Spectral Theory
Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
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Certified interval arithmetic proves a noncircular real-analytic domain with a boundary-constant Neumann eigenfunction.
· “A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures”
The weight is null-critical and fixes the sharp constant for n^{-2σ}.
· “An optimal fractional Hardy inequality on the discrete half-line”
A bifurcation branch on a real-symmetry relaxation reaches integer symmetries, refuting Schiffer and Pompeiu together.
Under spectral duality the coupling constant equals the classical midpoint value for any bias, any number of sites, and every resetting rate
The three-dimensional case settles the 1/6 conjecture for the Neumann–Poincaré operator.
· “A sharp isoperimetric inequality for the Neumann--Poincar\'e operator in every dimension”
Worst-case determinant-inverse loss grows as sqrt(e n), making Schäffer's 1970 upper bound optimal in constant.
· “Sch\"affer's matrix inequality: the exact asymptotic constant”
Proof resolves the maximizer conjecture for n≥123: every graph with 2(n−2) edges has algebraic connectivity at most 2.
· “Maximizing Algebraic Connectivity with 2(n-2) Edges: The Large Vertex Number Case”
A trace-class operator with trace π/2 shows a positive commutator need not fit Kato's sufficient condition.
· “A counterexample to the Kato conjecture for positive commutators”
Settles 60-year-old conjectures of Carl, Pietsch, Mityagin and Henkin up to the constant e.
· “On bounds between all s-numbers and widths of convex sets”
For every small diffusivity, one fixed velocity field produces an exponentially growing magnetic mode at a uniform rate.
Lacunary hopping weights make wave packets flatten like |t|^{-N} for every N, with no universal speed limit.
· “Arbitrarily Fast Quantum Dispersion in Long-Range Crystals”
Area below 8π unlocks Willmore minimizers of every topology in R^4.
· “Nonorientable Minimal Surfaces Embedded in the Round 4-sphere”
A one-dimensional solvable model proves positive defects trap modes below and negative defects above.
A new proof gives explicit remainder terms and uniform eigenfunction bounds for every fractional order between 0 and 2.
For Schrödinger operators with jump singularities, derivative jumps of the new indicator locate all resonance strings.
A coarse Poincaré inequality, not a manifold, is enough: error is 1/√(n v(h)) for simple eigenvalues.
· “Spectral stability of empirical metric-measure Laplacians”
Conformal ball metrics have positive Ricci and convex boundary, yet their first Steklov eigenvalue dips below 1.
The slope is the zero-frequency spectral density of the inverse sampling function; when it is positive, Anderson localization follows.
· “One-Dimensional Divergence-Type Jacobi Operators Driven by The Doubling Map”
A generalized spectral-averaging identity extends the rank-one spectral-shift phenomenon to infinite rank.
· “Analysis of the Singular Spectrum for General Perturbations”
A sequence of smooth planar domains with identical Steklov spectra always has a smoothly converging subsequence, settling a long-open…
· “The Steklov Determinant and Compactness of Isospectral Planar Domains”
A logarithmic saving (log T)^{-r/2} on thick quotients; strict o(T^{(n-r)/2}) locally.
A small interaction strength keeps the relaxation rate of the N-particle gas bounded away from zero.
A finite group's power graph is a friendship graph exactly when every nonidentity element has order 3.
· “Characterisations of finite groups with exponent q via their power graphs”
The side of accumulation flips at a critical coupling; the rate is set by the curve's logarithmic capacity.
· “The Landau-Dirac operator with shell interactions: self-adjointness and clustering”
For every invertible boundary condition on the power-kernel fractional Sturm–Liouville operator, the inverse's singular values follow…
One spectral condition replaces uniform discreteness; the previous multiplication-operator result is the zero-propagation case.
· “Spectral convergence of hyperbolic surfaces and quantum ergodicity”
Sharp bound: eigenvalue gaps never exceed sqrt(Lambda2 - Lambda1); counterexamples show geometry is essential.
