Self-adjoint Boundary Conditions Research Articles

Determination of the time-dependent thermal grooving coefficient

Changes in morphology of a polycrystalline material may occur through interface motion under the action of a driving force. An important special case that is considered in this paper is the thermal grooving that occurs when a grain boundary intersects the flat surface of a recently solidified metal slab giving rise to the formation of a thin symmetric groove. In case the transient surface diffusion is the main forming mechanism this yields a fourth-order time-dependent partial differential equation with unknown time-dependent surface diffusivity. In order to determine it, the profile of the free grooving surface at a fixed location is recorded in time. The grooving boundaries are supported by self-adjoint boundary conditions. We provide sufficient conditions on the input data for which the resulting coefficient identification problem is proved to be well-posed. Furthermore, we develop a predictor–corrector finite-difference spline method for obtaining an accurate and stable numerical solution to the nonlinear coefficient identification problem. Numerical results illustrate the performance of the inversion of both exact and noisy data.

Non-real eigenvalues of a class of fourth order discontinuous differential operators with indefinite weight function

The spectral theory of differential operators is one of the basic problems of differential operator theory. It has been an important tool in quantum mechanics and other fields. For the Sturm-Liouville Problems with weighted functions, the spectral theory of right-definite problems with self-adjoint boundary condition has been accomplished. But the spectral structure of indefinite problems, is quite different from and more complicated than that of right-definite problems. Compare with second order case, much less known for the fourth order indefinite differential operators. The present paper discusses a class of fourth order discontinuous differential operators. We obtained the estimate on the non-real eigenvalues, under separated boundary condition and the condition that weighted functions change signs for only once or any times.

On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from below

We extend the classical boundary values(0.1)g(a)=−W(ua(λ0,⋅),g)(a)=limx↓a⁡g(x)uˆa(λ0,x),g[1](a)=(pg′)(a)=W(uˆa(λ0,⋅),g)(a)=limx↓a⁡g(x)−g(a)uˆa(λ0,x)ua(λ0,x), for regular Sturm–Liouville operators associated with differential expressions of the type τ=r(x)−1[−(d/dx)p(x)(d/dx)+q(x)] for a.e. x∈(a,b)⊂R, to the case where τ is singular on (a,b)⊆R and the associated minimal operator Tmin is bounded from below. Here ua(λ0,⋅) and uˆa(λ0,⋅) denote suitably normalized principal and nonprincipal solutions of τu=λ0u for appropriate λ0∈R, respectively.Our approach to deriving the analog of (0.1) in the singular context employing principal and nonprincipal solutions of τu=λ0u is closely related to a seminal 1992 paper by Niessen and Zettl [58]. We also recall the well-known fact that the analog of the boundary values in (0.1) characterizes all self-adjoint extensions of Tmin in the singular case in a manner familiar from the regular case.We briefly discuss the singular Weyl–Titchmarsh–Kodaira m-function and finally illustrate the theory in some detail with the examples of the Bessel, Legendre, and Kummer (resp., Laguerre) operators.

Constructive solution of the inverse spectral problem for the matrix Sturm–Liouville operator

An inverse spectral problem is studied for the matrix Sturm–Liouville operator on a finite interval with the general self-adjoint boundary condition. We obtain a constructive solution based on the method of spectral mappings for the considered inverse problem. The nonlinear inverse problem is reduced to a linear equation in a special Banach space of infinite matrix sequences. In addition, we apply our results to the Sturm–Liouville operator on a star-shaped graph.

Global Exact Controllability of Bilinear Quantum Systems on Compact Graphs and Energetic Controllability

The aim of this work is to study the controllability of the bilinear Schrödinger equation on compact graphs. In particular, we consider the equation $(BSE)$ $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$, with $\mathscr{G}$ being a compact graph. The Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator, and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We provide a new technique leading to the global exact controllability of $(BSE)$ in $D(|\Delta|^{s/2})$ with $s\geq 3$. Afterwards, we introduce the “energetic controllability,” a weaker notion of controllability useful when the global exact controllability fails. In conclusion, we develop some applications of the main results involving, for instance, star graphs.

Canonical forms of self-adjoint boundary conditions for regular differential operators of order three

Canonical forms of self-adjoint boundary conditions for regular differential operators of order three

On the Boundary Conditions for the 1D Weyl–Majorana Particle in a Box

In (1+1) space-time dimensions, we can have two particles that are Weyl and Majorana particles at the same time---1D Weyl-Majorana particles. That is, the right-chiral and left-chiral parts of the two-component Dirac wave function that satisfies the Majorana condition, in the Weyl representation, describe these particles, and each satisfies their own Majorana condition. Naturally, the nonzero component of each of these two two-component wave functions satisfies a Weyl equation. We investigate and discuss this issue and demonstrate that for a 1D Weyl-Majorana particle in a box, the nonzero components, and therefore the chiral wave functions, only admit the periodic and antiperiodic boundary conditions. From the latter two boundary conditions, we can only construct four boundary conditions for the entire Dirac wave function. Then, we demonstrate that these four boundary conditions are also included within the most general set of self-adjoint boundary conditions for a 1D Majorana particle in a box.

