Papers

We study of the long-term behavior of a coupled wave-heat system. The system consists of a wave equation and a heat equation on two adjacent Lipschitz domains coupled by a common interface, with the heat equation being allowed to incorporate spatially dependent coefficients. We first establish strong asymptotic stability independent of the domains and coefficients. To this end, we employ the framework of closure relations, which reduces the spectral analysis of the coupled system to that of a wave equation. Secondly, we analyze non-uniform decay rates for classical solutions to the coupled system. Using a non-orthogonal decomposition of the state space, we reduce the problem to a resolvent-type estimate which is independent of the heat domain and heat coefficients. With this, we extend the known non-uniform decay rates to spatially dependent heat coefficients, yielding logarithmic decay under no assumptions and polynomial decay under the Geometric Control Condition.

We analyse magnetic Robin Laplacian in curvilinear half-plane with a smooth boundary. It is well known that the spectrum of the Robin Laplacian is unstable with respect to boundary deformations. This means that if the boundary is a straight line then the spectrum of the Robin Laplacian is purely essential. From the other hand, the perturbation of the boundary produces eigenvalues below the essential spectrum. In this paper, the Robin-Laplace operator with a compactly supported magnetic field is considered. We prove that the spectrum of the magnetic Robin Laplacian is stable under small and local deformations of the boundary.

In this paper, we consider one-dimensional discrete Schrödinger operators \begin{align} Hu(n)=(\Delta+V)u(n)\nonumber \end{align} on $\ell^2(\mathbb{N})$ with a self-adjoint boundary condition at $n=0$, where $\Delta$ denotes the discrete Laplacian and $V(n)$ is a real-valued perturbation satisfying $$V(n)=\frac{O(1)}{1+n}.$$ We determine the sharp transition for the asymptotic coefficient of \(V\) governing the existence and nonexistence of embedded eigenvalues.

The exact N-soliton solutions are derived for the coupled negative first order Klein-Gordon (CNKG) equation subject to vanishing boundary conditions by using the inverse scattering method via Gelfand-Levitan-Marchenko equation. Based on the zero-curvature representation, the conservation laws, integrals of motion, and Hamiltonian structure of the aforementioned coupled nonlinear equations are constructed. The Jost functions and their analyticity properties for the Manakov spectral problem are recalled. The integral equations for the eigenfunctions are then used to formulate the Gel'fand-Levitan-Marchenko equations. Solving these equations establishes a direct correspondence between the kernel functions and the potential, yielding the general N-soliton expressions.

We prove a sharp comparison, with Obata-type rigidity, for all Neumann eigenvalues of one-dimensional $\mathrm{CD}(1,2)$ spaces against the Legendre model, under a convexity condition on the density. It follows that minimizers of the $k$-th Laplace eigenvalue among closed surfaces of Gaussian curvature at least $1$ cannot collapse in the measured Gromov-Hausdorff completion, for every $k\ge2$. We also give a variational proof that smooth minimizers are round.

We prove that among convex planar quadrilaterals of equal area, the square uniquely maximizes the first nonzero Laplace–Neumann eigenvalue. This is the quadrilateral case of the Neumann analogue of the long-standing Pólya–Szegő conjecture, which asserts that the regular $n$-gon minimizes the first Laplace–Dirichlet eigenvalue among $n$-gons of equal area. The proof uses a Rayleigh–Ritz lower bound, obtained from the span of the first five nonconstant Neumann modes of the square, on a smooth auxiliary functional whose minimization implies the original maximization. Local maximality at the square follows from a symmetry-reduced Hessian, computed in closed form, together with a certified second-order difference-quotient test that establishes the local inequality on an explicit ball around the square. Global maximality is then established by a certified box covering of the parameter space, certifying on each box a direct upper bound on the first Rayleigh–Ritz eigenvalue. Both certifications operate entirely on integrals over a fixed reference square and avoid a posteriori finite-element bounds on the perturbed quadrilateral.

