Almost Interlacing in Spectral Theory

• Almost interlacing is defined as a near-alternation phenomenon where ordered spectra or zero sets display non-strict or boundary-only interlacing instead of a full alternating chain.
• It appears in diverse fields including matrix theory (e.g. T-matrices and Kotelyansky classes), reversible Markov dynamics, q-orthogonal polynomials, and modular forms, demonstrating consistent structural rigidity.
• Methodologies such as auxiliary point completion and algebraic transforms (e.g. Veronese decompositions) are employed to recover complete interlacing from partially interlaced systems.

Searching arXiv for papers on “almost interlacing” and related interlacing formulations. I’ll look for recent and foundational arXiv papers that explicitly discuss interlacing, weak interlacing, or “almost interlacing.” Almost interlacing denotes a family of weaker-than-complete interlacing phenomena in which two ordered spectra or zero sets remain highly constrained, but the classical alternating chain is replaced by monotonicity, boundary-only interlacing, non-strict inequalities, or completion by one or two additional points. The phrase is not uniform across the literature: in some papers it is only an intuitive description, while in others it names a precise defect of full interlacing. Across matrix theory, reversible Markov dynamics, orthogonal and $q$-orthogonal polynomials, combinatorial transforms, and modular forms, the recurring mechanism is that positivity, sign-regularity, reversibility, or mixed recurrences impose enough structure to force near-alternation without always giving the full Cauchy or Sturm pattern (Kushel, 2012, Hartich et al., 2018, Jordaan et al., 28 Apr 2026).

1. Terminological scope

The literature does not use a single formal definition of almost interlacing. In the matrix-theoretic study of Kotelyansky classes, the term is not formal, but the relevant results concern eigenvalue relations that are weaker than full Cauchy interlacing and still very strong: monotonicity of the smallest real eigenvalues across principal submatrices and strict interlacing only for boundary deletions (Kushel, 2012). In reversible Markov dynamics, the distinction is explicit: strict interlacing means $\lambda_{k-1}<\mu_k<\lambda_k$, whereas the “slightly weaker interlacing condition” is $\lambda_{k-1}\le \mu_k\le \lambda_k$, with possible equalities caused by degeneracies or vanishing local spectral weights at the target (Hartich et al., 2018).

For $q$-hypergeometric polynomials, the phrase is encoded geometrically rather than spectrally. The logarithmic mesh $\delta_{\log}$ measures how closely zeros follow a geometric $q$-lattice, and weak interlacing appears as membership in the closed class $\overline{P_n^q}$, equivalently $p(x)\preccurlyeq p(qx)$, rather than the strict class $P_n^q$ with $p(x)\prec p(qx)$ (Martinez-Finkelshtein et al., 5 Jun 2025). In mixed-recurrence theories for orthogonal polynomials, almost interlacing means that two zero sets fail to interlace by one or two points, and that multiplication by $\lambda_{k-1}<\mu_k<\lambda_k$0 or $\lambda_{k-1}<\mu_k<\lambda_k$1 restores complete interlacing (Jordaan et al., 4 Apr 2026, Jordaan et al., 28 Apr 2026). In the modular-forms setting, the phrase has an asymptotic and geometric meaning: zeros interlace for almost all members of a canonical basis and on most of the lower boundary arc, namely on a truncated arc $\lambda_{k-1}<\mu_k<\lambda_k$2 rather than the full boundary (Jenkins et al., 2013).

A plausible implication is that almost interlacing is best understood as a regime of near-rigidity rather than as a single invariant. What remains stable from paper to paper is not a fixed definition, but a structural pattern: sign control localizes zeros or eigenvalues to prescribed gaps, while some obstruction prevents a completely alternating chain.

2. Weak spectral interlacing in matrix theory

A central matrix-theoretic realization of almost interlacing arises from T-matrices and Kotelyansky classes. For an $\lambda_{k-1}<\mu_k<\lambda_k$3 real matrix $\lambda_{k-1}<\mu_k<\lambda_k$4, the quantity

$\lambda_{k-1}<\mu_k<\lambda_k$5

is the minimal real eigenvalue, with $\lambda_{k-1}<\mu_k<\lambda_k$6 if $\lambda_{k-1}<\mu_k<\lambda_k$7 has no real eigenvalues. A T-matrix is defined by positivity of these minimal real eigenvalues on principal submatrices together with eigenvalue monotonicity: $\lambda_{k-1}<\mu_k<\lambda_k$8 A strict T-matrix, or $\lambda_{k-1}<\mu_k<\lambda_k$9-matrix, satisfies strict inequalities whenever $\lambda_{k-1}\le \mu_k\le \lambda_k$0 (Kushel, 2012).

