Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom

We consider a periodic magnetic Schrödinger operator Hh, depending on the semiclassical parameter h>0, on a noncompact Riemannian manifold M such that H¹(M,R)=0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface S. First, we prove upper and lower estimates for the bottom λ₀(Hh) of the spectrum of the operator Hh in L²(M). Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrarily large number of spectral gaps for the operator Hh in the region close to λ₀(Hh), as h→0. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.     

The resolvent of a dilation-analytic three-particle system

If the potential in a three-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has a dilation-analytic continuation H(φ). The continuous spectrum of H(φ) consists of half-lines Y(λ_p, φ) starting at the thresholds λ_p of scattering channels and making angles 2φ with the positive real axis. If the interaction is the sum of local two-body potentials in suitable $\text{L}$^p-spaces, each half-line Y(λ_p, φ) is associated with an operator P(λ_p, φ) that projects onto an invariant subspace of H(φ). Suppose Y(λ_p, φ) does not pass through any two- or three-particle eigenvalues λ ≠ λ_p when φ runs through some interval $\text{0 < α ? φ ? β <}\text{π}\text{2}$. For φ in [α, β], this paper shows that the resolvent R(λ, φ) has smoothness properties near Y(λ_p, φ) that are sufficient for P(λ_p, φ)[H(φ) ? λ_p] e^?2iφ to be spectral and to generate a strongly differentiable group. The projection, the group, and the spectral resolution operators are norm continuous in φ. These results are not affected by any spurious poles of the resolvent equation. At a spurious pole λ = λ_p + ze^2iφ, the resolvent R(λ_p + ze^2iφ,φ) is examined by a method that uses two resolvent equations in succession and shows that there is norm continuity in z, φ. The case of spurious poles on Y(λ_p, φ) is included.     

The Birman-Schwinger principle on the essential spectrum

Let H₀ and H be self-adjoint operators in a Hilbert space. We consider the spectral projections of H₀ and H corresponding to a semi-infinite interval of the real line. We discuss the index of this pair of spectral projections and prove an identity which extends the Birman-Schwinger principle onto the essential spectrum. We also relate this index to the spectrum of the scattering matrix for the pair H₀, H.     

Spectral concentration near embedded eigenvalues

Techniques of Rayleigh-Schrödinger perturbation theory usually employed for perturbation of isolated eigenvalues are used to obtain theorems on spectral concentration near eigenvalues which are not assumed to be isolated. If {H_ϰ} is a family of self adjoint operators convergent in the strong resolvent sense to a self adjoint operator H₀ and λ₀ is an eigenvalue of finite multiplicity of H₀, then the spectrum of {H_κ} is concentrated near λ₀. Moreover, conditions under which concentration still occurs near λ₀ without the assumption of finite multiplicity are obtained in the semibounded case.     

Analytic scattering theory of two-body Schrödinger operators

We consider the self-adjoint analytic family of operators H(z) in L²(R^m) defined for $\text{z ? S}{}_{\mathrm{\alpha }}\text{= {z ∥}\text{Arg}\text{z ¦ < α}}$, associated with the operator H = H(1) = H₀ + V, where H₀ = ?Δ and V is a dilation-analytic short-range potential. The analytic connection between the local wave and scattering operators associated with the operators H(e^i?) is established. The scattering matrix S(?) of H has a meromorphic continuation S(z) to S_α with poles precisely at the resolvent resonances of H, and the local scattering operators of e^?2i?H(e^i?) have representations in terms of the analytically continued scattering matrix S(?e^i?).     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).     

Singular continuous Floquet operator for systems with increasing gaps

Consider the Floquet operator of a time-independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e^−iH₀Te^{−iκT|φ〉〈φ|}, where T and κ are the period and the coupling constant, respectively. Assume the spectrum of the self-adjoint operator H₀ is pure point, simple, bounded from below and the gaps between the eigenvalues (λ_n) grow like λ_n+1−λ_n∼Cn^d with d?2. Under some hypotheses on the arithmetical nature of the eigenvalues and the vector φ, cyclic for H₀, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum.     

