Papers

Pseudorandom structures are structures that behave like random ones, without necessarily being random themselves. It is a prominent, and evidently very difficult, conjecture that $$ V(n) = \lambda \cos \Big( 2 \pi \Big( x_1 + n x_2 + \frac{n(n-1)}{2} \alpha \Big) \Big) $$ is pseudorandom from a Schrödinger operator perspective in that the Schrödinger operator in $\ell^2(\mathbb{Z})$ with potential $V$ displays Anderson localization, that is, pure point spectrum with exponentially decaying eigenfunctions for almost all parameter values — the same spectral features as those produced by random potentials. We show that $V$ is pseudorandom in terms of its diffraction properties, that is, the associated diffraction measure is purely absolutely continuous for all $\lambda \not= 0$, all irrational $\alpha$, and all $x_1,x_2$ — which is the case as well for the random case. Our result gives further evidence for the conjecture in the Schrödinger case and it elucidates the apparent dual behavior of Schrödinger spectral measures and diffraction measures.

We prove a similarity result for non-self-adjoint non-elliptic quadratic $\mathcal{PT}$-symmetric operators with real spectrum satisfying a dynamical averaging condition. In particular, we show that such an operator is weakly similar to a self-adjoint operator precisely when the fundamental matrix of the associated quadratic form is diagonalizable.

These lectures are aimed at junior researchers seeking to enter the field of applications of complex analysis to harmonic analysis, Fourier analysis, spectral and scattering theory for differential operators and related fields. Starting with classical completeness problems posed by Bernstein, Beurling and Malliavin, Wiener, Kolmogorov and Krein, the course progresses to the solutions of the Gap and Type Problems, inverse spectral problems for Schrödinger operators and canonical Hamiltonian systems, and pointwise convergence of the non-linear Fourier transform. Each lecture is supplemented with exercises and references.

In this paper, we provide a general Sunada–Pesce–Sutton-type method that produces pairs of isospectral metric measure spaces. This construction relies on a representation theoretic assumption, and we give examples of spaces satisfying this condition, such as RCD spaces. As an application, we construct a pair of simply connected isospectral, non-isometric RCD non-Alexandrov spaces and a pair of Alexandrov non-orbifold spaces.

Perturbed plane waves are fundamental objects in scattering theory on Euclidean space and asymptotically Euclidean spaces. In this paper, we investigate the structure of perturbed plane waves in the $\textit{forward}$ direction, in which the outgoing spherical wave is typically singular and conjoined to the incoming plane wave. Melrose & Zworski provided a microlocal description (in the more general setting of asymptotically conic manifolds) using their notion of Lagrangian distributions associated to pairs of intersecting Legendrian submanifolds, on the way to proving that the S-matrix is an FIO. Here, we revisit the problem in the asymptotically Euclidean case, for which the oscillatory integrals used by Melrose–Zworski attain their most complicated form (relative to the more general asymptotically conic case). We seek a more elementary description in terms of physical-space asymptotics. These are specified using a two-faced compactification $X\hookleftarrow \mathbb{R}^d$, with one face for each asymptotic regime. We prove full polyhomogeneity. A transport equation arises as a model problem at the main face (`bf'). The quantum inverted harmonic oscillator arises as a model problem at the front face (`ff').

We determine the complete eigenvalue region of the following Markov chains on $2q$ states, for $q\ge2$. The states are $A_0,\ldots,A_{q-1}$ and $B_0,\ldots,B_{q-1}$. Each transition $A_j\to A_{j+1}$ and $B_j\to B_{j+1}$, for $0\le j<q-1$, has probability one. From $A_{q-1}$, the chain moves to $A_0$ with probability $a$ and to $B_0$ with probability $1-a$. From $B_{q-1}$, it moves to $B_0$ with probability $b$ and to $A_j$ with probability $(1-b)p_j$. Here $a,b\in[0,1]$ and $p_j\ge0$ with $\sum_{j=0}^{q-1}p_j=1$; these parameters vary over all permitted values. Every segment from zero to an attainable eigenvalue lies in the region. Its unit-circle points are exactly the roots of unity of order at most $2q$, as for unrestricted stochastic matrices of the same order, yet the full region is strictly smaller. It excludes $0<|\operatorname{Arg}\lambda|<\pi/q$. For odd $q$, the final nonreal boundary arc ends at a negative real point inside the unit disk; the remaining interval to $-1$ is also boundary. Every transition matrix has the same characteristic polynomial as one with $a=b$. Among $q+1$ fixed matrices in this balanced family, convex combinations of at most three realize every attainable eigenvalue, and two suffice on the boundary. Both bounds are sharp within this set. Consecutive reduced fractions in $[1/(2q),1/2]$ with denominators at most $2q$ select the upper nonreal boundary constructions. Each boundary radius is the unique solution of an equation whose left-hand side increases strictly with the radius.

We study linear cocycles over ergodic translations of the two-torus acting on Hermitian vector bundles described by unitary sewing matrices. For a continuous invariant complex line subbundle $L$, we prove that $c_1(L) \ne 0$ obstructs every nonzero measurable eigensection under a measurable phase-coboundary condition. The abstract obstruction holds in every rank, while for rank-two analytic cocycles with a dominated splitting over a Diophantine translation the phase condition is automatic. In that regime the criterion is exact: $L$ carries a nonzero measurable eigensection if and only if $c_1(L)=0$ and the winding vector of the multiplier phase vanishes. When these conditions hold, the realized eigenvalues form a dense coset on a circle; for each such eigenvalue, the eigenspace carried by $L$ is one-dimensional and has a real-analytic nowhere-vanishing generator. We also construct a Liouville example showing that the Diophantine hypothesis cannot be removed. The proof converts the Chern number into a magnetic charge on the universal cover and applies covariance for irrational magnetic translations.

In this thesis, we study the spectral properties of dynamically defined aperiodic metric and discrete graphs. Our goal is to determine to what extent spectral properties of discrete one-dimensional ergodic Schrödinger operators persist when the aperiodicity is manifested through the geometry rather than through a potential. The graphs considered here are inspired by one-dimensional aperiodic tilings, and are called tiling graphs and decorated $\mathbb{Z}$-graphs. For a large family of metric tiling graphs equipped with the standard Laplacian, we show that the spectrum is of zero Lebesgue measure, and is a generalized Cantor set up to a possible discrete set of energies. For decorated $\mathbb{Z}$-graphs, we further show that for a Baire-generic and Lebesgue almost-sure choice of the decoration edge lengths, the spectrum is a generalized Cantor set. We then study the integrated density of states (IDS) for metric and discrete decorated $\mathbb{Z}$-graphs. We prove a gap labelling theorem, which characterizes the set of possible values taken by the IDS inside spectral gaps. We show that the gap labels are contained in the Schwartzman group associated with the dynamical system generating the graph, up to a geometric scaling factor. Lastly, we consider the Dry Ten Martini Problem for discrete Sturmian decorated $\mathbb{Z}$-graphs, asking whether all possible values predicted by the gap labelling theorem are indeed attained by the IDS inside spectral gaps. We answer this question negatively, by identifying a large set of gap labels which are not attained due to jump discontinuities of the IDS. We then show that away from these jump discontinuities, the periodic approximants for Sturmian graphs display the same combinatorial structure as the standard Sturmian Hamiltonians, and use this to obtain an explicit characterization of the realized gap labels for Sturmian comb graphs.

A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian Zikang Deng Levitin and Yagudin conjectured in Conjecture 4.3 of LMS J. Comput. Math. 6 (2003) that, if the third Dirichlet eigenvalue of a connected planar domain is simple, then not all nodal lines of a corresponding eigenfunction can be closed. We construct a counterexample: a bounded connected planar domain with smooth boundary whose third Dirichlet eigenvalue is simple and for which the entire interior nodal set of every corresponding nonzero real eigenfunction is compactly contained in the domain. In fact, the nodal set consists of exactly two disjoint real-analytic simple closed curves. Starting from the Payne-conjecture counterexample of Dahne, Gomez-Serrano, and Hou, we connect two reflected copies by a symmetric thin neck. Reflection splits the spectrum into mixed Dirichlet-Neumann and Dirichlet spectra on a half-domain. A transverse Poincare inequality and the min-max principle yield convergence of both spectra to the base-domain spectrum, while boundary unique continuation gives strict separation at equal indices and places the even second branch at the third spectral position. Local eigenfunction convergence preserves strict sign inequalities; Courant's theorem confines the nodal set; and an Euler count determines its topology. A smooth inner exhaustion completes the construction.

