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Let (X , d , μ) be a space of homogeneous type in the sense of Coifman–Weiss, which satisfies a n-doubling property with n > 1 , and supports an L 2 -Poincaré inequality. Consider the time independent Schrödinger operator L = L + V on X, where L is a non-negative operator generalized by a Dirichlet form, and V is a non-negative Muckenhoupt weight which admits a reverse Hölder inequality of order q for some q > max { 1 , n / 2 } . Without the C 1 -regularity hypothesis, we derive that a solution u to the parabolic Schrödinger equation ∂ t u + L u = 0 on X × R + satisfies the Carleson measure condition sup B (x B , r B) 1 μ (B (x B , r B)) ∫ 0 r B 2 ∫ B (x B , r B) (| t ∂ t u | 2 + | t ∇ x u | 2) d μ d t t < ∞ if and only if u can be represented as the Gaussian integral of a BMO L -function f. As an application, some limiting behaviors of the Carleson measure/BMO function are also considered. [ABSTRACT FROM AUTHOR]

In the study of the electronic structure of heavy atoms, relativistic effects cannot be neglected and the Dirac operator naturally appears in place of the Schrödinger operator, raising a number of additional difficulties. The complexity of these systems has been addressed by various approximations. We consider a model consisting of differential equations coupling the time evolution of a finite number of relativistic electrons with the Newtonian dynamics of finitely many nuclei. The electrons are described by the Bogoliubov-Dirac-Fock (BDF) equation of quantum electrodynamics (QED). This system can be seen as a first step in the study of molecular dynamics phenomena, when both relativistic and quantum effects have to be taken into account for the electrons. We address the Cauchy problem for this model and prove global well-posedness. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of solutions as |x|→∞ of Schrödinger operators with homogeneous potentials. For this purpose, we use methods from semiclassical analysis and investigate semiclassical defect measures. We prove the localization in directions for the semiclassical defect measures. We apply the localization results to obtain a necessary observability condition. [ABSTRACT FROM AUTHOR]

We study the semiclassical distribution of resonances of a 2 × 2 matrix Schrödinger operator (1.1) near a fixed energy level where the underlying classical trajectories Γ1 of P1 and Γ2 of P2 are, respectively, periodic and non-trapping. The aim is to compute the imaginary part of the resonances appearing near the eigenvalues created by P1 when Γ1 intersects Γ2 with finite contact order m. Recent results [M. Assal, S. Fujiié and K. Higuchi, Semiclassical resonance asymptotics for systems with degenerate crossings of classical trajectories, Int. Math. Res. Not. 2024 (2024) 6879–6905] and [M. Assal, S. Fujiié and K. Higuchi, Transition of the semiclassical resonance widths across a tangential crossing energy-level, J. Math. Pures Appl. 191 (2024) 103634] assert that the width of resonances is of polynomial order in the semiclassical parameter with exponent 1 + 2/(m + 1), if the interaction U = r0(x) + ihr1(x)Dx is elliptic at the crossing point. Here, we remove this ellipticity assumption and prove that the exponent is 1 + 2(k + 1)/(m + 1), where k is a “vanishing order” of the interaction at the crossing points, suitably defined in terms of the vanishing orders of r0 and r1 depending on whether the crossing point is a turning point or not. [ABSTRACT FROM AUTHOR]

We give a sharp estimate for the first eigenvalue of the Schrödinger operator $L:=-\Delta -\sigma $ which is defined on the closed minimal submanifold $M^{n}$ in the unit sphere $\mathbb {S}^{n+m}$ , where $\sigma $ is the square norm of the second fundamental form. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study pseudospectral properties of the one-dimensional Schrödinger operator perturbed by a complex steplike potential. By constructing the resolvent kernel, we show that the pseudospectrum of this operator is trivial if and only if the imaginary part of the potential is constant. As a by-product, a new method to obtain a sharp resolvent estimate is developed, answering a concern of Henry and Krejčiřík, and a way to construct an optimal pseudomode is discovered, answering a concern of Krejčiřík and Siegl. This article also analyzes the impact of a complex point interaction on the spectrum and the resolvent norm. [ABSTRACT FROM AUTHOR]