Self-Adjoint Local Boundary Problems on Compact Surfaces. I. Spectral Flow

The paper deals with first order self-adjoint elliptic differential operators on a smooth compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on paths in the space of such operators connecting two operators conjugated by a unitary automorphism. The first result is the computation of the spectral flow for such paths in terms of the topological data over the boundary. The second result is the universality of the spectral flow: we show that the spectral flow is a universal additive invariant for such paths, if the vanishing on paths of invertible operators is required. In the next paper of the series we generalize these results to families of such operators parametrized by points of an arbitrary compact space instead of an interval. The integer-valued spectral flow is replaced then by the family index taking values in the $K^1$-group of the base space.

Non-real eigenvalues of nonlocal indefinite Sturm\u2013Liouville problems

The present paper deals with non-real eigenvalues of regular nonlocal indefinite Sturm–Liouville problems. The existence of non-real eigenvalues of indefinite Sturm–Liouville differential equation with nonlocal potential K(x,t) associated with self-adjoint boundary conditions is studied. Furthermore, a priori upper bounds of non-real eigenvalues for a class of indefinite differential equation involving nonlocal point interference potential function is obtained.

Controllability of localised quantum states on infinite graphs through bilinear control fields

In this work, we consider the bilinear Schrödinger equation (BSE) in the Hilbert space with an infinite graph. The Laplacian is equipped with self-adjoint boundary conditions, B is a bounded symmetric operator and with T>0. We study the well-posedness of the (BSE) in suitable subspaces of preserved by the dynamics despite the dispersive behaviour of the equation. In such spaces, we study the global exact controllability and the ‘energetic controllability’. We provide examples involving for instance infinite tadpole graphs.

Inverse spectral problem for the matrix Sturm-Liouville operator with the general separated self-adjoint boundary conditions

In this work, we study the matrix Sturm-Liouville operator with the separated self-adjoint boundary conditions of general type, in terms of two unitary matrices. Some properties of the eigenvalues and the normalization matrices are given. Uniqueness theorems for determining the potential and the unitary matrices in the boundary conditions from the Weyl matrix, two characteristic matrices or one spectrum and the corresponding normalization matrices are proved.

Eigenvalues of a class of fourth-order boundary value problems with transmission conditions using matrix theory

ABSTRACT A class of fourth-order boundary value problems with general self-adjoint boundary and transmission conditions is investigated. It is shown that such boundary value problems with a finite number of eigenvalues are equivalent to certain types of matrix eigenvalue problems. Moreover, the results are given under the general self-adjoint transmission conditions and the self-adjoint canonical forms of the fourth-order problems which are classified as separated, mixed and coupled, respectively.

Donoghue-type m-functions for Schrödinger operators with operator-valued potentials

Given a complex, separable Hilbert space $$\mathcal{H}$$ , we consider differential expressions of the type τ = −(d2/dx2) $$I_\mathcal{H}$$ + V(x), with x ∈ (x0,∞) for some x0 ∈ ℝ, or x ∈ ℝ (assuming the limit-point property of τ at ±∞). Here V denotes a bounded operator-valued potential V(·) ∈ $$\mathcal{B}(\mathcal{H})$$ such that V(·) is weakly measurable, the operator norm $$||V(\cdot)||_{\mathcal{B}(\mathcal{H})}$$ is locally integrable, and V(x) = V(x)* a.e. on x ∈ [x0,∞) or x ∈ ℝ. We focus on two major cases. First, on m-function theory for self-adjoint half-line L2-realizations H+,α in L2((x0,∞); dx; $$\mathcal{H}$$ ) (with x0 a regular endpoint for τ, associated with the self-adjoint boundary condition sin(α)u′(x0) + cos(α)u(x0) = 0, indexed by the selfadjoint operator α = α* ∈ $$\mathcal{B}(\mathcal{H})$$ ), and second, on m-function theory for self-adjoint full-line L2-realizations H of τ in L2(ℝ; dx; $$\mathcal{H}$$ ). In a nutshell, a Donoghue-type m-function $$M^{Do}_{A,\mathcal{N}{_i}}(\cdot)$$ associated with self-adjoint extensions A of a closed, symmetric operator $$\dot{A}$$ in $$\mathcal{H}$$ with deficiency spaces Nz = ker ( $$\dot{A}$$ * −zI $$\mathcal{H}$$ ) and corresponding orthogonal projections $${P_{{N_z}}}$$ onto Nz is given by $$\begin{gathered} M_{A,{\mathcal{N}_i}}^{Do}(z) = {P_{{\mathcal{N}_i}}}(zA + {I_\mathcal{H}})(A) - z{I_\mathcal{H}}{)^{ - 1}}{P_{{\mathcal{N}_i}|{\mathcal{N}_i}}} \hfill \\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = z{I_{{\mathcal{N}_i}}} + ({z^2} + 1){P_{{N_i}}}{(A - z{I_\mathcal{H}})^{ - 1}}{P_{{N_i}|{\mathcal{N}_i}}},\;\;\;z \in \mathbb{C}\backslash \mathbb{R}. \hfill \\ \end{gathered}$$ In the concrete case of half-line and full-line Schrodinger operators, the role of $$\dot{A}$$ is played by a suitably defined minimal Schrodinger operator H+,min in L2((x0,∞); dx; $$\mathcal{H}$$ ) and Hmin in L2(ℝ; dx; $$\mathcal{H}$$ ), both of which will be proven to be completely non-self-adjoint. The latter property is used to prove that if H+,α in L2((x0,∞); dx; $$\mathcal{H}$$ ), respectively, H in L2(ℝ; dx; $$\mathcal{H}$$ ), are self-adjoint extensions of H+,min, respectively, Hmin, then the corresponding operator-valued measures in the Herglotz–Nevanlinna representations of the Donoghue-type m-functions $$M^{Do}_{H+,\alpha},\mathcal{N_{+,i}}(\cdot)$$ and $$M^{Do}_{H,\mathcal{N}{_i}}(\cdot)$$ encode the entire spectral information of H+,α, respectively, H.