We construct the first examples of one-dimensional long-range crystals exhibiting arbitrarily fast polynomial dispersive bounds. The Floquet functions of our Hamiltonians are highly oscillatory Weierstrass functions, whose rough autosimilar structure drives the fast dispersion. The proof develops a new Fourier decay theory for $C^\alpha$ images of the Lebesgue measure, based on a Dolgopyat-inspired transfer operator method, and yields a van der Corput lemma for Weierstrass functions. As a consequence, the local time of classical Weierstrass functions of sufficiently large lacunarity exists and is $C^k$, answering a question raised by Geman and Horowitz in 1980.

For a graph \(G\) admitting a real symmetric realization with exactly two distinct eigenvalues, \(MB(G)\) is the minimum, over all such realizations, of the smaller of the two eigenvalue multiplicities. Adm, Fallat, Meagher, Nasserasr, Plosker, and Yang asked for this parameter for the complement of a path on at least eight vertices. We answer their question completely by proving $$ MB(\overline{P_n})=3 \qquad (n\ge 6). $$ In particular, this resolves the previously unresolved orders \(n\ge 9\) divisible by three. The proof is exact and constructive. We exhibit six vectors in \(\mathbb{R}^3\) whose mutual inner products vanish exactly for consecutive indices, whose rank-one outer products form a basis of \(\mathbb{S}^3\), and which admit a strictly positive Parseval scaling. An elementary absorption lemma then permits any finite faithful orthogonal extension of this vector chain to be added with small positive weights while the six original weights are corrected to retain the Parseval identity. The resulting Gram matrix is a rank-three orthogonal projection in \(\mathcal{S}(\overline{P_n})\). A local two-dimensional orthogonality obstruction gives the matching lower bound. For completeness, we include self-contained proofs of the exceptional small orders: \(MB(\overline{P_3})=1\), whereas \(q(\overline{P_4})=4\) and \(q(\overline{P_5})=3\).

In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $\Delta$, we show that the second-smallest eigenvalue of its Laplacian matrix satisfies $\lambda_2(L_G)\lesssim\Delta\frac g n$. We also obtain an improved eigenvalue bound for $K_h$-minor-free graphs of $\lambda_2(L_G)\lesssim\Delta\frac{h^2(\log h)^2}n$. In fact, our results directly prove much stronger results for reweighted eigenvalues, including higher reweighted eigenvalues. As a consequence, we obtain bounds not just on Laplacian eigenvalues, but also on normalized Laplacian eigenvalues and Steklov eigenvalues. Our results for genus-$g$ graphs are optimal for all of these kinds of eigenvalues, while our results for $K_h$-minor-free graphs are optimal up to $\log(h)$ factors. Our techniques for genus-$g$ graphs bootstrap bounded-degree bounds of normalized eigenvalues for entire classes to bounds for reweighted eigenvalues for the same classes without the bounded-degree limitation, while our techniques for $K_h$-minor-free graphs generalize an argument of Korhonen and Lokshtanov, making use of the Lovász local lemma.

In this paper, we consider the spectrum of the one dimensional non-self-adjoint Dirac operator with matrix-valued coefficients whose entries are PT-symmetric periodic functions. We prove that the spectrum of the considered operator contains a large part of the real axis.

We consider a Toeplitz realisation of the fractional discrete Laplacian $(-\Delta)^{\alpha}$ on the half-line $\mathbb{N}$ as a compression of the full-line fractional discrete Laplacian to $\ell^{2}(\mathbb{N})$. For all $\alpha>0$, we prove that the fractional Hardy inequality $$(-\Delta)^{\alpha}\geq\frac{4^{\alpha}\Gamma^2(\alpha+1/2)}{\pi}\frac{\Gamma(2\,\cdot\,-1)}{\Gamma(2\,\cdot\,-1+2\alpha)}$$ holds on $\ell^{2}(\mathbb{N})$ and is optimal in a strong sense. In particular, we show that the inequality cannot be improved and equality is not attained by any nonzero element of $\ell^{2}(\mathbb{N})$. As a consequence, we deduce a fractional generalisation of the discrete Birman inequality.

In this paper, we obtain the dispersive estimates and global well-posedness of the nonlinear magnetic Schrödinger equation in $\mathbb{R}^{3}$ with nonlinearity $|u|^{p-1}u$ with exponent $\frac{5}{3}< p<5$ when the initial value stays in a suitable space $\Sigma_{s}$. By proving the resolvent estimates with weight functions and the \textquotedblleft almost equivalence\textquotedblright between $(-\Delta_{A})^{\frac{s}{2}}$ and $(-\Delta)^{\frac{s}{2}}$, we obtain the Strichartz estimates of $|J_{A}(t)|^{s}u:=e^{\frac{i|x|^{2}}{4t}}(-t^{2}\Delta_{A})^{\frac{s}{2}}e^{\frac{-i|x|^{2}}{4t}}u$, therefore the dispersive decay estimates are obtained.

