(H)\in I_j$ for every $j$ and preserving the stated ordering. The exact Hessian $H$, interval arithmetic or another validated spectral method, and all rounding assumptions used by the certificate are part of that external certificate; a Ritz value or residual alone is not, by definition, such a certificate.

For $d\ge 2$ and $\gamma >0$, the enclosure certifies index one with gap $\gamma$ when $u_1\le -\gamma$ and ${\mathrm{\ell }}_j\ge \gamma$ for every $j\ge 2$. For $d=1$, the condition is $u_1\le -\gamma$, with no positive-eigenvalue clauses. A validated enclosure records the input evidence; this definition does not assert that any particular eigensolver produces it.