The relevant matrix classes are built from minors. A Kotelyansky matrix has all principal minors positive and all almost principal minors nonnegative; a strictly Kotelyansky matrix has all principal and all almost principal minors strictly positive. Almost principal minors are obtained from a principal minor by deleting one row and one different column. Their positivity feeds directly into positivity of compound matrices and exterior powers, which is the key to the spectral arguments (Kushel, 2012).

For an $\lambda_{k-1}\le \mu_k\le \lambda_k$1 SK matrix $\lambda_{k-1}\le \mu_k\le \lambda_k$2, if $\lambda_{k-1}\le \mu_k\le \lambda_k$3 is the principal submatrix obtained by deleting the $\lambda_{k-1}\le \mu_k\le \lambda_k$4-th row and column, Theorem 4 gives the extremal inequalities

$\lambda_{k-1}\le \mu_k\le \lambda_k$5

Theorem 5 then shows that every SK matrix is a $\lambda_{k-1}\le \mu_k\le \lambda_k$6-matrix. This is one-sided interlacing: as principal submatrices become smaller, the smallest real eigenvalue strictly increases. It is strong spectral ordering, but not full alternating interlacing for every eigenvalue and every deletion (Kushel, 2012).

The same paper proves a sharper boundary theorem. If $\lambda_{k-1}\le \mu_k\le \lambda_k$7 is an SK-matrix with positive simple eigenvalues

$\lambda_{k-1}\le \mu_k\le \lambda_k$8

and $\lambda_{k-1}\le \mu_k\le \lambda_k$9 is the principal submatrix obtained by deleting row $q$0 and column $q$1, then for $q$2 or $q$3,

$q$4

Thus full strict interlacing survives along the first or last row/column, whereas for interior deletions such a global chain can fail even for totally positive matrices. This boundary-only phenomenon is one of the clearest matrix manifestations of almost interlacing (Kushel, 2012).

The weak interlacing mechanism extends beyond pure positivity. Strictly totally J-sign-symmetric matrices and strictly J-sign-symmetric Kotelyansky matrices satisfy the same extremal inequalities and are also $q$5-matrices. The paper does not claim full Cauchy-style interlacing for arbitrary principal submatrices in these generalized classes; the preserved structure is again monotonicity of minimal eigenvalues and control of spectral extremes, not a complete alternating pattern (Kushel, 2012).

3. Relaxation and first-passage spectra in reversible Markov dynamics

In reversible Markovian dynamics, almost interlacing appears as a duality between the relaxation spectrum and the first-passage spectrum. For a finite reversible Markov jump process with generator $q$6, the relaxation eigenvalues are ordered

$q$7

while after making a state $q$8 absorbing, the absorbing generator $q$9 has eigenvalues

$\delta_{\log}$0

The first-passage density admits the spectral expansion

$\delta_{\log}$1

with $\delta_{\log}$2 and $\delta_{\log}$3 (Hartich et al., 2018).

The key structural fact is that the first-passage poles are encoded as zeros of the diagonal resolvent. In Laplace space,

$\delta_{\log}$4

so the poles of $\delta_{\log}$5 at $\delta_{\log}$6 are zeros of $\delta_{\log}$7, whereas the poles of $\delta_{\log}$8 itself are the relaxation eigenvalues $\delta_{\log}$9. This immediately sets up an alternation of poles and zeros on the real axis (Hartich et al., 2018).

For reversible finite chains, the paper proves the non-strict interlacing inequality

$q$0

If all relaxation eigenvalues are non-degenerate and all eigenfunctions are non-vanishing at the target, the inequalities sharpen to

$q$1

Strict interlacing is guaranteed for one-dimensional birth–death chains with the absorbing state at an outer boundary. In more general graphs, internal targets, or fine-tuned systems, equalities may occur, producing interlacing-with-possible-touching rather than strict alternation. This is the paper’s precise sense of almost interlacing (Hartich et al., 2018).

The same structure persists for effectively one-dimensional diffusions. With reflecting boundaries for relaxation and a Dirichlet condition at the absorbing point, the diagonal resolvent factorizes as

$q$2

so the first-passage zeros and relaxation poles interlace exactly as in a Sturm–Liouville problem. When geometry ceases to be effectively one-dimensional, or when modes vanish at the absorbing set, strict inequalities can collapse to equalities (Hartich et al., 2018).