This paper studies some fundamental problems in spectral geometry in the $p$-adic setting. By viewing the Vladimirov-Taibleson operator $D^{\alpha}$ as the $p$-adic counterpart of the fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ in the Archimedean setting, we prove Weyl's law for the Dirichlet operator and establish it for the Neumann operator outside an exceptional set with zero Lebesgue asymptotic density. For the Dirichlet operator, we derive the sharp estimate for the remainder and prove that the Weyl-Berry conjecture fails. Furthermore, we show that Pólya's conjecture fails in general, and we give geometric necessary and sufficient conditions for it to hold.

We prove that for any given frequency resonance exponent and phase resonance exponent, there exist a frequency and a phase exactly realizing these values such that the Almost Mathieu operator exhibits Anderson localization for $\ln|\lambda|>\max\{\beta(\alpha),\delta(\alpha,\theta)\}$. This resolves the conjecture in \cite{MR4686650}.

Let $H_0$ be the standard discrete Laplacian on $\mathbb Z^d$, $d\geq4$, let $R_0(z)=(H_0-z)^{-1}$, and let $p'$ denote the Hölder conjugate of $p$, with $p'=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from $\ell^p(\mathbb Z^d)$ to $\ell^{p'}(\mathbb Z^d)$ by proving local Fourier-decay bounds for surface measure on the Fermi surfaces of the lattice dispersion relation. At a regular point, the decay is governed by the number of coordinate directions in which the quadratic term vanishes: after a local change of variables, these directions produce cubic one-dimensional phases, while the remaining directions are quadratic. For $d\geq5$, we prove the sharp global range \[ 1\leq p\leq\frac{2(d+2)}{d+5}. \] Away from the threshold energies, this range remains sharp in odd dimensions, whereas for even $d\geq6$ the sharp range improves to \[ 1\leq p\leq\frac{2(2d+5)}{2d+11}. \] In dimension four, we obtain uniform estimates for $1\leq p<4/3$ and an $\ell^{4/3}$-to-$\ell^4$ endpoint estimate with a square-root logarithmic loss. Matching anisotropic Knapp-type constructions give the corresponding necessary conditions. We also derive Birman–Schwinger and Kato smoothing bounds, complex-eigenvalue estimates, and improved thin spectral projection estimates.

We study the Cauchy problem for the defocusing mKdV equation with almost periodic initial data. If the initial data corresponds to a reflectionless Dirac operator which satisfies certain Craig-type conditions on the spectrum, we prove that the Cauchy problem has solution that is almost periodic in spacetime, and that this is the only solution which is locally bounded in a suitable sense. In particular, the result applies to small analytic quasiperiodic initial data with Diophantine frequencies.

We determine the sharp constant in the nearest-neighbor Hardy inequality on $\Z^3$ with the Euclidean inverse-square weight. For every finitely supported function $u:\Z^3\to\C$, we prove \[ \sum_{x\in\Z^3}\sum_{j=1}^3 |u(x+e_j)-u(x)|^2 \geq \frac14\sum_{x\in\Z^3\setminus\{0\}} \frac{|u(x)|^2}{|x|^2}. \] The coefficient $1/4$ is sharp, and equality is not attained by a nonzero finitely supported function. The proof uses an explicit reciprocal edge field and an edgewise completion of squares, together with a concavity argument for the associated vertex weight.

Let $F_k(G)$ be the $k$-token graph of a finite graph with nonnegative edge weights, and let $A_k$ and $D_k$ be its weighted adjacency and degree matrices. For every $-1\leq\vartheta\leq1$, we prove that $k\mapsto\lambda_{\max}(A_k+\vartheta D_k)$ is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At $\vartheta=1$ and $\vartheta=0$, this gives the signless-Laplacian and adjacency spectral-radius monotonicity conjectures of Apte, Parekh, and Sud. The spectral result follows from a finite-time theorem: for every $t\geq0$, the heat contents $\binom{n}{k}^{-1}\mathbf{1}^{*}e^{t(A_k+\vartheta D_k)}\mathbf{1}$ are log-concave in $k$. We encode all token levels in one Lorentzian polynomial. A four-variable operator symbol proves preservation by each edge heat gate, and the Lie-Trotter formula passes this preservation to the full semigroup. Large-time growth rates recover the top eigenvalues. The same construction also yields Lorentzian polynomials from top spectral projections. Finally, we show that the local symbol certifies exactly the parameter range $[-1,1]$.