Invariant subspaces of analytic multiparticle Hamiltonians

The quantum mechanics of n particles interacting through analytic two-body interactions can be formulated as a problem of functional analysis on a Hilbert space $\text{G}$ consisting of analytic functions. On $\text{G}$, there is an Hamiltonian H with resolvent R(λ). These quantities are associated with families of operators H(?) and R(λ, ?) on $\text{L}$, the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) consists of possible isolated points, plus a number of half-lines starting at the thresholds of scattering channels and making an angle 2? with the real axis.Assuming that the two-body interactions are in the Schmidt class on the two-particle space $\text{G}$, this paper studies the resolvent R(λ, ?) in the case ? ≠ 0. It is shown that a well known Fredholm equation for R(λ, ?) can be solved by the Neumann series whenever ¦λ¦ is sufficiently large and λ is not on a singular half-line. Owing to this, R(λ, ?) can be integrated around the various half-lines to yield bounded idempotent operators P_p(?) (p = 1, 2,…) on $\text{L}$. The range of P_p(?) is an invariant subspace of H(?). As ? varies, the family of operators P_p(?) generates a bounded idempotent operator P_p on a space $\text{G}$. The range of this is an invariant subspace of H. The relevance of this result to the problem of asymptotic completeness is indicated.     

Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval

The matrix-valued Weyl-Titchmarsh functions M(λ) of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M(λ)) and the residues of M(λ) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N×N Weyl-Titchmarsh functions) corresponding to N×N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct.     

First order spectral perturbation theory of square singular matrix polynomials

We develop first order eigenvalue expansions of one-parametric perturbations of square singular matrix polynomials. Although the eigenvalues of a singular matrix polynomial P(λ) are not continuous functions of the entries of the coefficients of the polynomial, we show that for most perturbations they are indeed continuous. Given an eigenvalue λ₀ of P(λ) we prove that, for generic perturbations M(λ) of degree at most the degree of P(λ), the eigenvalues of P(λ)+?M(λ) admit covergent series expansions near λ₀ and we describe the first order term of these expansions in terms of M(λ₀) and certain particular bases of the left and right null spaces of P(λ₀). In the important case of λ₀ being a semisimple eigenvalue of P(λ) any bases of the left and right null spaces of P(λ₀) can be used, and the first order term of the eigenvalue expansions takes a simple form. In this situation we also obtain the limit vector of the associated eigenvector expansions.     

The spectral projections of dissipative operators

We exhibit a way of imbedding the spectral theory of abstract dissipative operators in the spectral theory of a model operator. Using this imbedding we construct projections of the absolutely continuous spectrum in terms of the original space. Such projections are computed for certain differential operators.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 109–114.     

A constructive procedure to recover asymptotics of short-range or long-range potentials

We study an inverse scattering problem for a pair of Hamiltonians (H,H₀) on L²(Rⁿ), where H₀=-Δ and H=H₀+V, V being a short- or long-range potential. By an elementary constructive method, we show that the scattering operator S, which is localized near a fixed energy λ>0, determines the asymptotics of the potential V at infinity, in dimension n?3. This is done by studying the action of the scattering operator on suitable wave packets.     

An infinite-dimensional Evans function theory for elliptic boundary value problems

An infinite-dimensional Evans function E(λ) and a stability index theorem are developed for the elliptic eigenvalue problem in a bounded domain Ω⊂Rm. The number of zero points of the Evans function in a bounded, simply connected complex domain D is shown to be equal to the number of eigenvalues of the corresponding elliptic operator in D. When the domain Ω is star-shaped, an associated unstable bundle E(D) based on D is constructed, and the first Chern number of E(D) also gives the number of eigenvalues of the elliptic operator inside D.     

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n_±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A^?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.     