The interlacing is computationally productive. The paper constructs $q$3 from relaxation data by a Newton series whose coefficients are organized by almost triangular matrices, and applies the method to an Ornstein–Uhlenbeck process and to an 8-state protein-folding model. A plausible implication is that almost interlacing here is not merely descriptive; it is the analytic device that makes full first-passage reconstruction possible from relaxation eigendata (Hartich et al., 2018).

4. Geometric zero spacing and $q$4-orthogonal polynomials

For little $q$5-Jacobi polynomials, almost interlacing is formalized through the logarithmic mesh. With $q$6, the polynomials

$q$7

are orthogonal on $q$8 when $q$9 and $\overline{P_n^q}$0. In that regime all zeros are real, simple, and lie in $\overline{P_n^q}$1, and each lattice interval $\overline{P_n^q}$2 contains at most one zero (Martinez-Finkelshtein et al., 5 Jun 2025).

For a polynomial $\overline{P_n^q}$3 with positive zeros $\overline{P_n^q}$4, the logarithmic mesh is

$\overline{P_n^q}$5

This quantity measures how close the zero set is to a geometric progression. The decisive characterization is

$\overline{P_n^q}$6

and similarly $\overline{P_n^q}$7 is equivalent to weak interlacing $\overline{P_n^q}$8. In this framework, almost interlacing is the closed or limiting situation in which the geometric alternation is non-strict or only attained in closure (Martinez-Finkelshtein et al., 5 Jun 2025).

The main structural theorem states that for $\overline{P_n^q}$9 and $p(x)\preccurlyeq p(qx)$0,

$p(x)\preccurlyeq p(qx)$1

so

$p(x)\preccurlyeq p(qx)$2

The same theorem yields interlacing with the $p(x)\preccurlyeq p(qx)$3-derivative through

$p(x)\preccurlyeq p(qx)$4

hence

$p(x)\preccurlyeq p(qx)$5

Further strict interlacing holds under parameter transforms such as

$p(x)\preccurlyeq p(qx)$6

and, for $p(x)\preccurlyeq p(qx)$7,

$p(x)\preccurlyeq p(qx)$8

These are strong interlacing statements rather than almost ones (Martinez-Finkelshtein et al., 5 Jun 2025).

Almost interlacing enters through closure, degeneration, and loss of orthogonality. Limit families such as Stieltjes–Wigert inherit only the closed mesh bound $p(x)\preccurlyeq p(qx)$9, while non-orthogonal specializations such as $P_n^q$0 yield factorizations

$P_n^q$1

with $P_n^q$2. The zero set then decomposes into an exact geometric block and a smaller orthogonal family, so the full polynomial remains in $P_n^q$3. This is almost interlacing in a precise geometric sense: the zeros are still organized by the $P_n^q$4-lattice, but the strongest strict inequalities may only survive in the reduced factor or in the limit family (Martinez-Finkelshtein et al., 5 Jun 2025).

5. Completion by one or two added points

A different theory treats almost interlacing as an incomplete alternating pattern that can be repaired by adding auxiliary zeros. In the one-point version, the basic object is a mixed recurrence with a linear factor,

$P_n^q$5

where $P_n^q$6 and $P_n^q$7 already interlace and $P_n^q$8 on the relevant interval. The extra point $P_n^q$9 is then appended to the zero set of $p(x)\prec p(qx)$0 through $p(x)\prec p(qx)$1, and the resulting $p(x)\prec p(qx)$2 zeros fully interlace with those of $p(x)\prec p(qx)$3 (Jordaan et al., 4 Apr 2026).

The paper “Separating zeros of polynomials using an added interlacing point” proves several general theorems of this type. In the case

$p(x)\prec p(qx)$4

if $p(x)\prec p(qx)$5, then $p(x)\prec p(qx)$6; if instead $p(x)\prec p(qx)$7, then $p(x)\prec p(qx)$8. Moreover, when $p(x)\prec p(qx)$9 lies outside the extreme zeros of $\lambda_{k-1}<\mu_k<\lambda_k$00, full interlacing of $\lambda_{k-1}<\mu_k<\lambda_k$01 with $\lambda_{k-1}<\mu_k<\lambda_k$02 is recovered. A second theorem gives $\lambda_{k-1}<\mu_k<\lambda_k$03 when $\lambda_{k-1}<\mu_k<\lambda_k$04, again with full $\lambda_{k-1}<\mu_k<\lambda_k$05–$\lambda_{k-1}<\mu_k<\lambda_k$06 interlacing once $\lambda_{k-1}<\mu_k<\lambda_k$07 moves past the leftmost or rightmost zero. Theorem 2.3 is explicitly partial: at least $\lambda_{k-1}<\mu_k<\lambda_k$08 zeros of $\lambda_{k-1}<\mu_k<\lambda_k$09 occupy distinct gaps between consecutive zeros of $\lambda_{k-1}<\mu_k<\lambda_k$10, and the remaining two are classified by the position of $\lambda_{k-1}<\mu_k<\lambda_k$11. This is an exact formulation of one-point almost interlacing (Jordaan et al., 4 Apr 2026).

The same framework yields explicit completion points for Krawtchouk, Meixner, Narayana, Jacobi, and Laguerre polynomials. For example, the Jacobi parameter shift produces the extra point

$\lambda_{k-1}<\mu_k<\lambda_k$12

and the Laguerre shift produces $\lambda_{k-1}<\mu_k<\lambda_k$13. In each case the position of $\lambda_{k-1}<\mu_k<\lambda_k$14 determines whether full interlacing of the original pair holds or whether only the completed object $\lambda_{k-1}<\mu_k<\lambda_k$15 interlaces (Jordaan et al., 4 Apr 2026).

The two-point theory extends this mechanism to pairs that fail to interlace by exactly two points. The mixed recurrence now has quadratic correction,

$\lambda_{k-1}<\mu_k<\lambda_k$16

with $\lambda_{k-1}<\mu_k<\lambda_k$17. The main theorem shows that

$\lambda_{k-1}<\mu_k<\lambda_k$18

in the minus-sign case, and classifies the permitted locations of $\lambda_{k-1}<\mu_k<\lambda_k$19 in the plus-sign case. If $\lambda_{k-1}<\mu_k<\lambda_k$20 lies to the left of the smallest zero of $\lambda_{k-1}<\mu_k<\lambda_k$21 and $\lambda_{k-1}<\mu_k<\lambda_k$22 to the right of the largest, then full interlacing of $\lambda_{k-1}<\mu_k<\lambda_k$23 with $\lambda_{k-1}<\mu_k<\lambda_k$24 follows (Jordaan et al., 28 Apr 2026).

This quadratic completion improves earlier partial results for Jacobi polynomials and resolves an open question for Meixner–Pollaczek polynomials. In the Jacobi case, the two explicit added points arise as the roots of a quadratic obtained by eliminating an intermediate term from a mixed recurrence. The same pattern appears for Pseudo-Jacobi polynomials. The conceptual content is uniform: almost interlacing means that a complete alternation exists after inserting precisely those auxiliary points encoded by the mixed recurrence (Jordaan et al., 28 Apr 2026).

6. Algebraic and combinatorial frameworks

Interlacing also appears as an exact preservation property under algebraic transforms. For a formal power series $\lambda_{k-1}<\mu_k<\lambda_k$25, the Veronese decomposition

$\lambda_{k-1}<\mu_k<\lambda_k$26

collects coefficients by congruence classes modulo $\lambda_{k-1}<\mu_k<\lambda_k$27. Zhang defines

$\lambda_{k-1}<\mu_k<\lambda_k$28

and proves that if the sequence

$\lambda_{k-1}<\mu_k<\lambda_k$29

is interlacing, then so is

$\lambda_{k-1}<\mu_k<\lambda_k$30

This supplies an interlacing-based proof of the real-rootedness part of Beck–Stapledon’s conjecture for Ehrhart $\lambda_{k-1}<\mu_k<\lambda_k$31-polynomials and recovers interlacing families for colored Eulerian polynomials studied by Savage–Visontai (Zhang, 2018).

That paper does not define almost interlacing, but it identifies several directions in which exact interlacing might weaken. Degree thresholds, log-concavity hypotheses, and partial interlacing of Veronese components are presented as plausible routes toward weaker phenomena. In particular, the paper notes that log-concavity can force full interlacing in a moderate degree range, and suggests that larger-degree settings may lead naturally to partial or asymptotic interlacing after repeated Veronese iterations. These are interpretive extensions rather than proved almost-interlacing theorems (Zhang, 2018).