Tipping points—abrupt, potentially irreversible reorganizations of a system's statistical state—are commonly anticipated through critical slowing down: recovery slows, autocorrelation and variance rise, and spectra redden. This paradigm is powerful near simple equilibrium bifurcations but is not a general theory for stochastic, multistable, or metastable systems. We develop such a theory from the Ruelle–Pollicott (RP) spectrum of Kolmogorov generators of hypoelliptic Itô diffusions. Resolving the sensitivity of invariant statistics on RP spectral blocks shows that each contribution factorizes into a spectral denominator and a residue coupling the block to both the observable and perturbation direction. Small denominators permit large responses; residues produce them. Thus a closing RP gap is not sufficient for an early warning, while growing residues can generate the classical signature with no gap closure. An early-warning signal is therefore a property of a triple: RP block, observable, and perturbation direction. We demonstrate this on a stochastic non-normal system whose RP spectrum is exactly frozen, yet classical indicators become more alarming than during genuine gap closure. The RP decomposition attributes this to residue growth and yields an index $\mathcal{N}(f)$ satisfying $\mathcal{N}(f)\leq 1$ for reversible dynamics; hence $\mathcal{N}(f)>1$ certifies residue-driven amplification. For metastable systems, one killed problem yields two complementary spectral objects: the Doob $Q$-process isolates in-well recovery, while escape clocks and committor-weighted destination probabilities govern interwell transitions. In a one-dimensional fold, these scale as $(\epsilon_c-\epsilon)^{1/2}$ and $(\epsilon_c-\epsilon)^{3/2}$, separating bifurcation-induced from noise-induced tipping. Doob drift also connects to optimal Girsanov sampling.

We study three identical bosons with contact attraction of strength mu on the square lattice at total quasimomentum K = 0 and K = pi. First, at K = 0, for all sufficiently large mu there are exactly two eigenvalues below the essential spectrum. The trimer branch follows from Kato perturbation theory at the simple isolated eigenvalue 3 of the pair-contact operator, uniformly in K; the atom-dimer branch follows from a Feshbach-Schur reduction whose limiting equation coincides with the second Birman-Schwinger (Faddeev) eigenvalue of the finite-rank principal part. The spectral gap between the two branches grows linearly in mu. Second, at K = pi the trimer branch is unchanged up to O(mu^-2), the two-particle threshold equals -mu + 4 exactly, and the gap to it is 2mu - 2 plus lower order corrections. The second bound state disappears because the exchange operator changes sign, S_pi = -S_0: the s-channel eigenvalue that exceeds 1 at K = 0 becomes negative, while the only positive atom-dimer channel at K = pi is the odd p-channel, bounded by 2 - 4/pi < 1. Consequently the operator has exactly one eigenvalue below the threshold minus 36/(mu - 10). Third, the ground state is exponentially localized in the relative coordinates. The decay exponent is positive unconditionally; under a non-cancellation hypothesis it is bounded above by log(3mu) plus lower order terms, while finite-volume computations indicate the slowest tail is the atom-dimer dissociation channel with decay rate one half log(2mu). A dictionary between the principal-part method, Kato theory, and the atom-dimer threshold matrix is given, together with the three-dimensional constants for comparison.

We consider eigenvalue counting functions of Robin Laplace operators on cuboids where the Robin parameter and the spectral cut-off $\lambda$ are coupled. Our main focus is the critical regime, in which the Robin parameter is proportional to $\sqrt{\lambda}$. We obtain a two-term asymptotic expansion for the counting functions in this coupled setting. In the critical regime, the second term in the asymptotic expansion depends non-trivially on the proportionality constant and interpolates continuously between the corresponding second terms for the Dirichlet and Neumann Laplacians. Outside the critical regime, one recovers the corresponding Dirichlet or Neumann asymptotics. We also establish a Pólya-type inequality for the counting function whenever the ratio of the Robin parameter and $\sqrt{\lambda}$ exceeds a dimension-dependent threshold.

We investigate inverse scattering at a fixed energy for semiclassical Schrödinger operators with smooth potentials. For compactly supported potentials satisfying a natural virial condition, we prove that $ \liminf_{h\to0} \|S_1(\lambda,h)-S_2(\lambda,h)\|<\sqrt{2} $ implies $V_1=V_2$, provided the fixed energy $\lambda$ lies above both potentials. We then consider radial short-range repulsive potentials. Under a monotonicity assumption on the radial force, we show that the semiclassical differential cross section at a single fixed energy, up to $o(1)$ as $h\to0$, determines the potential throughout the classically accessible region. Finally, we obtain an analogous rigidity result for nontrapping compactly supported perturbations of the Euclidean metric under a strict convexity assumption.