For Schrödinger operators with potentials that are asymptotically homogeneous of degree - 2 , the size of the coupling determines whether it has finite or infinitely many negative eigenvalues. In the latter case, the asymptotic accumulation of these eigenvalues at zero has been determined by Kirsch and Simon. A similar regime occurs for potentials that are not asymptotically monotone but oscillatory. In this case, when the ratio between the amplitude and frequency of oscillation is asymptotically homogeneous of degree - 1 , the coupling determines the finiteness of the negative spectrum. We present a new proof of this fact by making use of a ground-state representation. As a consequence of this approach, we derive an asymptotic formula analogous to that of Kirsch and Simon. [ABSTRACT FROM AUTHOR]

We provide a probabilistic proof of the fundamental gap estimate for Schrödinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo [Potential Anal. 44 (2016), pp. 423–442] for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei [J. Differential Geom. 112 (2019), pp. 347–389], as well as by C. He, G. Wei, and Qi S. Zhang [Amer. J. Math. 142 (2020), pp. 1161–1191]. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We investigate the boundedness of "vertical" Littlewood–Paley–Stein square functions for the nonlocal fractional discrete Laplacian on the lattice Z , where the underlying graphs are not locally finite. When q ∈ [ 2 , ∞) , we prove the l q boundedness of the square function by exploring the corresponding Markov jump process and applying the martingale inequality. When q ∈ (1 , 2 ] , we consider a modified version of the square function and prove its l q boundedness through a careful in on the generalized carré du champ operator. A counterexample is constructed to show that it is necessary to consider the modified version. Moreover, we extend the study to a class of nonlocal Schrödinger operators for q ∈ (1 , 2 ] . [ABSTRACT FROM AUTHOR]

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori 핋 d , where d ≥ 3 . The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter, The proof of the l 2 decoupling conjecture, Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong, Strichartz estimates for N-body Schrödinger operators with small potential interactions, Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate. [ABSTRACT FROM AUTHOR]

We prove an abstract criterion on spectral instability of nonnegative selfadjoint extensions of a symmetric operator and apply this to self-adjoint Neumann Laplacians on bounded Lipschitz domains, intervals, and graphs. Our results can be viewed as variants of the classical weak coupling phenomenon for Schrödinger operators in L^2(\mathbb {R}^n) for n=1,2. [ABSTRACT FROM AUTHOR]

Asymmetry plays a significant role in the transmission dynamics of novel fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed Schrödinger models that incorporate different symmetry developmental stages. The Laplace transform, which can be used to discover the analytic (exact) solution to linear fractional differential equations. It is suggested in this work as a technique for resolving the fractional order Schrödinger equation with boundary conditions where the fractional derivatives of Caputo and Riemann-Liouville are applied. This can be used to resolve fractional and ordinary differential equations. After that, discovered the precise solution to a specific fractional differential equation example. The results show that the novel transform "Laplace Transform" when applied to the provided fractional differential equations boundary value problem, yields accurate solutions without the need for lengthy calculations. In this study, we investigate a class of fractional boundary value problems with two boundary value conditions that involve orders of fractional derivative: σ∈ (2,3] and x ∈ (0, a]. We illustrate our primary findings with several cases. We provide multiple examples to highlight our main conclusions. We proved the solution using the Laplace transformation after obtaining the existence using the fractional integral and integral operator methods.

Let L=−△+VL=-\bigtriangleup +V be the Schrödinger operator on Rn{{\mathbb{R}}}^{n}, where V≠0V\ne 0 is a non-negative function satisfying the reverse Hölder class RHq1R{H}_{{q}_{1}} for some q1>n⁄2{q}_{1}\gt n/2. △\bigtriangleup is the Laplacian on Rn{{\mathbb{R}}}^{n}. Assume that bb is a member of the Campanato space Λνθ(ρ){\Lambda }_{\nu }^{\theta }\left(\rho ) and that the fractional integral operator associated with LL is ℐβL{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}. We study the boundedness of the commutators [b,ℐβL]\left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] with b∈Λνθ(ρ)b\in {\Lambda }_{\nu }^{\theta }\left(\rho ) on local generalized mixed Morrey spaces. Generalized mixed Morrey spaces Mp→,φα,V{M}_{\overrightarrow{p},\varphi }^{\alpha ,V}, vanishing generalized mixed Morrey spaces VMp→,φα,VV{M}_{\overrightarrow{p},\varphi }^{\alpha ,V}, and LMp→,φα,V,{x0}L{M}_{\overrightarrow{p},\varphi }^{\alpha ,V,\left\{{x}_{0}\right\}} are related to the Schrödinger operator, in that order. We demonstrate that the commutator operator [b,ℐβL]\left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] is satisfied when bb belongs to Λνθ(ρ){\Lambda }_{\nu }^{\theta }\left(\rho ) with θ>0\theta \gt 0, 0