We give an explicit polynomial deformation of the Euclidean metric on the unit three-ball which has positive Ricci curvature and strictly convex boundary but violates Escobar's conjectured lower bound for the first nonzero Steklov eigenvalue. Our proof is fully human-verifiable.

Weil's criterion states that the Riemann Hypothesis (RH) is equivalent to the non-negativity of an explicit quadratic form Q(f) on test functions. For f supported in [-L,L] we study the normalized infimum lambda_min(L) = inf Q(f)/||f||^2 from both sides. Lower bounds (unconditional): positivity on bounded support is classical for supp f in [-(log 2)/2,(log 2)/2] (Yoshida; Connes-Consani). We prove by certified computation that Q(f) >= 8.9e-18 ||f||^2 for all supp f in [-0.8,0.8], i.e. autocorrelation support 1.6, 2.3 times the classical range. The proof rests on a one-stroke reduction: a pointwise envelope for the Weil symbol, with a comb constant shown optimal by Weyl equidistribution, converts positivity on the whole window into positive semidefiniteness of a single finite matrix. The odd parity sector shows that the certified positivity holds for arbitrary complex test functions of support 1.6, and that the window ground state is simple and even, the spectral hypothesis required by the operator-theoretic program of Connes, Consani, Moscovici and van Suijlekom. Upper bounds (unconditional): variational bounds are evaluated on the geometric side in interval arithmetic, with no appeal to zeros or RH, down to 3.2e-283 at L=2. They follow the empirical law -ln lambda_min(L) ~ 2 pi^2 N(T*)/ln N(T*), T* = 2 pi e^{2L}, with N the zero-counting function; 2 pi^2 matches the Landau-Widom eigenvalue-plunge rate. Under RH, lambda_min(L) <= exp(-L e^L) for all large L. Synthesis: at L=0.8 the two halves enclose the profile, 8.9e-18 <= lambda_min(0.8) <= 2.27e-17, both certified. We show why the positivity route cannot reach RH unassisted: any one-stroke certificate must resolve frequencies up to 2 pi e^{A_L}, A_L ~ 4e^L, a doubly exponential threshold that no pointwise bound on the prime comb lowers, while the spectral margin collapses at the Landau-Widom rate.

We show that every smooth manifold $M$ of positive dimension admits fully supported distributions $u,v\in \mathcal D'(M)$ with $\mathrm{WF}(u)\cap({-}\mathrm{WF}(v))=\emptyset$ and $uv=0$. Neither factor can be continuous on any non-empty open set, but $u$ may be chosen of optimal Sobolev order $\dim M/2$, with wavefront set contained in any prescribed closed conical set $\Gamma\subset T^\ast M\setminus 0$ with non-empty and non-maximal fibers. Simultaneously, the wavefront set of $v$ may be confined to any compatible open conical set or, in geometric situations, to closed conical sets such as conormal bundles of foliations or ray bundles generated by locally conformally closed one-forms. We give positivity criteria and show that full-support zero divisor pairs accumulate at $(1,0)$ in Sobolev-microlocal topologies. The one-dimensional case uses a Fourier null series of Kozma-Olevskiĭ.

Hypercube graphs are fundamental model spaces of positive curvature in discrete comparison geometry. Let $G$ be a finite, connected, simple, unweighted graph with Bakry–Émery curvature bounded below by $K$. We call $G$ Lichnerowicz-sharp if its first non-zero non-normalized Laplacian eigenvalue $\lambda_1=K$. We prove that, after removing a canonical collection of edges on which every $K$-eigenfunction is constant, the resulting graph has a canonical bundle structure. Its fibers are regular, have similar structure with hypercubes, and are Laplacian-cospectral with hypercubes, although they need not themselves be hypercubes. If the base graph is nontrivial, then it satisfies $\mathrm{CD}(K,\infty)$ and has first eigenvalue strictly greater than $K$. As a consequence, if the vertex degree in $G$ is constant along each canonical fiber, then every fiber is a hypercube and $G$ is a hypercube bundle. Conversely, for every $d\geq 4$, we construct Lichnerowicz-sharp graphs with non-hypercube canonical fibers of degree $d$.