We derive optimized upper bounds for the first Dirichlet eigenvalue of a bounded domain under lower bounds for the conformal Ricci tensor of Shen and Ye. The method also yields sharp estimates for the bottom spectrum on complete noncompact manifolds, a four-dimensional application involving $Q$-curvature, and local and global spectral-Ricci extensions of Cheng's estimate. As a consequence, we obtain a sharp comparison between the spectra of the Laplacian and the conformal Laplacian in terms of the smallest Schouten eigenvalue.

Continuous-time quantum walks on a lattice spread ballistically and converge to a limiting distribution for the rescaled position. On the hard-wall half line the boundary reflects the walker but does not change the bulk dispersion, so the ballistic front remains set by the maximal group velocity. We ask how close one can get to this front when the initial state is constrained to occupy only the first $M$ sites. We show that optimizing the moment-generating function of the limiting-velocity distribution over this finite-support class reduces to the principal eigenvalue of an explicit $M\times M$ Hermitian matrix, which yields the optimal state by direct diagonalization. For large $M$, a near-front scaling limit produces a continuum description that controls the entire peak region. In particular, the maximal mean drift approaches the front with an inverse-square deficit in $M$ and a constant prefactor $\pi^2/4$. Numerical results verify both the finite-$M$ spectral formulation and the predicted scaling.

We prove a Faber-Krahn inequality for the first eigenvalue of the $p$-Hermite operator (the weighted $p$-Laplacian with Gaussian weight) on Lipschitz domains in $\R^n$ under Robin boundary conditions with positive Robin parameter. The main result states that, among all domains of given Gaussian measure, the first eigenvalue is minimized by a half-space, and equality holds only for half-spaces. This extends the classical Faber-Krahn inequalities for the $p$-Laplacian \cite{BucurCV} to the $p$-Hermite operator and generalizes the linear case \cite{ChiacchioMathann} to the full nonlinear regime $p>1$.

Let $\lambda_1(\Omega)$ and $T(\Omega)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $\Omega\subset\mathbb{R}^2$, and let $M=\max_\Omega u$, where $u$ is the torsion function. We prove the sharp inequalities $\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$. This proves the two-dimensional case of Conjecture 4.2 proposed by van den Berg, Buttazzo, and Pratelli. Its planar formulation was later restated as Conjecture 1.1 by Bañuelos and Mariano, who proved it for triangles and rectangles. The lower bound follows by combining Payne's strict estimate for $\lambda_1M$ with the sharp torsion-efficiency inequality $T(\Omega)\geq |\Omega|M/3$. For smooth strictly convex domains, we use an Airy stress potential to construct a convex body whose surface-area measure is the torsional first-variation measure. A sharp one-dimensional inequality for directional maximum profiles then yields a stronger geometric containment. For the upper bound, we factor the Pólya functional through the second torsion moment. A level-set torsion–perimeter inequality gives the factor $5/6$, while a sharp weighted one-dimensional estimate gives the factor $\pi^2/10$. Collapsing triangles and elongating rectangles show that both endpoint constants, as well as the efficiency constant $1/3$, are sharp.

We construct a real self-adjoint semigroup on an $\ell^2$-space which is individually eventually strongly positive with respect to a strictly positive vector, but whose operators fail to be positive at arbitrarily large times. Moreover, the generator of this semigroup has compact resolvent. Consequently, individual and uniform eventual positivity are not equivalent for self-adjoint semigroups on $L^2$-spaces, even under the additional assumption that the generator has compact resolvent. The construction consists of a positive self-adjoint rank-one perturbation of a diagonal operator and a sequence of small spectral shifts on antisymmetric two-point modes.

We investigate the discrete spectrum of finitely many point interactions for Neumann and Robin Laplacians on special unbounded and exterior $C^{1,1}$ domains in dimensions two and three. The operators are realized as self-adjoint extensions through an ordinary boundary triple whose gamma field and Weyl matrix are constructed from the Robin Green kernel. Eigenvalues below the background spectrum are characterized by the Weyl matrix, yielding an exact finite-dimensional counting formula. If the background operator is non-negative, this also gives the number of negative eigenvalues without assuming a finite zero-energy limit of the Weyl matrix. In the one-centre case we identify the critical coupling and prove that the unique eigenvalue branch is real analytic, strictly increasing, and strictly concave. For scalar multicentre couplings $\Theta=\alpha I_N$, sufficiently strong attraction produces exactly $N$ eigenvalues below the background spectrum, and all of them have universal leading asymptotics coinciding with the whole-space laws. If the bottom of the background spectrum is an isolated eigenvalue, the branch exists for every finite coupling and we determine its leading decoupling asymptotics as the coupling tends to $+\infty$. Explicit exterior-sphere and exterior-disk models illustrate the critical couplings and their threshold behaviour.