Abstract With b belonging to a new B M O θ ( ρ ) $BMO_{\theta}(\rho )$ space, L = − △ + V $L=-\triangle +V$ is a Schrödinger operator on R n ${\mathbb{R}^{n}}$ with nonnegative potential V belonging to the reverse Hölder class R H n / 2 $RH_{n/2}$ . The fractional integral operator associated with L is denoted by I β L ${\mathcal{I}}_{\beta}^{L}$ . We investigate the boundedness of I β L ${\mathcal{I}}_{\beta}^{L}$ and [ b , I β L ] $[b,{\mathcal{I}}_{\beta}^{L}]$ , which are its commutators with b θ ( ρ ) $b_{\theta}(\rho )$ on vanishing generalized mixed Morrey spaces V M p → , φ α , V $VM_{\vec{p},\varphi}^{\alpha ,V}$ related to Schrödinger operation and generalized mixed Morrey spaces M p → , φ α , V $M_{\vec{p},\varphi}^{\alpha ,V}$ . The boundedness of the operator I β L ${\mathcal{I}}_{\beta}^{L}$ is ensured by finding sufficient conditions on the pair ( φ 1 , φ 2 ) $(\varphi _{1},\varphi _{2})$ , which goes from M p → , φ 1 α , V $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to M q → , φ 2 α , V $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , and from V M p → , φ 1 α , V $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to V M q → , φ 2 α , V $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , ∑ i = 1 n 1 p i − ∑ i = 1 n 1 q i = β $\sum \limits _{i=1}^{n}\frac{1}{p_{i}}-\sum \limits _{i=1}^{n}\frac{1}{q_{i}}=\beta $ . When b belongs to B M O θ ( ρ ) $BMO_{\theta}(\rho )$ and ( φ 1 , φ 2 ) $(\varphi _{1},\varphi _{2})$ satisfies some conditions, we also show that the commutator operator [ b , I β L ] $[b,{\mathcal{I}}_{\beta}^{L}]$ is bounded from M p → , φ 1 α , V $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to M q → , φ 2 α , V $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ and from V M p → , φ 1 α , V $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to V M q → , φ 2 α , V $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ .

We consider the one-dimensional Schrödinger equation for the sextic anharmonic oscillator potential within the Dunkl formalism. Solutions of bound state type are constructed, and results are compared to the conventional scenario. [ABSTRACT FROM AUTHOR]

We consider the discrete Schrödinger operators with potentials whose values are read along the orbits of a shift of finite type. We study a certain subset of the collection of energies at which the Lyapunov exponent is zero and prove monotonicity of this set with respect to the shift embeddings. Then we introduce a certain function J (A, μ) determined by the position of these zeros and prove monotonicity of J (A, μ) with respect to embeddings [ABSTRACT FROM AUTHOR]

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, H ħ , V : = - ħ - 2 L ħ + V on the lattice ħ Z n , where V is a positive multiplication operator and L ħ is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit ħ → 0 . [ABSTRACT FROM AUTHOR]

This article is devoted to the construction of spectral series of the Schrödinger operator with double delta potential of the form , , where are the poles of 2- or 3-surface of revolution , in the semiclassical limit as . The operator is considered to be an arbitrary self-adjoint extension of the Laplace–Beltrami operator. [ABSTRACT FROM AUTHOR]

We give a description of finite-zone -potentials in terms of explicit theta-functional formulas. [ABSTRACT FROM AUTHOR]

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these inequalities both for Schrödinger and Pauli operators. Pauli operators with Aharonov-Bohm magnetic field are discussed as well. [ABSTRACT FROM AUTHOR]

In this article, we define Perelman's functionals on manifolds with non-isolated conical singularities by starting from a spectral point of view for the Perelman's λ -functional. (Our definition of non-isolated conical singularities includes isolated conical singularities.) We prove that the spectrum of Schrödinger operator - 4 Δ + R on manifolds with non-isolated conical singularities consists of discrete eigenvalues with finite multiplicities, provided that scalar curvatures of cross sections of cones have a certain lower bound. This enables us to define the λ -functional on these singular manifolds, and further, to prove that the infimum of W-functional is finite, with the help of some weighted Sobolev inequalities. Furthermore, we obtain some asymptotic behavior of eigenfunctions and the minimizer of the W-functional near the singularity, and a more refined optimal partial asymptotic expansion for eigenfunctions near isolated conical singularities. We also study the spectrum of - 4 Δ + R and Perelman's functionals on manifolds with more general singularities, i.e. the r α -horn singularities which serve as prototypes of algebraic singularities. [ABSTRACT FROM AUTHOR]