A broader matrix formulation is given by the notion of a fully interlacing matrix of formal power series. For a $\lambda_{k-1}<\mu_k<\lambda_k$32 matrix $\lambda_{k-1}<\mu_k<\lambda_k$33, the associated Lace matrix $\lambda_{k-1}<\mu_k<\lambda_k$34 is formed by interleaving Toeplitz matrices of the entries. The matrix $\lambda_{k-1}<\mu_k<\lambda_k$35 is fully interlacing when $\lambda_{k-1}<\mu_k<\lambda_k$36 is totally positive. This strengthens the usual notion of an interlacing sequence: row and column matrices are special cases, and pairwise interlacing need not imply full interlacing. Full interlacing is preserved under matrix products, flips across the reverse diagonal, and Veronese sections (Athanasiadis et al., 2024).

This suggests a graded notion of almost interlacing in terms of partial total positivity. The paper itself points toward “total positivity up to a certain order or level” as a natural extension, although it does not formulate a definitive theory. A plausible implication is that almost interlacing in algebraic settings may be viewed as the passage from full total positivity of the Lace matrix to weaker positivity conditions on only lower-order minors (Athanasiadis et al., 2024).

7. Modular forms and asymptotic interlacing on boundary arcs

For weakly holomorphic modular forms on $\lambda_{k-1}<\mu_k<\lambda_k$37, almost interlacing acquires a geometric meaning. The Duke–Jenkins canonical basis $\lambda_{k-1}<\mu_k<\lambda_k$38 is characterized by

$\lambda_{k-1}<\mu_k<\lambda_k$39

where $\lambda_{k-1}<\mu_k<\lambda_k$40 with $\lambda_{k-1}<\mu_k<\lambda_k$41. The gap functions are

$\lambda_{k-1}<\mu_k<\lambda_k$42

holomorphic forms with maximal possible initial gap in the $\lambda_{k-1}<\mu_k<\lambda_k$43-expansion. If $\lambda_{k-1}<\mu_k<\lambda_k$44, all zeros of $\lambda_{k-1}<\mu_k<\lambda_k$45 in the standard fundamental domain lie on the circular arc

$\lambda_{k-1}<\mu_k<\lambda_k$46

(Jenkins et al., 2013).

The exact interlacing theorem in this setting is that the zeros of $\lambda_{k-1}<\mu_k<\lambda_k$47 interlace on $\lambda_{k-1}<\mu_k<\lambda_k$48 with the zeros of $\lambda_{k-1}<\mu_k<\lambda_k$49. The proof compares the normalized boundary values of $\lambda_{k-1}<\mu_k<\lambda_k$50 with trigonometric approximants such as

$\lambda_{k-1}<\mu_k<\lambda_k$51

and then controls the zero shifts by explicit error estimates (Jenkins et al., 2013).

For the full basis, the result is weaker and therefore closer to the article’s subject. For any fixed $\lambda_{k-1}<\mu_k<\lambda_k$52, let

$\lambda_{k-1}<\mu_k<\lambda_k$53

If $\lambda_{k-1}<\mu_k<\lambda_k$54 is fixed, then the zeros of $\lambda_{k-1}<\mu_k<\lambda_k$55 interlace with those of $\lambda_{k-1}<\mu_k<\lambda_k$56 on $\lambda_{k-1}<\mu_k<\lambda_k$57 for sufficiently large $\lambda_{k-1}<\mu_k<\lambda_k$58. If $\lambda_{k-1}<\mu_k<\lambda_k$59 is fixed, then the zeros of $\lambda_{k-1}<\mu_k<\lambda_k$60 interlace with those of $\lambda_{k-1}<\mu_k<\lambda_k$61 on $\lambda_{k-1}<\mu_k<\lambda_k$62 for sufficiently large $\lambda_{k-1}<\mu_k<\lambda_k$63. Thus interlacing holds for almost all forms in the basis and on most of the lower boundary, but not uniformly on the full arc. This is the paper’s concrete realization of almost interlacing (Jenkins et al., 2013).

Taken together, these results suggest that almost interlacing is a spectrum of near-alternation principles. In matrix classes it appears as monotone smallest eigenvalues and boundary-only alternation; in reversible stochastic dynamics as interlacing with possible touching; in $\lambda_{k-1}<\mu_k<\lambda_k$64-orthogonal families as closed logarithmic-mesh bounds; in mixed recurrences as interlacing completed by one or two extra points; in algebraic transforms as potential weakening of full total positivity; and in modular forms as asymptotic interlacing on a truncated geometric locus. The unifying content is not exact alternation itself, but the persistence of strong gap constraints after exact interlacing has partially broken.