Asymptotic completeness and S-matrix for singular perturbations

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple (A0,A), where A0 and A are semi-bounded self-adjoint operators in L2(M,B,m) such that the set {u∈dom(A0)∩dom(A):A0u=Au} is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which A0 corresponds to the free Laplacian in L2(Rn) and A describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

Eigenvalues of Fourth-Order Boundary Value Problems with Self-Adjoint Canonical Boundary Conditions

Fourth-order boundary value problems with general real self-adjoint boundary conditions are investigated. It is obtained that the eigenvalues of the problem depend not only continuously on all parameters of the problem but also smoothly on the boundary conditions. Furthermore, the derivatives of the eigenvalues with respect to boundary conditions are given in the sense of the fundamental canonical forms of fourth-order self-adjoint boundary conditions including each type of separated, mixed and coupled cases.

Optimal bounds on the fundamental spectral gap with single-well potentials

We characterize the potential-energy functions $V(x)$ that minimize the gap $\Gamma$ between the two lowest Sturm-Liouville eigenvalues for \[ H(p,V) u := -\frac{d}{dx} \left(p(x)\frac{du}{dx}\right)+V(x) u = \lambda u, \quad\quad x\in [0,\pi ], \] where separated self-adjoint boundary conditions are imposed at end points, and $V$ is subject to various assumptions, especially convexity or having a form. In the classic case where $p=1$ we recover with different arguments the result of Lavine that $\Gamma$ is uniquely minimized among convex $V$ by the constant, and in the case of single-well potentials, with no restrictions on the position of the minimum, we obtain a new, sharp bound, that $\Gamma > 2.04575\dots$.

Existence Conditions of Negative Eigenvalues in the Regular Sturm–Liouville Boundary Value Problem and Explicit Expressions for Their Number

For the regular Sturm–Liouville boundary value problem with general nonseparated self-adjoint boundary conditions, conditions for the existence of zero and negative eigenvalues and expressions for their number are obtained. The conditions are expresses in a closed form, and the coefficient functions of the original equation appear in these conditions indirectly through a single numerical characteristic.

Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces

We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $\Sigma$ of) compact hypersurfaces $\Gamma=\partial\Omega$, $\Omega\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the free Laplacian on the whole space ${\mathbb{R}}^{n}$. Our results apply to all standard examples of boundary conditions, like Dirichlet, Neumann, Robin, $\delta$ and $\delta'$-type, either assigned on $\Gamma$ or on $\Sigma\subset\Gamma$.

Morse Index Theorem of Lagrangian Systems and Stability of Brake Orbit

In this paper, we prove Morse index theorem of Lagrangian systems with self-adjoint boundary conditions. Based on it, we give some nontrivial estimates on the difference of Morse indices. As an application, we get a new criterion for the stability problem of brake periodic orbit.

On traces and modified Fredholm determinants for half-line Schrödinger operators with purely discrete spectra

After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schrödinger operators ( − d 2 / d x 2 ) + q (- d^2/dx^2) + q on ( 0 , ∞ ) (0,\infty ) with purely discrete spectra. Roughly speaking, the class considered is generated by potentials q q that, for some fixed C 0 > 0 C_0 > 0 , ε > 0 \varepsilon > 0 , x 0 ∈ ( 0 , ∞ ) x_0 \in (0, \infty ) , diverge at infinity in the manner that q ( x ) ≥ C 0 x ( 2 / 3 ) + ε 0 q(x) \geq C_0 x^{(2/3) + \varepsilon _0} for all x ≥ x 0 x \geq x_0 . We treat all self-adjoint boundary conditions at the left endpoint 0 0 .