Even one Fourier mode sees eigenvalues accumulate at all three thresholds from both sides, with explicit $t^{-1}$ counts.
Settles the limit multiplicity conjecture for thin hyperbolic manifolds, with a geometric rate from shadow returns.
For 1≤p≤2, φ(A,B) is controlled by input commutators; for p>2 even finite-rank counterexamples block any such bound.
· “Commutator estimates for functions of noncommuting self-adjoint operators”
Past separation N^{2/3}, two simultaneous extremes occur at most as often as the product of the individual chances.
· “One-Sided Product-Scale Upper Bounds for Nested Complex Wishart Extremes”
Rigorous proof at positive temperature ties the quantum many-body grand potential to a classical one.
· “A semiclassical limit of reduced Hartree-Fock theory at positive temperature”
Resolvent estimates and Weyl-invariant polynomials give a purely analytic route to the temperedness criterion.
· “Spectra of elliptic invariant differential operators on locally symmetric spaces”
If the characteristic polynomial has repeated factors, reducibility is checked by graph connectivity and a bounded extension tower.
· “Reducibility of linear representations, free ideals, and Kippenhahn's conjecture”
The overlap is a Legendre ratio that yields the unitary intertwiner and exact trace series.
· “Intertwining the line bundle and Grauert-tube Hardy quantizations of the round 2-sphere”
For a family of circle maps, decay is none, exponential, or double exponential, set by one multiplier.
· “Mixing for Free Semigroup Actions of Blaschke Products on the Circle”
For any lattice tile, the shortest dual vector and Cheeger constant obey sharp dimension-free lower bounds.
For a fixed set of growth rates, the bell arrangement raises the top eigenvalue and the front speed for every tilt.
A two-term residue trace proves the ball is the only 3-manifold with its Steklov spectrum
For star-shaped mean-convex domains in hyperbolic space, the equal-area ball has the largest first Steklov eigenvalue.
An exact determinant benchmark plus a new criterion fixes the O(1) constant that leading-order asymptotics miss
New proof: off-diagonal eigenfunction sums decay with energy-gap width, even on a flat torus.
· “Quantum mixing for eigenfunctions of rational polygons in configuration space”
The free complex Banach lattice extension of any bounded operator has a spectrum of one of just three shapes
· “The spectrum of operator extensions to free Banach Lattices”
For matrices with distinct eigenvalues, one program now outputs the complete list instead of brute-force search.
Proof settles the last open dimension, n=3, with a one-crossing slope comparison.
· “The Weinstock inequality for convex domains in hyperbolic space”
Simply connected, strictly convex, real-analytic drums that are not congruent but have equal Steklov spectra.
Graph limits, hypergraphs, manifolds and metric measure spaces obey the same inequality.
For fermionic wavefunctions the tail is even faster, k^{-10/3}, and both rates are fixed by the electron-electron cusp.
All Dirichlet eigenvalues of L^n_{π/p} are known; large-order Polya and sharp two-term asymptotics follow.
Sharpness q of an outward peak sets the leading energy scale, replacing linear growth on smooth and cornered domains.
· “Eigenvalues of the Magnetic Neumann Laplacian on Domains with Peaks”
A verified 13-fold symmetric region satisfies u=0 and ∂νu=1 on its boundary without being a disc.
· “A computer-assisted counterexample to the planar Berenstein conjecture”
Negative eigenvalues of a graph with short core and infinite leads leave at exactly two rates; vertex data counts them.
· “Two escape rates for negative eigenvalues of quantum graphs with a shrinking core”
Fiber modes decay exponentially; the Gaussian profile and every coefficient come from the Abelian cover's heat kernel.
For radial log-concave measures, the ball uniquely minimizes the sum of the first n reciprocal Witten-Laplacian eigenvalues.
· “An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures”
For the Hermite operator in two dimensions, the L2-to-L10/3 projection norm is O(lambda^{-1/10}) at every eigenspace.
· “Sharp Endpoint Eigenfunction Estimates for the Two-Dimensional Hermite Operator”
The oscillatory Helmholtz part dominates; the exponential boundary part shrinks as λ^{-1/4} and decays with distance.
