68 problems

Let ${{\mathscr M}}_{\epsilon}$ be a physical spacetime, let ${{\mathscr M}}$ be its background spacetime, and let $h_{ab}$ be a perturbative metric tensor pulled back from…

In perturbative $phi^4$ theory, consider the beta function renormalized in the minimal-subtraction (MS) scheme, and call a Feynman graph primitive if it has no subdivergences. Prim…

Fermi golden rule formula. The identity

Braak's eigenvalue-count conjecture. For a given parity, determined by whether $N$ is even or odd, the interval $I_N$ contains at most two shifted eigenvalues. Moreover, two interv…

Let the nonrelativistic model be considered for solutions that are perturbations of plane waves, and let the corresponding relativistic model include the three extra field equation…

Let $\widetilde{E}_{\gamma_i}$ denote the perturbed energy appearing in the replacement … with … Although the upper bound of the summation index $a$ may differ from the approximati…

Let $f$ be a nonuniformly expanding map. Perturbation-persistence conjecture. Sufficiently small perturbations of $f$ have positive probability of also being nonuniformly expanding…

Let $f(x_1,\ldots,x_n)$ be a homogeneous polynomial and let $\mathcal D$ be as in the preceding conjecture. Thus condition $(A)$ means that $f$ can be written as … where each $f_m$…

Let $f(t)$ be the quasi-periodic function in the preceding perturbative construction, let $M(f)$ denote its mean, and let $g$ be defined by the expansion … with…

Let $S_N$ be the approximants to a perturbative series at order $N$, and suppose that the observed behavior has $c(N)<1$ for the given terms of the series. In that case, the higher…

Consider the perturbative eigenvalue expansion $E=E^{[0]}+\sum_{k=1}^{\infty}\la^{-k}E^{[k]}$ for the Hamiltonian $H=H_0+\la^{-1}H_1$. For the in-state sector indexed by $n,\ol{n}$…

Leading-order cutoff-error conjecture. At leading order, a similar situation is encountered for higher-order terms:

The paper considers a perturbative procedure for constructing gauge-invariant variables in a two-parameter perturbation scheme, and confirms its applicability through third order.…

Consider a periodically driven two-level system with effective Hamiltonian expanded perturbatively in the driving frequency, whose leading terms involve the Pauli matrices…

In the perturbative expansion in the coupling $g$, let the $n$th-order contribution denote the term of order $g^n$ to the total area within all loops in a system of linear size $L$…

Feenberg-scaling invariance conjecture. Since the true characteristic polynomials, which depend only on the total Hamiltonian, are invariant under Feenberg scaling, estimates obtai…

Resonance-order conjecture. It is conjectured that $p=(2j-1)/2$ first enters at order $k=2j-1$, one new resonance per two additional orders, consistent with the tongue widths…

Let $L$ be a maximal dissipative operator and let $M$ be a dissipative operator such that ${\rm Dom}(M)\subset {\rm Dom}(L)$ and $K=M-L$ satisfies … for some $c\geq 0$ and $d<1$. T…

Let an ALF gravitational instanton be a Hermitian instanton with asymptotically locally flat asymptotic structure, and consider its gravitational perturbations. ALF rigidity conjec…

Let $H(\lambda)$ have lowest eigenvalue $E(\lambda)$ with corresponding eigenvector $\phi(\lambda)$. For each nonnegative integer $\ell$, let $\mathcal{P}^\ell$ be the orthogonal p…

Abelian Higgs correlator theorem. The correlator may be obtained by applying $\mathcal{T}_{\mathrm{cont}}$ to

The ground state energy and reduced density matrices admit a perturbative expansion whose first terms are denoted by $\gamma_0$ and $\gamma_1$. On the torus, translation invariance…

The induced quantity $Q_{\mathrm{ind}}$ is considered in the infinite-radius limit, where the finite-radius circle is replaced by the line and expressions are treated perturbativel…

Usual-properties conjecture. Each black vertex is the starting point of exactly two arrows labeled by “$\mathrm{e}$” and “$\mu$”, is the endpoint of exactly two arrows also labeled…

Let $\langle H \rangle_n^{(k)}$ denote the variational approximation to the energy at state $n$ and order $k$, and let $Z$ be the variational parameter. Higher-order flatness conje…