We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $\delta \in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^\delta := \left( \frac{\sin((1+\delta)\pi)}{\pi(j+k+1+\delta)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbert matrix. For $\delta$ not a half-integer, we prove the large-$N$ power-law asymptotics of the determinant \[ \log \text{det}(I_N \pm H_N^\delta) = -\frac{\theta_\delta^2 \pm \theta_\delta}{2} \log N + O(1) \] where $\theta_\delta := \delta$ if $\delta < -1/2$, and $\theta_\delta := \frac 1 \pi \arcsin(\sin(\delta\pi))$ if $\delta \ge -1/2$ and $\arcsin \colon [-1,1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]$ denotes the principal branch of $\arcsin$. For $\delta \geq -\frac 12$ the asymptotics is known. The novelty of the present paper is the regime $\delta < - \frac 12$ and understanding the dichotomy in the decay exponent. This stems from the $\pm 1$ edge eigenvalues of the limiting operator appearing for $\delta < - \frac 12$. Most notably, we obtain for $\delta < -\frac12$ \[ \log \det\bigl(I_N - (H_N^\delta)^2\bigr) = -\delta^2 \log N + O(1). \] Such determinants arise in the study of Anderson's orthogonality catastrophe.

The chiral limit of the Bistritzer–MacDonald model for twisted bilayer graphene has exactly flat bands at magic angles. We show that exact flatness can persist beyond the chiral limit with nonchiral tunnelling potentials in the usual symmetry class. We construct a family of nonchiral tunnelling potentials for which the Hamiltonian has at least two zero-energy states at every Bloch momentum and for every real value of the nonchiral coupling. This gives a family of counterexamples to Open Problem~3 of Zworski's survey \cite{ZworskiSurvey}. Conversely, we prove that every admissible potential with this property belongs to this family, thereby obtaining a complete characterization. By contrast, we show that the standard BM model, namely, BM Hamiltonians with first harmonic tunnelling potentials, does not admit exactly flat bands at the magic angle except possibly at a discrete set of nonchiral coupling strengths, with no finite accumulation point. To prove this, we introduce an explicit invertibility criterion. The criterion requires the projected perturbation to have a nonzero scalar coefficient at some Bloch momentum, obstructing exact flatness for general nonchiral tunnelling. A local Taylor expansion argument verifies the criterion for the standard BM potentials. For the family constructed above, however, the projected perturbation vanishes at every Bloch momentum, so the invertibility criterion fails.

In this paper, we consider the Rosenthal – Halmos problem of almost commuting matrices with respect to the normalized Hilbert – Schmidt norm $\|\cdot\|_{2,d}=d^{-1/2}\|\cdot\|_2$. We show that if $X$ and $Y$ are self-adjoint matrices with $\|X\|\le 1$, $\|Y\|\le 1$, then there exist commuting self-adjoint matrices $X',Y'$ such that $\|X-X'\|_{2,d}+\|Y-Y'\|_{2,d}\le 5\|[X,Y]\|_{2,d}^{1/3}$, and $[X,X']=0$. Within these constraints, the exponent $1/3$ cannot be improved.

Sharp Hardy–Rellich constants are spectral thresholds for operators with critical inverse-power potentials. Let $C_\ell(N)$ denote the optimal constant in the $\ell$th-order discrete Hardy–Rellich inequality on $\Z^N$. Recent independent work established the leading high-dimensional behavior $C_\ell(N)\sim2^\ell N^\ell$ for every fixed $\ell$. We determine the next two orders and prove \[ C_\ell(N)=2^\ell N^\ell+\gamma_\ell N^{\ell-1} +\delta_\ell N^{\ell-2}+O_\ell(N^{\ell-3}), \] with explicit coefficients $\gamma_\ell$ and $\delta_\ell$. The central difficulty is a degeneracy that grows with the dimension: after conjugation, the leading operator is scalar on the $2N$ nearest neighbours of the origin. We resolve this cluster using signed-permutation symmetry and an effective operator on four lattice orbits. Weighted torus estimates and Feshbach–Schur reduction justify the finite-dimensional expansion inside the full operator, while a residual bound and Temple's inequality give the stated remainder. In particular, $C_1(N)=2N-4-20/(3N)+O(N^{-2})$.