· “A Vibrating Clamped Plate Resembles a Vibrating Membrane at High Frequency”
Spherical averaging turns the PDE into an ODE, yielding exponential lower bounds along a sequence of points going to infinity.
· “Unique continuation at infinity for potentials with arbitrary radial growth”
A Banach-space analogue of the Hilbert-space spectral theorem: eigenvalues stay real, the sum converges in operator norm.
· “Spectral theorem for compact self-adjoint operators on smooth Banach spaces”
Fixed-boundary case: no real wave numbers. Free-boundary case: one, $\alpha=0$, with four explicit modes.
· “Steady state oscillations of a half cylinder with a pseudo isotropic structure”
Spectral endpoints obey the same identity, and face-scale corrections keep shape memory beyond universality.
For every odd p≥3, a W^(2p-1,∞) potential gives a global classical KdV flow.
A survey of a recent proof shows typical surfaces mix as fast as their geometry allows, driving chaos.
Even near agreement with a ball's Steklov eigenvalues, not just exact equality, forces the boundary to be round.
Settles an open conjecture and extends the parity rule to all non-integer fractional orders.
The result settles a 2008 conjecture and adds two sharp inequalities to the short list of bounded determinantal ratios.
For weak constant fields, the square locally minimizes the lowest magnetic eigenvalue among equal-area rectangles.
· “Spectral inequalities for magnetic Neumann Laplacian on rectangles”
A volume-aware graph Laplacian keeps eigenvalues comparable even as surfaces pinch and collapse.
· “A Weighted Discretization of Riemannian Manifolds with Lower Ricci Bounds”
Exact formulas for the Bolza surface and Klein quartic fill a long-missing gap in low-genus spectral theory.
· “Spectral determinants of the Bolza surface and the Klein quartic”
Proof that spectra, Berry curvature and quantum metric vary while the topological class stays fixed.
· “Beyond K-Theory: Geometry and Holomorphy in Hyperbolic Band Theory”
First coarse Courant and Bézout bounds for eigenfunction sums, with exponent within ε of the homogeneous dimension.
Capacity of thin cylinders pins where eigenfunction growth stops; in 3D only a log-log factor is lost.
· “Capacity estimates and improved lower bounds for the inner radius of nodal domains”
Perturbations too singular to be functions or measures still leave a complete eigenfunction basis, for any power decay.
· “Eigensystems of Dirac operators with singular potentials”
Penetrable solenoids with a thin low-conductivity layer converge strongly to the magnetic Robin Hamiltonian.
· “Remarks on Regular Approximations to the Robin Aharonov-Bohm Hamiltonian”
Closed hyperbolic 3-manifolds of growing volume get two-torsion line bundles whose smallest Laplace eigenvalue tends to 1.
· “Near optimal spectral gaps for line bundles on hyperbolic three-manifolds”
Curved guides start the half-line at the lower of two thresholds; a negative 1D potential signals bound states.
· “Spectral Analysis of a Soft Quantum Waveguide Coupled to a Massive Quantum Scalar Field”
New error bounds on ℓ²(G,Y) make spectra of infinite matrices computable from finite pieces, with convergence as patches grow.
Large frequencies become lattice counts, low frequencies variational estimates; certified computation closes gaps in dimensions 3–12.
· “P\'{o}lya's conjecture for higher-dimensional Neumann balls”
A purely distributional Radon-transform framework absorbs the spectral singularities of ReLU-like activations, and under an even/odd…
· “Radon Measure Representations for Infinite-Width Neural Networks with Singular Activations”
New proof ties logarithmic L^p gains to the geodesic flow's singular values, going beyond nonpositive curvature.
· “Nonconcentration of eigenfunctions in Microlocal Kakeya-Nikodym norms: a phase space approach”
On complete graphs with n≥5, CNOT and B-gate match iSWAP at the second moment, independent of c.
· “Spectral gaps of ironed two-qubit gadgets matching the iSWAP gap”