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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 intrinsic and dynamic kinetic energies and the potential energies of electron states in the hydrogen atom were determined using the operator formalism in Schrödinger's nonrelativistic equation. Intrinsic energies were determined using the momentum operator, while for ℓ ≠ 0, the additional dynamic energies of the spinning fields were determined using the angular momentum operator. All 10 states up to the principal quantum number n = 3 and all 4 m states of n = 7, l = 3 were analyzed. The two forms of kinetic energy can only be explained with an electron field representation. All total kinetic and potential energies conformed with the well-known 1/n2 rule. Angular momentum analysis of the 2P1/2 state provided a field spinning rate; in addition, the dynamic kinetic energy of the spinning field determined by both operator analysis and explicit calculation based on the spinning rate gave the same energy results. [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]

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]

In this paper we investigate the well-posedness of the Cauchy problem for a Schrödinger operator with singular lower order terms. We allow distributional coefficients and we approach this problem via the regularising methods at the core of the theory of very weak solutions. We prove that a very weak solution exists and it is unique modulo negligible perturbations. Very weak solutions converge to classical solutions when the equation coefficients are regular enough. [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]

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 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]

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]

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]

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]

Let Ω ⊂ ℝ d and consider the magnetic Laplace operator given by H (A) = (– i ∇ – A (x))2, where A : Ω → ℝ d , subject to Dirichlet boundary conditions. For certain vector fields A , this operator can have eigenfunctions, H (A)ψ = λψ, that are highly localized in a small region of Ω. The main goal of this paper is to show that if |ψ| assumes its maximum at x 0 ∈ Ω, then A behaves 'almost' like a conservative vector field in a 1 / λ -neighborhood of x 0 in a precise sense. In particular, we expect localization in regions where |curl A | is small. The result is illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

We study the local maximal oscillatory integral operator T α , β ∗ ⁢ (f) ⁢ (x) = sup 0 < t < 1 ⁡ | ∫ ℝ n e i ⁢ | t ⁢ ξ | α | t ⁢ ξ | β ⁢ Ψ ⁢ (| t ⁢ ξ |) ⁢ f ^ ⁢ (ξ) ⁢ e 2 ⁢ π ⁢ i ⁢ 〈 x , ξ 〉 ⁢ 푑 ξ | , where α ∈ (0 , 1) , β > 0 , and Ψ is a cutoff function that vanishes in a neighborhood of the origin. First, in the case 0 < p < 1 , we obtain the H p ⁢ (ℝ n) → L p ⁢ (ℝ n) boundedness of T α , β ∗ with the sharp relation among α , β and p. Then, using interpolation, we obtain the L p ⁢ (ℝ n) boundedness on T α , β ∗ when p > 1 , which is an improvement of the recent result by Kenig and Staubach. At the critical case p = 1 and β = n ⁢ α 2 , we show T α , β ∗ : B q ⁢ (ℝ n) → L 1 , ∞ ⁢ (ℝ n) , where B q ⁢ (ℝ n) is the block space introduced by Lu, Taibleson and Weiss in order to study the almost every convergence of the Bochner–Riesz means at the critical index. As a further application, we obtain the convergence speed of a combination to the fractional Schrödinger operators { e i ⁢ t ⁢ k ⁢ | △ | α } . [ABSTRACT FROM AUTHOR]

We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective. [ABSTRACT FROM AUTHOR]

The inverse Sturm–Liouville problem with smooth potentials is considered. The high-order estimate of the entire function associated with two Sturm–Liouville problems is established. Applying this estimate expression to inverse Sturm–Liouville problems, we proved that the conclusion in [L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math. Phys. 50 2009, 3, Article ID 033505] remains true for more general case. [ABSTRACT FROM AUTHOR]

We will explore, in any space dimension d ≥ 4 , the decay in the energy space for the damped magnetic Schrödinger equation with non-local nonlinearity and radial initial data in H 1 (R d) . We will also display new Morawetz identities and corresponding localized Morawetz estimates. [ABSTRACT FROM AUTHOR]

We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb–Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of (−Δ + V + M)uM = 1 in R d ; here M ∈ R is chosen so that the operator is positive. We further prove that the infimum of ( u M − 1 − M) is a lower bound for the ground state energy E0 and derive a simple iteration scheme converging to E0. [ABSTRACT FROM AUTHOR]

In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet–Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians. [ABSTRACT FROM AUTHOR]

Cardinal series representations for solutions of the Sturm–Liouville equation −y′′+q(x)y=ρ2y,x∈(0,L)$$ -{y}^{\prime \prime }+q(x)y={\rho}^2y,x\in \left(0,L\right) $$ with a complex‐valued potential q(x)$$ q(x) $$ are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to ρ$$ \rho $$ in any strip Imρ

In this paper, we study a class of conformal metric deformations in the quasi-radial coordinate parametrizing the three-sphere in the conformally compactified Minkowski spacetime S 1 × S 3 . Prior to reduction of the associated Laplace–Beltrami operators to a Schrödinger form, a corresponding class of exactly solvable potentials (each one containing a scalar and a gradient term) is found. In particular, the scalar piece of these potentials can be exactly or quasi-exactly solvable, and among them we find the finite range confining trigonometric potentials of Pöschl–Teller, Scarf and Rosen–Morse. As an application of the results developed in the paper, the large compactification radius limit of the interaction described by some of these potentials is studied, and this regime is shown to be relevant to a quantum mechanical quark deconfinement mechanism. [ABSTRACT FROM AUTHOR]

We establish semiclassical resolvent estimates for Schrödinger operators with long-range matrix-valued potentials. As an application, we prove a resonance-free domain in trapping situations. These results generalize the well-known results of [4] [N. Burq, Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math. 124(4) (2002) 677–735] in the case of scalar Schrödinger operators. In addition, we give an estimate on the scattering amplitude for Schrödinger operators with compactly supported matrix-valued potentials. [ABSTRACT FROM AUTHOR]

This paper is devoted to N -wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators L , whose potentials Q (x , t) tend to constants Q ± for x → ± ∞. For special choices of Q ± , we outline the spectral properties of L , the direct scattering transform and construct its fundamental analytic solutions. We generalize Wronskian relations for the case of CBC — this allows us to analyze the mapping between the scattering data and the x -derivative of the potential Q x . Next, using the Wronskian relations, we derive the dispersion laws for the N -wave hierarchy and describe the NLEE related to the given Lax operator. [ABSTRACT FROM AUTHOR]