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure 1 - o (1) and polynomial diameter. Both bounds rely on spectral gaps—of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second—which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator , associated with a system of two particles on a two-dimensional lattice. It is shown that the subspaces of even as well as odd functions are invariant under operator The sets of quasimomenta and the class of potentials are described, for which the operator has an infinite number of eigenvalues , , for , . The explicit form of and the rate of convergence of the sequence to the bottom of the essential spectrum are found. [ABSTRACT FROM AUTHOR]

The del Rio–Gesztesy–Simon's theorem (Int Math Res Notices 15:751–758, 1997) states that, under certain conditions, two partially known spectra and partial information about the potential uniquely determine the potential. We generalize the above result to the case where the potential function is locally smooth at the right endpoint. [ABSTRACT FROM AUTHOR]

We consider the discrete Schrödinger operator H=−Δ+V$$ H=-\Delta +V $$ with a sparse potential V$$ V $$ and find conditions guaranteeing either existence of wave operators for the pair H$$ H $$ and H0=−Δ$$ {H}_0=-\Delta $$ or presence of dense purely point spectrum of the operator H$$ H $$ on some interval [λ0,0]$$ \left[{\lambda}_0,0\right] $$ with λ0<0$$ {\lambda}_0<0 $$. [ABSTRACT FROM AUTHOR]

In the context of the tadpole graph R$$ \mathcal{R} $$, we examine the damped Schrödinger semigroup e−itd2dx2$$ {e}&#x0005E;{- it\frac{d&#x0005E;2}{d{x}&#x0005E;2}} $$. The first step in our analysis involves a meticulous examination of the spectrum and a suitable decomposition of the resolvent's kernel. By doing so, we establish that the generalized eigenfunctions constitute a Riesz basis of a specific subspace of L2(R)$$ {L}&#x0005E;2\left(\mathcal{R}\right) $$. As a result, we demonstrate that the energy corresponding to the system decays exponentially. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence and multiplicity of stationary solutions to the nonlinear Schrödinger equation with concave-convex nonlinearities. The main contribution of this research lies in the fact that zero lies in the boundary of the spectrum of the strongly indefinite Schrödinger operator. Our approach utilizes a novel perturbing method, specifically designed for cases where zero is situated within the spectral gap of the Schrödinger operator. By applying a generalized linking argument concerning the strongly indefinite functional, we establish the existence of two sequences of solutions to the perturbing equation, each corresponding to positive and negative energy states respectively. Finally, we employ the perturbing method to demonstrate the existence of two solutions to our equation. [ABSTRACT FROM AUTHOR]

We construct closed-form solutions to the one-dimensional, stationary Dunkl–Schrödinger equation for the inverted oscillator potential with an inverse quadratic singularity at the origin. Conditions are derived for our solutions to represent bound states, and an example is given. [ABSTRACT FROM AUTHOR]

In this paper, we study block Jacobi operators on ℤ$$ \mathrm{\mathbb{Z}} $$ with quasi‐periodic meromorphic potential. We prove the nonperturbative Anderson localization for such operators in the large coupling regime. [ABSTRACT FROM AUTHOR]

Let H = (−Δ)m + V be a higher-order elliptic operator on L2(ℝn), where V is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with H. In particular, we first establish sharp global Kato smoothing estimates for eitH, based on uniform resolvent estimates of Kato–Yajima type for the absolutely continuous part of H. As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig–Ruiz–Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates. [ABSTRACT FROM AUTHOR]

We prove rotations-reducibility for close to constant quasi-periodic S L (2 , R) cocycles in one frequency in the finite regularity and smooth cases, and derive some applications to quasi-periodic Schrödinger operators. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator in a bounded domain $ \mathbf{R}^n (n=2,3) $ R n (n = 2 , 3) with Neumann boundary condition. We suppose that this domain contains small deformable inclusions, i.e. regions where the potentials do not have the same values as the exterior medium. Our goal is to construct an asymptotic formula for the case of multiple and simple eigenvalue problems. We find an expansion that highlights the relation between the deformation parameters and the eigenvalues. The problem is devoted to a domain containing a finite number of inhomogeneities. Also, we focus on the interface of a small inclusion. [ABSTRACT FROM AUTHOR]

Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and potentials. In this review, we present some generalisations of substitutions, with a focus on substitutions on compact alphabets, and with an outlook towards their spectral theory. Guided by two main examples, we will illustrate what changes when one moves from finite to compact (infinite) alphabets, and discuss under which assumptions do we recover the usual geometric and statistical properties which make them viable models of materials with almost periodic order. We also present a planar example (which is a two‐dimensional generalisation of the Thue−Morse substitution), whose diffraction is purely singular continuous. [ABSTRACT FROM AUTHOR]

Periodic approximations of quasicrystals are a powerful tool in analyzing spectra of Schrödinger operators arising from quasicrystals, given the known theory for periodic crystals. Namely, we seek periodic operators whose spectra approximate the spectrum of the limiting operator (of the quasicrystal). This naturally leads to study the convergence of the underlying dynamical systems. We treat dynamical systems which are based on one‐dimensional substitutions. We first find natural candidates of dynamical subsystems to approximate the substitution dynamical system. Subsequently, we offer a characterization of their convergence and provide estimates for the rate of convergence. We apply the proposed theory to some guiding examples. [ABSTRACT FROM AUTHOR]

We prove a reverse Lieb--Thirring inequality with a sharp constant for the matrix Schrödinger equation on the half-line. [ABSTRACT FROM AUTHOR]

We consider the fractional relativistic Schr¨odinger-Choquard equation ... where ε > 0 is a small parameter, s ∈ (0, 1), m > 0, N > 2s, μ ∈ (0, 2s), (-Δ + m2)s is the fractional relativistic Schr¨odinger operator, V: RN → R is a continuous potential having a local minimum, f: R → R is a continuous nonlinearity with subcritical growth at infinity and F(t) = t 0 f(τ)dτ. Exploiting appropriate variational arguments, we construct a family of solutions concentrating around the local minimum of V as ε → 0. [ABSTRACT FROM AUTHOR]

Let L = - Δ ℍ n + V be a Schrödinger operator on Heisenberg groups ℍ n , where Δ ℍ n is the sub-Laplacian, the nonnegative potential V belongs to the reverse Hölder class B 풬 / 2 . Here 풬 is the homogeneous dimension of ℍ n . In this article, we introduce the fractional heat semigroups { e - t ⁢ L α } t > 0 , α > 0 , associated with L. By the fundamental solution of the heat equation, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K α , t L ⁢ (⋅ , ⋅) , respectively. As an application, we characterize the space BMO L γ ⁢ (ℍ n) via { e - t ⁢ L α } t > 0 . [ABSTRACT FROM AUTHOR]

Pseudomodes of non-self-adjoint Schrödinger operators corresponding to large pseudoeigenvalues are constructed. The approach is non-semiclassical and extendable to other types of models including the damped wave equation and Dirac operators. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of multiple solutions in the Bessel Potential space for a new class of nonlinear fractional Schr¨odinger-Poisson systems involving the distributional Riesz fractional derivative. To reach our goal, we use the symmetric mountain pass theorem under some suitable assumptions on nonlinearity f(x, u) and potential V (x). [ABSTRACT FROM AUTHOR]

The problem of describing invariant extensions of the 3D Schrödinger operator H with finite number of point interactions leads to the need for studying matrices of a special type, the permutation matrices. A large class of such extensions considered in a certain boundary triplet is in one-to-one correspondence with a set of the so-called boundary operators (matrices). The extension of H with point interactions concentrated on X = {x1,... , xm} ⊂ R 3 is invariant under the symmetry group of X (or its subgroup) if and only if the corresponding boundary matrix commutes with the set of m-by-m permutation matrices induced by the symmetry group, i.e., belongs to its commutant. The bicommutant theorem for such a set of matrices is proven for an arbitrary finite point set. For the special cases of regular polygon, tetrahedron, and cube, the basis of the bicommutant, considered as a vector space, is specified explicitly. [ABSTRACT FROM AUTHOR]

We briefly review the analytical continuation method for determining quasinormal modes (QNMs) and the associated frequencies in open systems. We explore two exactly solvable cases based on the Pöschl-Teller potential to show that the analytical continuation method cannot determine the full set of QNMs and frequencies of a given problem starting from the associated bound state problem in quantum mechanics. The root of the problem is that many QNMs are the analytically continued counterparts of solutions that do not belong to the domain where the associated Schrödinger operator is self-adjoint, challenging the application of the method for determining full sets of QNMs. We illustrate these problems through the physically relevant case of BTZ black holes, where the natural domain of the problem is the negative real line. [ABSTRACT FROM AUTHOR]

With b belonging to a new B M O θ (ρ) space, L = − △ + V is a Schrödinger operator on R n with nonnegative potential V belonging to the reverse Hölder class R H n / 2 . The fractional integral operator associated with L is denoted by I β L . We investigate the boundedness of I β L and [ b , I β L ] , which are its commutators with b θ (ρ) on vanishing generalized mixed Morrey spaces V M p → , φ α , V related to Schrödinger operation and generalized mixed Morrey spaces M p → , φ α , V . The boundedness of the operator I β L is ensured by finding sufficient conditions on the pair (φ 1 , φ 2) , which goes from M p → , φ 1 α , V to M q → , φ 2 α , V , and from V M p → , φ 1 α , V to V M q → , φ 2 α , V , ∑ i = 1 n 1 p i − ∑ i = 1 n 1 q i = β . When b belongs to B M O θ (ρ) and (φ 1 , φ 2) satisfies some conditions, we also show that the commutator operator [ b , I β L ] is bounded from M p → , φ 1 α , V to M q → , φ 2 α , V and from V M p → , φ 1 α , V to V M q → , φ 2 α , V . [ABSTRACT FROM AUTHOR]

We define spectral (Bloch) varieties of multidimensional differential operators on non-simply connected manifolds. In their terms we give a description of the analytic dependence of the spectra of magnetic Laplacians on non-simply connected manifolds on the values of the Aharonov–Bohm fluxes, construct analogs of spectral curves for two-dimensional Dirac operators on Riemann surfaces, and thereby find new conformal invariants of immersions of surfaces into three- and four-dimensional Euclidean spaces. [ABSTRACT FROM AUTHOR]

A family of quasi-1D Schrödinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibits changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed C∗-algebra. The quotient of this algebra by the ideal of compact operators is studied, and an index theorem is deduced from these investigations. This result corresponds to a topological version of Levinson’s theorem in the presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the last two sections of the paper, the K-theory of the main C∗-algebra and the dependence on an external parameter are carefully analyzed. In particular, a surface of resonances is exhibited, probably for the first time. The contents of these two sections are of independent interest, and the main result does not depend on them. [ABSTRACT FROM AUTHOR]

We develop a general approach to study three-dimensional Schrödinger operators with confining potentials depending on the distance to a surface. The main idea is to apply parallel coordinates based on the surface but outside its cut-locus in the Euclidean space. If the surface is asymptotically planar in a suitable sense, we give an estimate on the location of the essential spectrum of the Schrödinger operator. Moreover, if the surface coincides up to a compact subset with a surface of revolution with strictly positive total Gauss curvature, it is shown that the Schrödinger operator possesses an infinite number of discrete eigenvalues. [ABSTRACT FROM AUTHOR]

We present a review of recent results about a class of Schrödinger operators usually called soft quantum waveguides, with the focus on relations between their spectral properties and the geometry of the confinement. We also mention a number of open problems in this area. [ABSTRACT FROM AUTHOR]

Let (X , d , μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we first establish several weighted norm estimates for various maximal functions. Then we show the weighted boundedness of the fractional integral I β associated with admissible functions and its commutators. Similarly to I β , corresponding results for Calderón–Zygmund operators T associated with admissible functions are also included in this article. [ABSTRACT FROM AUTHOR]

We study a one-dimensional nonstationary Schrödinger equation with a potential slowly depending on time. The corresponding stationary operator depends on time as on a parameter. It has finitely many negative eigenvalues and absolutely continuous spectrum filling . The eigenvalues move with time to the edge of the continuous spectrum and, having reached it, disappear one after another. We describe the asymptotic behavior of a solution close at some moment to an eigenfunction of the stationary operator, and, in particular, the phenomena occurring when the corresponding eigenvalue approaches the absolutely continuous spectrum and disappears. [ABSTRACT FROM AUTHOR]