In this paper, we establish isodiametric-type upper bounds for Steklov eigenvalues of bounded Lipschitz domains $\Omega\subset \mathbb{R}^n$, $n\geq 2$, valid for every $k\in \mathbb{N}$: \[ D(\Omega)\sigma_k(\Omega)\leq C(n)k, \] \[ D(\Omega)\sigma_k(\Omega)\leq C(n)k^2\left(\frac{|\Omega|^{\frac{1}{n}}}{D(\Omega)}\right)^{\frac{n}{n-1}}, \] where $C(n)>0$ depends only on $n$. The first estimate is sharp in its linear dependence on $k$, while the second is sharp both in the exponent of the volume-diameter ratio and, once that exponent is fixed, in its quadratic dependence on $k$. In particular, the second estimate gives an affirmative answer to Open Question 4.37 in [5].

We investigate the eigenfrequency asymptotics of a bounded acoustic cavity containing a shrinking high-contrast inclusion. This configuration is motivated by contrast-enhanced ultrasound imaging, in which microbubbles are employed as acoustic contrast agents. We identify dimension-dependent material scalings that keep the inclusion-induced resonance in a fixed order-one frequency regime as the inclusion shrinks. Our main result gives a complete asymptotic description of the spectrum in any prescribed bounded frequency window. Two distinct spectral mechanisms arise: the background Dirichlet eigenfrequencies persist as perturbed eigenvalue clusters, while the singular material contrast creates an additional Minnaert eigenvalue branch. The limiting Minnaert frequency is explicit in both dimensions, with a capacitance-based expression in three dimensions and an area-based expression in two dimensions. Two-dimensional numerical experiments confirm both the perturbation of the background Dirichlet eigenfrequencies and the emergence of the additional Minnaert branch.

We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $\delta$-shell interaction of strength $\tau\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $\Sigma\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $\Sigma$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $\Sigma$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $\tau$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.

We prove the refined Koornwinder–Kostenko–Teschl conjecture. For the normalized weighted Jacobi function and all $n\in\mathbb N\_0$, $\alpha,\beta\ge0$, and $-1\le x\le1$, $$ |g\_n^{(\alpha,\beta)}(x)| \le \left[ \frac{(n+1)(n+\alpha+\beta+1)} {(n+\alpha+1)(n+\beta+1)} \right]^{1/4}\le1. $$ The proof combines a central contour estimate with Sturm–Sonin localization and an exact inverse moment. It also yields an $n$-uniform extreme-lobe theorem and a sharp canonical-product first-lobe principle. Applied to the discrete Laguerre evolution, the estimate gives the optimal positive-parameter decay. For $-1<\alpha\le0$, a complementary one-sided Jacobi inequality gives the exact norm $\|\mathrm{e}^{-\mathrm{i} tH_\alpha}\|_{\ell^1\to\ell^\infty} =(1+t^2)^{-(1+\alpha)/2}$. Thus the large-time decay exponent is $\min\{1,1+\alpha\}$ for every $\alpha>-1$. Bessel, Laguerre, and Darboux scaling limits show that the temporal and fixed-diagonal exponents are optimal.

For $N\geq 3$ and a potential phase $\vartheta\in\mathbb{R}$, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle $\varphi\in\mathbb{R}$, $A_N(\varphi,\vartheta)=e^{i\varphi}(S+S^{-1})/2+\operatorname{diag}(\cos(2\pi j/N+\vartheta))_{j\in\mathbb{Z}/N\mathbb{Z}}$, where $S e_j=e_{j+1}$ is the periodic shift on $\mathbb{C}^N$. We derive a Chambers formula and isolate the part $Q_{N,\varphi}$ of the characteristic polynomial that depends only on $N$ and $\varphi$, but not on $\vartheta$ or on a change of boundary conditions for the shift operator. We then show, for every $N$, that the zeros of $Q_{N,\varphi}$ lie on the two perpendicular lines $e^{i\varphi/2}\mathbb{R}\cup e^{i(\varphi/2+\pi/2)}\mathbb{R}$. For even $N$, the same property holds for the matrices $A_N(\varphi,\vartheta)$ with $\vartheta\in 2\pi\mathbb{Z}/N$, and we compute their limiting eigenvalue measure explicitly. For $\varphi\in[-\pi,\pi]$, the eigenvalue distribution approximates elliptic-integral densities with masses $1-|\varphi|/\pi$ and $|\varphi|/\pi$, and maximal radii $2|\cos(\varphi/2)|$ and $2|\sin(\varphi/2)|$, respectively. At $\varphi=\pi/2$, the central polynomial $Q_{N,\varphi}$ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.

For periodic Schrödinger operators $H_V=-\Delta+V$ with bounded real-valued potentials on $\mathbb R^d$ with $d\ge2$, we show that for generic potentials, each endpoint of every spectral gap is attained by a single Bloch band at only finitely many quasimomenta, and has a nondegenerate Hessian at every attaining point. This proves the Spectral Edge Conjecture for periodic Schrödinger operators.

We study the high-energy spectral effect of Robin boundary perturbations on complete Laplace eigenspaces and its consequences for eigenfunction non-localization. Highly degenerate Neumann levels generally split under Robin perturbation, but this splitting need not produce a simple spectrum: distinct modal classes may still contribute to the same Robin eigenvalue. We prove that, for every fixed positive Robin parameter, the number of modal classes contributing to any one eigenvalue is uniformly bounded over the spectrum on the equilateral triangle and on every rectangle with rational squared aspect ratio. On the equilateral triangle, for all sufficiently small Robin parameters, the Neumann complete-eigenspace observation estimate persists with a constant uniform both in the eigenvalue and in the boundary parameter. For an arbitrary fixed positive Robin parameter, a high-frequency shell-splitting analysis gives a spectrum-wide bound on the number of modal classes contributing to any one Robin eigenvalue. In the nonsquare rectangular case, the corresponding splitting is governed by a strongly convex profile on weighted quadratic shells. Since each modal class has uniformly boundedplane-wave complexity, the spectral complexity bound implies observation on every measurable set of positive measure through a multidimensional Turán–Nazarov inequality, without a frequency-separation assumption. Thus, spectral simplicity is not required for complete-eigenspace non-localization: uniformly bounded spectral coincidence complexity is sufficient.

We introduce and study equivariant analytic spectral invariants associated with the Aç{\i}kmeşe lift of a graph with self-loops $G_S$. The canonical $\mathbb{Z}_2$-action yields an orthogonal isotypic decomposition of the lifted Laplacian into the anti-symmetric part $\mathcal{L}(G_S)$ and the symmetric part $M_{sym}$, from which we prove that the spectrum of $\mathcal{L}(G_S)$ is exactly the even-indexed spectrum of the lifted Laplacian. We express twisted moments of these blocks as traces of the generalised twisted moment operator against the corresponding $\mathbb{Z}_2$-projections, and obtain a tight upper bound under certain restriction with a characterization of the equality case. We introduce the equivariant heat character and establish a trace-norm stability estimate. For the matrix element of the resolvent $(pI+\mathcal{L}(G_S))^{-1}$ associated with the loop vector, we derive a Laplace-transform identity for the equivariant heat character, a determinant formula, a spectral representation, and a Laurent expansion. We further characterize the case where $S$ is a union of connected components through resolvent and equivariant heat-character identities, and obtain explicit equivariant heat characters for joins of full-loop graphs and the line graph of a full-loop connected regular graph. Finally, we introduce the regularized equivariant heat integral and derive spectral and trace formulas for it.

For the bilaplacian in spherical and hyperbolic spaces, clamped vibrating and buckling eigenvalues on geodesic disks are shown to satisfy scaling inequalities analogous to the standard scale invariance of Euclidean plate eigenvalues. These results extend curved-space scaling results from the Laplacian to fourth-order eigenvalue problems. In addition, the limiting behavior of the scaled eigenvalues is determined as the disks expand to fill the ambient space. Depending on the geometry and the choice of scaling, the eigenvalues tend to zero, positive or negative infinity, or finite values associated with the bottom of the hyperbolic spectrum. The negative divergence in one spherical buckling limit is related by conformal inversion to an exterior buckling problem.

Dynamical localization requires all position moments of a quantum wavepacket to remain bounded in time, but for quasiperiodic Schrödinger operators such bounds are generally not uniform in phase. In the positive Lyapunov exponent regime, the best known phase-uniform estimates instead grow on a logarithmic scale. We prove that both the logarithmic scale and the dependence on the moment order are sharp for a class of one-frequency quasiperiodic Schrödinger operators with even potentials. Our main ingredient is a reflective version of semi-uniformly localized eigenfunctions, adapted to the two localization centers forced by a completely resonant phase, from which we obtain matching logarithmic lower bounds along sequences of times.