We prove an optimal Hardy inequality for every aperiodic, symmetric random walk in $\mathbb{Z}^2$ with finite variance. In particular, we verify null-criticality, and thus, optimality of the underlying Hardy weight. Under suitable moment conditions, we use fine asymptotics of the potential kernel due to Fukai and Uchiyama in order to derive the asymptotics of the weight. For the standard Laplacian, we recover the expected first order term in the asymptotics but also show that next order term is negative. Thus, our result shows that the constant in the Hardy inequality proven by Kapitanski and Laptev cannot be larger than $1/4$, which is the optimal constant in the continuum. The proof of null-criticality rests on a new criterion for general graphs beyond the locally finite case. We also recover the situation of $\mathbb{Z}^d$ with $d \geq 3$ which can also be also treated by our new method.

Using an analogue of the Mandelstam-Giddings-Wolpert diagrams, we introduce a new canonical representative of the conformal class of a bounded domain of arbitrary connectivity in $\mathbb{C}$ as a flat conical surface with geodesic boundary (a "truncated light-cone diagram", simply LC-diagram in the sequel). The space of diagrams carries natural coordinates. We derive variational formulas for the determinant of the Dirichlet Laplacian on a LC-diagram with respect to these coordinates. Then, passing to the Schottky double of the LC-diagram, making use of the Burghelea-Friedlander-Kappeler formula and the known variational formulas for determinants of Laplacians on the moduli space of holomorphic differentials, we compute the Dirichlet-to-Neumann (DN) conformal invariant $\frac{{\rm det}'\Lambda}{|\Gamma|}$ (here $\Lambda$ is the DN operator on the boundary, $\Gamma$, of a multiply connected domain and $|\Gamma|$ is the length of the boundary). The resulting formula (which uses the periods of the Schottky double of the domain only) provides an elementary counterpart to the formulas of Guillarmou and Guillopé who had expressed the DN invariant through the Ruelle and Selberg zeta-functions. Our formula agrees with the recent result of Wentworth on the asymptotics of the DN invariant as all but one boundary components shrink; we have used this result to fix the undetermined constant of integration in our formula. As a corollary, we derive an explicit formula for the determinant of the Dirichlet Laplacian in a multiply connected domain.

We show that every closed, nonorientable surface can be minimally embedded in $\mathbb{S}^4$, providing in particular the first known examples of embedded, nonorientable minimal surfaces in $\mathbb{S}^4$ with negative Euler characteristic. All of the surfaces we obtain have area below $8\pi$, which has applications to the existence of nonorientable surfaces minimizing the Willmore functional with prescribed topology in $\mathbb{R}^n$ for $n\geq 4$. Moreover, the number of geometrically distinct embeddings of each nonorientable genus is shown to grow at least exponentially with respect to the genus. Among these surfaces, we identify a distinguished, highly symmetric family which converges in the large-genus limit to a union of four half-spheres, meeting along a great circle, whose poles are vertices of a regular tetrahedron. Rescalings of this family converge to a new singly-periodic nonorientable minimal surface in $\mathbb{R}^4$, which seems to provide the first example of a complete, embedded, nonorientable minimal surface in $\mathbb{R}^4$ without continuous symmetry group. The proofs further develop the equivariant eigenvalue optimization methodology from our earlier work, applied to carefully chosen families of symmetry groups acting on nonorientable surfaces. Interestingly, we also find natural pairs of surfaces and group actions for which there is no metric maximizing the first normalized eigenvalue of the Laplacian.

We study a one-dimensional scalar model for an effective-mass quasiparticle field coupled to a continuum of two-level atoms with at most one excitation present. We prove a general theorem on the existence of bound states under sign-definite spatially localized perturbations of constant atomic density and analyze the bound states for several exactly solvable examples.

We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian on the bounded interval $(-1,1)$. This improves the eigenvalue asymptotics of Kulczycki–Kwaśnicki–Małecki–Stós and Kwaśnicki, and confirms the conjectural $O_\alpha(n^{-2})$ remainder suggested by the numerical simulations of Kaleta–Kwaśnicki–Małecki. Moreover, we prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $\alpha$. This settles the conjecture proposed by Kwaśnicki through numerical experiments. Furthermore, we prove that the $n$-th eigenfunction has exactly $n-1$ zeros in the interval $(-1,1)$ and every zero is simple, and hence there are exactly $n$ nodal domains. A key ingredient in the proof is an explicit representation of the eigenfunction.