Maps preserving operator pairs whose products are projections

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dimH?2. It is proved that a surjective map φ on B(H) preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti-unitary operator U on H such that φ(A)=λU^∗AU for all A in B(H) for some constants λ with λ²=1. Related results for surjective maps preserving operator pairs whose triple Jordan products are nonzero projections in both directions are also obtained. These show that the operator pairs whose products or triple Jordan products are nonzero projections are isometric invariants of B(H).     

Eigenvalues of the reference operator and semiclassical resonances

We prove that the estimate of the number of the eigenvalues in intervals , of the reference operator L^#(h) related to a self-adjoint operator L(h) is equivalent to the estimate of the integral over [λ−δ,λ+δ] of the sum of harmonic measures associated to the resonances of L(h) lying in a complex neighborhood Ω of λ>0 and the number of the positive eigenvalues of L(h) in [λ−δ,λ+δ]. We apply this result to obtain a Breit-Wigner approximation of the derivative of the spectral shift function near critical energy levels.     

Spectral structure of Anderson type Hamiltonians

We study self adjoint operators of the form?H _ω = H ₀ + ∑λ_ω(n) <δ _n ,·>δ _n ,?where the δ _n ’s are a family of orthonormal vectors and the λ_ω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ _n and δ _m are not completely orthogonal, then the restrictions of H _ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases. Oblatum 27-V-1999 & 6-I-2000?Published online: 8 May 2000     

Complex dynamical variables for multiparticle systems with analytic interactions. I

The n-body problem is formulated as a problem of functional analysis on a Hilbert space $\text{G}$ whose elements are analytic functions of complex dynamical variables. It is assumed that the two-body interaction is local and spherically symmetric, and belongs to the two-particle space $\text{G}$. The n-body resolvent R(λ) is constructed with the help of Fredholm methods. The operator R(λ) on $\text{G}$ is associated with a family of operators R(λ, ?) on $\text{L}$² which are resolvents of closed linear operators H(?), the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) contains a set of parallel half-lines starting at the thresholds of scattering channels and making an angle 2? with the positive real axis. The half-lines are branch cuts of R(λ, ?), but matrix elements of R(λ, ?) can be continued analytically across these. The operator R(λ, ?) may have isolated poles. The location of these does not depend on ?. Each pole is associated with one or more eigenvectors of H(?) belonging to spaces $\text{G}$. There may be poles off the real axis, the location of a pole determining for which values of ? it is on the physical sheet of H(?). It is shown how poles off the real axis give rise to resonances in the scattering cross section, the shape of a resonance being as one would expect on the basis of a model in which the scattering takes place via a decaying compound state having an eigenvector of H(?) with complex energy as its wave function.     

Continuity of the spectrum on a class of upper triangular operator matrices

Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A^○∈B(K) denote the Berberian extension of an operator A∈B(H). It is proved that the set theoretic function σ, the spectrum, is continuous on the set C(i)⊂B(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and at every non-zero λ∈σp(A^○) for some operators X and B such that λ∉σp(B) and σ(A^○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where Aii∈C(i) and Aii has SVEP for all 1?i?m, then σ is continuous on CS(m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP.     

Perturbation theory for eigenvalues and resonances of Schrödinger hamiltonians

Suppose that e^2_?|x|V ∈ ReL^P(R³) for some p > 2 and for g ∈ R, H(g) = ? Δ + gV, H(g) = ?Δ + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues λ(g) as they are absorbed by the continuous spectrum, that is $\text{λ(g) ↗6 0}\text{as}\text{g ↘5 g}{}_0\text{> 0}$. We find a series expansion in powers of $\text{(g ? g}{}_0\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, λ(g) = ∑}{}_{\mathrm{n\; =\; 2}}{}^{\mathrm{\infty }}\text{a}{}_{\mathrm{n}}\text{(g ? g}{}_0\text{)}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$ whose values for g < g₀ correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues.