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# Schrödinger-Operators with Singular Interactions

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*in*Journal of Mathematical Analysis and Applications 184(1):112–139 · May 1994

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- ... for some 0 < a < 1 and b > 0, hence by KLMN theorem the form B is closed, semi-bounded, and defines a self-adjoint operator bounded from below, see also [3]. ...... Taking into account Eqs. (2.11) and (3.8) we have that 3) ). Taking into account the explicit form of the resolventˆRresolventˆ resolventˆR 0 (λ) in the p-coordinates, see, e.g., Eq. (2.7), and by Eq. (B.8) (together with the definition of ξ (ℓ) (λ)) we have ...... and defined C 123 to be the least of C 12 , C 23 and C 31 . Two analogous inequalities hold true for the terms involving ρ (2),ε − ξ (2) and ρ (3),ε − ξ (3) . ...Article
- Mar 2018
- J MATH PHYS

We consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in norm resolvent sense. The two-body rescaled potentials are of the form $v^{\varepsilon}_{\sigma}(x_{\sigma})= \varepsilon^{-1} v_{\sigma}(\varepsilon^{-1}x_\sigma )$, where $\sigma = 23, 12, 31$ is an index that runs over all the possible pairings of the three particles, $x_{\sigma}$ is the relative coordinate between two particles, and $\varepsilon$ is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials $v_\sigma$ with $\alpha_\sigma \delta_\sigma$, where $\delta_\sigma$ is the Dirac delta-distribution centered on the coincidence hyperplane $x_\sigma=0$ and $\alpha_\sigma = \int_{\mathbb{R}} v_\sigma dx_\sigma$. To prove the convergence of the resolvents we make use of Faddeev's equations. - ... cf. [3]. With these notations in hand the main result of this note can be formulated. ...... cf. [3]. As one expects, it suffices to compute the limits of (A 0 − (λ − mc 2 )) −1 , Φ λ−mc 2 , and C λ−mc 2 for c → ∞ to prove Theorem 1.1. ...In this note it is shown that the nonrelativistic limit of the Dirac operator with a Lorentz scalar δ‐shell interaction of constant strength supported on a C2‐smooth surface in ℝ3 is the Schrödinger operator with a δ‐potential of the same strength.
- ... In more detail, the expressions for the scattering matrix we provide in (5.38) relative to Dirichlet obstacles and in (5.39) relative to Neumann obstacles, extend to any dimension and to Lipschitz obstacles the similar ones obtained for two-dimensional obstacles with piecewise C 2 boundary in [12, Theorems 5.3 and 5.6] and [13, Theorems 4.2 and 4.3]; similar formulae are also given, in a smooth two dimensional setting in [7, Subsections 5.2 and 5.3] and in a smooth n-dimensional setting in [25, Subsections 6.1 and 6.2]. The construction of the operator ∆ α with semi-transparent boundary condition of δ-type provided in Theorem 5.9 extends, as regards the regularity of the boundary and/or the class of admissible strength functions, previous constructions given, for example, in [18], [16], [10], [30], [6], [24], [15]. Asymptotic completeness for the scattering couple (∆, ∆ α ) provided in Theorem 5.9 extend results on existence and completeness given, in the case the boundary is smooth and the strength are bounded, in [6] and [24]. ...... Remark 5.13. The conditions providing the self-adjoint operator ∆ α in Theorem 5.9 are weaker, as regards the regularity of the boundary and/or the class of admissible strength functions, than the ones assumed in previous works, see, for example, [18], [16], [10], [30], [6], [24], [15]. Asymptotic completeness for the scattering couple (∆, ∆ α ) provided in Theorem 5.9 extend results on existence and completeness given, in the case the boundary is smooth and the strength are bounded, in [6] and [24]. ...ArticleFull-text available
- Nov 2017

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given. - ... The goal of this paper is to determine a spectral shift function for the pair {H, H δ,α }, where H = −∆ is the usual self-adjoint Laplacian in L 2 (R n ), and H δ,α = −∆ − αδ C is a singular perturbation of H by a δ-potential of variable real-valued strength α ∈ C 1 (C) supported on some smooth, compact hypersurface C that splits R n , n ≥ 2, into a bounded interior and an unbounded exterior domain. Schrödinger operators with δ-interactions are often used as idealized models of physical systems with short-range potentials; in the simplest case point interactions are considered, but in the last decades also interactions supported on curves and hypersurfaces have attracted a lot of attention, see the monographs [2,4,26], the review [22], and, for instance, [3,5,9,12,13,18,23,24,25,27,35] for a small selection of papers in this area. It is known from [9] (see also [12]) that for an integer m > (n/2) − 1 the m-th powers of the resolvents of H and H δ,α differ by a trace class operator, ...... see [9, Proposition 3.7] and [18] for more details. For c ∈ R we shall also make use of the self-adjoint operator ...Article
- Oct 2017

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. Here $\delta_{\cal{C}}$ denotes a singular $\delta$-potential which is supported on a smooth compact hypersurface $\mathcal{C}\subset\mathbb{R}^n$ and $\alpha$ is a real-valued function on $\mathcal{C}$. - ... in this work is mainly based on the resolvent properties. In the following we will use the results of[7,5,24,25]where strongly as well as weakly singular potentials were considered. ...... Let ˘ H β stand for the operator associated to the above form in the sense of the first representation theorem, cf.[18, Chap.VI]. Following the arguments from[7]we conclude that I −βR ΣΣ (z) : L 2 (Σ) → L 2 (Σ) defines the Birman– Schwinger operator for ˘ H β. Using Thm. 2.2 of[25]one obtains ...Article
- May 2017
- J PHYS A-MATH THEOR

We consider a straight three dimensional quantum layer with singular potential, supported on a straight wire which is localized perpendicularly to the walls and connects them. We prove that an infinite number of embedded eigenvalues appear in this system. Furthermore, we show that after introducing a small surface impurity to the layer, the embedded eigenvalues turn to the second sheet resolvent poles, which state resonances. We discuss the asymptotics of the imaginary component of the resolvent pole with respect to the surface area. - ... The largest part of the existing literature (see, e.g. [33,62,64,66,67,110,117]) is devoted to the case of a real interaction strength α. However, there has been recent interest in non-real α; see, e.g. ...... Related conditions for absence of non-real eigenvalues in higher dimensions for Schrödinger operators with complex-valued regular potentials can be found in [68,69]. In the self-adjoint setting absence of negative eigenvalues for α ∞ small enough is also a consequence of the Birman-Schwinger bounds in [33]; see also [61]. ...The spectral properties of non-self-adjoint extensions $A_{[B]}$ of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator $B$. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for $A_{[B]}$ to have a non-empty resolvent set are provided in terms of the parameter $B$ and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for $A_{[B]}$ are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schr\"odinger operators with $\delta$-potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.
- ... As a third application, presented in Section 7, we consider the pair {H, H δ,α }, where H = −∆ is the usual self-adjoint realization of the Laplacian in L 2 (R n ), and H δ,α = −∆ − αδ C is a singular perturbation of H by a δ-potential of variable realvalued strength α ∈ C 1 (C) supported on some compact hypersurface C that splits R n , n ≥ 2, into a bounded interior and an unbounded exterior domain. Schrödinger operators with δ-interactions are often used as idealized models of physical systems with short-range potentials; in the simplest case point interactions are considered, but in the last decades also interactions supported on curves and hypersurfaces have attracted a lot of attention, see the monographs [2, 4, 36], the review [32], and, for instance, [3, 5, 10, 13, 25, 33, 34, 35, 37] for a small selection of papers in this area. It will be shown in Theorem 7.4 that the trace class condition ...... see [10, Proposition 3.7] and [25] for more details. For c ∈ R we shall also make use of the self-adjoint operator which coincides with the acoustic single-layer potential for the Helmholtz equation, that is, ...The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schr\"{o}dinger operators with compactly supported potentials, and finally, Schr\"{o}dinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.
- ... The self-adjointness and qualitative spectral properties of Schrödinger operators with δ-and δ 1 -interactions are well understood, see e.g. [6,7,11,15,16,29] and the references therein, and the discrete eigenvalues can be characterized via an abstract version of the Birman Schwinger principle. However, following the strategy from the point interaction model one arrives, instead of Key words and phrases. ...... We are interested in the eigenvalues of two kinds of perturbations of P as self-adjoint operators in L 2 pR n q which are formally given by A α :" P ` αδ Σ and B β :" P ` βxδ 1 Σ , ¨yδ 1 Σ , where δ Σ is the Dirac δ-distribution supported on Σ and the interaction strengths α, β are real valued functions defined on Σ with α, β ´1 P L 8 pΣq. For P " ´∆ these operators have been intensively studied e.g. in [7,11,15,16], for certain strongly elliptic operators and smooth surfaces several properties of A α and B β have been investigated in [6,29]. For the realization of A α as an operator in L 2 pR n q we remark that if the distribution A α f is generated by an L 2 -function, then f i/e :" f ae Ω i/e has to fulfill (1.1) γf i " γf e and B ν f e ´ B ν f i " αγf on Σ, as then the singularities at Σ compensate, cf. ...Preprint
- Jul 2019

In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $\delta$- and $\delta'$-interactions are studied. We show the self-adjointness of these operators and derive equivalent formulations for the eigenvalue problems involving boundary integral operators. These formulations are suitable for the numerical computations of the discrete eigenvalues and the corresponding eigenfunctions by boundary element methods. We provide convergence results and show numerical examples. - ... In the informal language, the operator H α is the distributional Laplacian in R 2 \ Γ with interface conditions [∂u] + αu = 0 on Γ, where [∂u] denotes a suitably defined jump of the normal derivative of u on Γ, see e.g. [2,6] for a more detailed discussion. ...... The present work is dedicated to the memory of Johannes F. Brasche . His first works on Schrödinger operators with measure potentials [1,6] served as a basis for the rigorous mathematical study of a large class of quantum-mechanical models, and the works of last years on large coupling convergence [3] suggested a far-reaching abstract generalization of strongly coupled δ-interactions, which will certainly lead to further progress in the domain. ...Preprint
- Sep 2019

Let $\Gamma\subset \mathbb{R}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues of the Schr\"odinger operator $H_\alpha$ with the attractive $\delta$-potential of strength $\alpha>0$ supported by $\Gamma$, which is defined by its quadratic form \[ H^1(\mathbb{R}^2)\ni u\mapsto \iint_{\mathbb{R}^2} |\nabla u|^2\,\mathrm{d}x-\alpha\int_\Gamma u^2\, \mathrm{d}s, \] where $\mathrm{d}s$ stands for the one-dimensional Hausdorff measure on $\Gamma$. It is shown that if $n\in\mathbb{N}$ is fixed and $\alpha$ is large, then the well-defined $n$th eigenvalue $E_n(H_\alpha)$ of $H_\alpha$ behaves as \[ E_n(H_\alpha)=-\alpha^2 + 2^{\frac{2}{p+2}} \mathcal{E}_n \,\alpha^{\frac{6}{p+2}} + \mathcal{O}(\alpha^{\frac{6}{p+2}-\eta}), \] where the constants $\mathcal{E}_n>0$ are the eigenvalues of an explicitly given one-dimensional Schr\"odinger operator determined by the cusp, and $\eta>0$. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when~$\Gamma$ is smooth or piecewise smooth with non-zero angles. - ... Let L; L 2 @ in , be smooth bounded curve without self-intersections. To switch on the interaction between in and the external domain ex = R 3 n in through line-like window L we use so-called "restriction-extension" procedure (see, e.g., [8,19,20,21,22,23,24]). Let us start from the Neumann Laplacian in in. ...ArticleFull-text available
- Dec 2017

A model of the Helmholtz resonator with narrow slit is considered. The con- struction is based on the theory of self-adjoint extensions of symmetric operators. Resonance states are described. It is proved that the system of the resonance states is complete in L2(), where is the convex hull of the resonator with window. - ... This extension is by no means straightforward as one needs to introduce weighted L 1 norms of the potential into the estimates. More upper estimates for N − (V ) in the case d = 2 can be found in [4,6,7,8,13,18,19] and the references therein. ...In this paper we present quantitative upper estimates for the number of negative eigenvalues of a two dimensional Schrödinger operator with potential supported by an unbounded Lipschitz curve. The estimates are given in terms of weighted and type Orlicz norms of the potential.
- ... Since the spectral analysis of Dirac operators with strongly localized potentials is a challenging problem, such potentials are often replaced in mathematical physics by singular δ-type potentials. This idea was successfully applied in nonrelativistic quantum mechanics, see, e.g., [3,8,14,18,23,24,30] and the references therein, and in the recent years also in the relativistic setting. In this paper we study singular perturbations of the free Dirac operator A 0 acting in L 2 (R 3 ; C 4 ) ∼ = L 2 (R 3 ) 4 , which are formally given by A η,τ = A 0 + (ηI 4 + τ β)δ Γ , β := I 2 0 0 −I 2 ; see Section 2.2 and Section 3 below for the precise definition and the main properties of the appearing objects. ...PreprintFull-text available
- Aug 2019

We provide a limiting absorption principle for self-adjoint realizations of Dirac operators with electrostatic and Lorentz scalar $\delta$-shell interactions supported on regular compact surfaces. Then we show completeness of the wave operators and give a representation formula for the scattering matrix. - ... These potentials, which are called δ-interactions, are supported on sets of Lebesgue measure zero and used as idealized replacements for regular potentials localized in thin neighborhoods of the interaction supports in the ambient Euclidean space. In nonrelativistic quantum mechanics these interactions were successfully studied in the case of Schrödinger operators with point interactions in [1] or with δ-interactions supported on hypersurfaces in R d , e.g., in [10,13,21]. In the relativistic setting also the one dimensional case, i.e. ...Preprint
- Jul 2019

This paper is devoted to the study of the two-dimensional Dirac operator with an arbitrary combination of an electrostatic and a Lorentz scalar $\delta$-interaction of constant strengths supported on a closed curve. For any combination of the coupling constants a rigorous description of the self-adjoint realization of the operators is given and the spectral properties are described. For a non-zero mass and a critical combination of coupling constants the operator appears to have an additional point in the essential spectrum, which is related to a loss of regularity in the operator domain, and the position of this point is expressed in terms of the coupling constants. - ... In Section 3, after supplying, in Section 2, some preliminary material, we introduce the rigorous definition of the self-adjoint operators A F by providing their resolvent through a Kre˘ ın's type formula expressed in terms of the free resolvent (i.e the resolvent of A ∅ ) (see Theorem 3.7). Let us remark here that the operator A F could be equivalently defined by quadratic form methods (see, e.g., [9], [17] and references therein); however, in order to study the Limiting Absorption Principle (LAP for short) and the Scattering Matrix, one needs a convenient resolvent formula. One more important remark about our use of resolvent formulae is the following: proceeding as in [15], one could try to provide a resolvent formula of A F expressed directly in terms of A 0 ; however this would imply the use of trace (evaluation) operators in the operator domain of A 0 , and these are less well-behaved than in the Sobolev space H 2 (R 3 ), the self-adjointness domain of ∆ ∅ (in particular is not clear what should be the correct trace space). ...PreprintFull-text available
- Jul 2018

We study scattering for the couple $(A_{0},A_{F})$ of Schr\"odinger operators in $L^2(\mathbb{R}^3)$ formally defined as $A_0 = -\Delta + \alpha\, \delta_{\pi_0}$ and $A_F = -\Delta + \alpha\, \delta_{\pi_F}$, $\alpha >0$, where $\delta_{\pi_F}$ is the Dirac $\delta$-distribution supported on the deformed plane given by the graph of the compactly supported function $F:\mathbb{R}^{2}\to\mathbb{R}$ and $\pi_{0}$ is the undeformed plane corresponding to the choice $F\equiv 0$. We show asymptotic completeness of the corresponding wave operators, provide a Limiting Absorption Principle and give a representation formula for the Scattering Matrix $S_{F}(\lambda)$. Moreover we show that, as $F\to 0$, $\|S_{F}(\lambda)-\mathsf 1\|^{2}_{\mathfrak{B}(L^{2}({\mathbb S}^{2}))}={\mathcal O}\!\left(\int_{\mathbb{R}^{2}}d\textbf{x}|F(\textbf{x})|^{\gamma}\right)$, $0<\gamma<1$. - ... Since, by the correspondence with von Neumann's theory (see [33], [35]), any self-adjoint extension of S can be defined through (1.1) assuming the hypothesis ran(τ ) = K (equivalently, using the corresponding ordinary boundary triplet, see [14], [38, Theorem 14.7]), these results seems to settle down our questions about Z Q (at least in the case V = π, W = π * ). However, in cases where the defect indices of S are not finite, in particular in applications to partial differential operators, it can be much more convenient to do not require ran(τ ) = K (and sometimes V = 1, W = 1) and so to do not use ordinary boundary triplets (see, e.g., [12], [7], [13], [32], [23], [16], [17], [18], [2], [3], [4], [6], [28], [9], [10]). While some results regarding the validity of (1.1) for any z ∈ ρ(A 0 ) ∩ ρ(A Q ) are known even for not ordinary boundary triplets (as generalized boundary triplets and quasi-boundary triplets, see e.g., [3], [15], [5]), some additional hypotheses are required in these cases (which moreover do not necessarily conform to our framework). ...PreprintFull-text available
- Sep 2018

Let $A_{Q}$ be the self-adjoint operator defined by the $Q$-function $Q:z\mapsto Q_{z}$ through the Krein-like resolvent formula $$(-A_{Q}+z)^{-1}= (-A_{0}+z)^{-1}+G_{z}WQ_{z}^{-1}VG_{\bar z}^{*}\,,\quad z\in Z_{Q}\,,$$ where $V$ and $W$ are bounded operators and $$Z_{Q}:=\{z\in\rho(A_{0}):\text{$Q_{z}$ and $Q_{\bar z }$ have a bounded inverse}\}\,.$$ We show that $$Z_{Q}\not=\emptyset\quad\Longrightarrow\quad Z_{Q}=\rho(A_{0})\cap \rho(A_{Q})\,.$$ We do not suppose that $Q$ is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that $Q$ is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity. - ... Quantum motion in a geometrically complicated background is often modeled by networks of leaky quantum wires, which are mathematically described by Schrödin- ger operators with singular potentials supported on families of curves, see, e.g., the monograph [34, Chapter 10], the papers [10,17,30,59,79], and the references therein. Such models based on PDEs are mathematically more involved than the alternative concept of quantum graphs [14] based on ODEs, but have serious advan- tages from the physical point of view since they do not neglect quantum tunnelling between parts of the network. ...Preprint
- Dec 2018

The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian $A_\alpha =(i \nabla + A)^2 + \alpha\delta$ in $L^2(R^2)$ with a $\delta$-potential supported on a finite $C^{1,1}$-smooth curve $\Sigma$ are studied. Here $A = \frac{1}{2} B (-x_2, x_1)^\top$ is the vector potential, $B>0$ is the strength of the homogeneous magnetic field, and $\alpha\in L^\infty(\Sigma)$ is a position-dependent real coefficient modeling the strength of the singular interaction on the curve $\Sigma$. After a general discussion of the qualitative spectral properties of $A_\alpha$ and its resolvent, one of the main objectives in the present paper is a local spectral analysis of $A_\alpha$ near the Landau levels $B(2q+1)$. Under various conditions on $\alpha$ it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of $\alpha$. Furthermore, the use of Landau Hamiltonians with $\delta$-perturbations as model operators for more realistic quantum systems is justified by showing that $A_\alpha$ can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials. - ... From the mathematical point of view the parameter which matters is the codimension of the interaction support. If the latter is one the operators can be treated naturally using the associated quadratic forms in the spirit of [3], for codimension two the problem is more subtle. The scattering problem in the codimension one case was discussed in [10] where we considered the situation where the singular interaction support is a curve Γ in the plane, or more generally a family of curves, which can regarded as a local deformation of a single straight line Σ . ...Preprint
- Nov 2018

We consider the scattering problem for a class of strongly singular Schr\"odinger operators in $L^2(\mathbb{R}R^3)$ which can be formally written as $H_{\alpha,\Gamma}= -\Delta + \delta_\alpha(x-\Gamma)$ where $\alpha\in\mathbb{R}$ is the coupling parameter and $\Gamma$ is an infinite curve which is a local smooth deformation of a straight line $\Sigma\subset\mathbb{R}^3$. Using Kato-Birman method we prove that the wave operators $\Omega_\pm(H_{\alpha,\Gamma}, H_{\alpha,\Sigma})$ exist and are complete. - ... For instance, these are used for describing the interaction of quantum particles with charged hypersurfaces, for approximating Hamiltonians that describe the propagation of electrons through thin barriers, among others. An extensive liter- ature devoted to Schrödinger operator with singular potentials is available (see, e.g., the papers 2,3,4,5,6,7,8,9,10,11,12 , and references therein). ...In this paper, we consider one‐dimensional Schrödinger operators Sq on with a bounded potential q supported on the segment and a singular potential supported at the ends h0, h1. We consider an extension of the operator Sq in defined by the Schrödinger operator and matrix point conditions at the ends h0, h1. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator . Moreover, we provide closed‐form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self‐adjoint and nonself‐adjoint problems involving general point interactions described in terms of δ‐ and δ′‐distributions.
- ... [21][22][23][24], as well as on surfaces in e.g. [12, 13]. In [14], the author also investigates Dirac operators related to links in S 3 , yet his setup is fundamentally different, as he studies Dirac operators in the complement of a link in S 3 endowed with a hyperbolic metric. ...Article
- Jan 2017

We define Dirac operators on $\mathbb{S}^3$ (and $\mathbb{R}^3$) with magnetic fields supported on smooth, oriented links and prove self-adjointness of certain (natural) extensions. We then analyze their spectral properties and show, among other things, that these operators have discrete spectrum. Certain examples, such as circles in $\mathbb{S}^3$, are investigated in detail and we compute the dimension of the zero-energy eigenspace. - ... When M " R n , it is well known (see e.g. [21]) that since S is bounded, L S,δ α is a relatively compact perturbation of´∆of´∆ on L 2 pR n q and then σ ess pL S,δ α q " r0, `8q. ...Preprint
- Feb 2019

This memoir is devoted to a part of the results from the author about two topics: in the first part, the asymptotics of the low-lying eigenvalues of Schr\"odinger operators in domains that may have corners, and in the second part, the analysis of the thresholds of a class of fibered operators. The main common object is the magnetic Laplacian, and the two parts are connected through the study of model problems in unbounded domains. - ... [1, 6, 21, 31] and the references therein. On the other hand Dirac operators with electrostatic δ-shell interactions are also interesting from the mathematical point of view, since it can be expected that their spectral properties depend on the geometry of the interaction support and/or the interaction strength; such effects are studied in the monograph [23] and, e.g., in [12, 20, 21, 22, 24] for Schrödinger operators with δ-potentials. The mathematical study of Dirac operators with singular interactions supported on a set of measure zero started in the 1980s. ...Article
- Dec 2016

In this paper we prove that the Dirac operator with an electrostatic $\delta$-shell interaction of critical strength $\eta = \pm 2$ supported on a $C^2$-smooth compact surface $\Sigma$ is self-adjoint in $L^2(\mathbb{R}^3;\mathbb{C}^4)$ and we describe its domain explicitly in terms of traces and jump conditions in $H^{-1/2}(\Sigma; \mathbb{C}^4)$. While the non-critical interaction strengths $\eta \not= \pm 2$ have received a lot of attention in the recent past, the critical case $\eta = \pm 2$ remained open. Our approach is based on abstract techniques in extension theory of symmetric operators, in particular, boundary triples and their Weyl functions. - ... In the case d = 2, this estimate fails, but there has been significant recent progress in obtaining estimates of the CLR-type in two- dimensions and the best known estimates have been obtained in [26]. Esti- mates for the number of negative eigenvalues of Schrödinger operators with potential of the form V µ, where µ is a Radon measure and V is an appropri- ate function, were obtained in [6], and results from [26] were extended to this setting in [13,16]. In the present paper, we obtain estimates for the number of eigenvalues below the essential spectrum of a Schrödinger operator with potential of the form V µ similar to those in [16] in a strip subject to bound- ary conditions of the Robin type. ...PreprintFull-text available
- Mar 2019

In this paper, quantitative upper estimates for the number of eigenvalues of lying below the essential spectrum of Schroedinger operators with potentials generated by Ahlfors regular measures in a strip subject to two different types of boundary conditions (Robin and Dirichlet respectively) are presented. The estimates are presented in terms of weighted L^1 norms and Orlicz norms of the potential. - ... In the case d = 2, this estimate fails, but there has been significant recent progress in obtaining estimates of the CLR-type in twodimensions and the best known estimates have been obtained in [26]. Estimates for the number of negative eigenvalues of Schrödinger operators with potential of the form V µ, where µ is a Radon measure and V is an appropriate function, were obtained in [6], and results from [26] were extended to this setting in [13,16]. In the present paper, we obtain estimates for the number of eigenvalues below the essential spectrum of a Schrödinger operator with potential of the form V µ similar to those in [16] in a strip subject to boundary conditions of the Robin type. ...Article
- Mar 2019
- J MATH ANAL APPL

In this paper, quantitative upper estimates for the number of eigenvalues lying below the essential spectrum of Schrödinger operators with potentials generated by Ahlfors regular measures in a strip subject to two different types of boundary conditions (Robin and Dirichlet respectively) are presented. The estimates are presented in terms of weighted L ¹ norms and Orlicz norms of the potential. - ... Here c ∈ R\{0} is, up to the factor 2, the potential strength and can be either attractive (c < 0) or repulsive (c > 0), but is considered to be time independent. In the context of solvable models in quantum mechanics the spectral theory of Schrödinger operators with δ-potentials has deserved a lot of attention in the past decades; for a first glance we refer the reader to the monograph [18] and the contributions [19][20][21]. The formal Eq. ...ArticleFull-text available
- Mar 2019

The main goal of this note is to study the time evolution of superoscillations under the 1D-Schrödinger equation with attractive or repulsive Dirac \(\delta \)-potential located at the origin of the real line. Such potentials are of particular interest since they simulate short range interactions and the corresponding quantum system is an explicitly solvable model. Moreover, we give the large time asymptotics of this solution, which turns out to be different for the repulsive and the attractive model. The method that we use to study the time evolution of superoscillations is based on the continuity of the time evolution operator acting in a space of exponentially bounded entire functions. - ... 1]); Dirichlet screens have been studied firstly, in a 2-dimensional setting, in [19]. Semi- transparent interface conditions appear, apart in quantum mechanical models (see, e.g., [8], [4] and references therein), in connections with acoustic models with gradient singularities, see [24]. Conditions of the type αγ 0 = [γ 1 ]u appear in [18] and [6] in a non self-adjoint setting (i.e. when α is complex-valued): this compels the use of different data operators. ...PreprintFull-text available
- Jan 2019

We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of self-adjoint operators $(\widetilde\Delta,\Delta)$, where $\Delta$ is the free Laplacian in $L^{2}({\mathbb R}^{3})$ and $\widetilde\Delta$ is one of its singular perturbations, i.e., such that the set $\{u\in H^{2}({\mathbb R}^{3})\cap \text{dom}(\widetilde\Delta)\, :\, \Delta u=\widetilde\Delta u\}$ is dense. Typically $\widetilde\Delta$ corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular our results apply to standard, either separating or semi-transparent, boundary conditions at $\Gamma=\partial\Omega$, where $\Omega\subset{\mathbb R}^{3}$ is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on $\Sigma\subset\Gamma$, a relatively open subset with a Lipschitz boundary. We show that either $\Gamma$ or $\Sigma$ are determined by the knowledge of the Scattering Matrix, equivalently of the Far Field Operator, at a single frequency. - The main objective of this paper is to systematically develop a spectral and scattering theory for self-adjoint Schrödinger operators with \(\delta \)-interactions supported on closed curves in \(\mathbb {R}^3\). We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten–von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix.
- Article
- Jan 2017

We discuss operators of the type $H = -\Delta + V(x) - \alpha \delta(x-\Sigma)$ with an attractive interaction, $\alpha>0$, in $L^2(\mathbb{R}^3)$, where $\Sigma$ is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and $V$ is a potential bias being a positive constant $V_0$ in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, $V_0=\alpha^2$. We show that $\sigma_\mathrm{disc}(H)$ is then empty if the bias is supported in the `exterior' region, while in the opposite case isolated eigenvalues may exist. - Article
- Jan 2017

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schr\"odinger operator with an attractive $\delta$-interaction of a fixed strength the support of which is a star graph with finitely many edges of an equal length $L \in (0,\infty]$. Under the constraint of fixed number of the edges and fixed length of them, we prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle, properties of the Macdonald function, and on a geometric inequality for polygons circumscribed into the unit circle. - ArticleFull-text available
- Feb 2017

In this paper we construct the resolvent and analyze scattering properties of the quantum two-body problem introduced in [arXiv:1504.08283] which describes two (distinguishable) particles moving on the half-line under the influence of singular two-particle interactions. Most importantly, due to the spatial localization of the interactions the two-body problem is of a non-separable nature. We will discuss the presence of embedded eigenvalues and using the detailed knowledge about the kernel of the resolvent we prove a version of the limiting absorption principle. Furthermore, by an appropriate adaptation of the Lippmann-Schwinger approach we are able to construct generalized eigenfunctions which consequently allow us to establish an explicit expression for the (on-shell) scattering amplitude. An approximation of the scattering amplitude in the weak-coupling limit is also derived. - In this paper we consider the three-dimensional Schr\"{o}dinger operator with a $\delta$-interaction of strength $\alpha > 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,\beta f(x))$, where $\beta \in [0,\infty)$ and $f\colon \mathbb{R}^2\rightarrow\mathbb{R}$, $f\not\equiv 0$, is a $C^2$-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schr\"odinger operator coincides with $[-\frac14\alpha^2,+\infty)$. We prove that for all sufficiently small $\beta > 0$ its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit $\beta \rightarrow 0+$. On a qualitative level this eigenvalue tends to $-\frac14\alpha^2$ exponentially fast as $\beta\rightarrow 0+$.
- Article
- Sep 2016
- Math Nachr

We show that a Schrödinger operator Aδ,α with a δ-interaction of strength α supported on a bounded or unbounded C²-hypersurface Σ⊂Rd,d≥2, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator Aδ,α with a singular interaction is regarded as a self-adjoint realization of the formal differential expression -Δ-α-δΣ,·(combining right arrow above)δΣ, where α:Σ→R is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result. - Article
- Sep 2017
- REV MATH PHYS

We analyze scattering in a system of two (distinguishable) particles moving on the half-line ℝ+ under the influence of singular two-particle interactions. Most importantly, due to the spatial localization of the interactions, the two-body problem is of a non-separable nature. We will discuss the presence of embedded eigenvalues and using the obtained knowledge about the kernel of the resolvent, we prove a version of the limiting absorption principle. Furthermore, by an appropriate adaptation of the Lippmann-Schwinger approach, we are able to construct generalized eigenfunctions which consequently allow us to establish an explicit expression for the (on-shell) scattering amplitude. An approximation of the scattering amplitude in the weak-coupling limit is also derived. - ArticleFull-text available
- Apr 2018

- Article
- Nov 2017
- ACTA APPL MATH

We find the exponential growth rate of the population outside a ball with time dependent radius for a branching Brownian motion in Euclidean space. We then see that the upper bound of the particle range is determined by the principal eigenvalue of the Schr\"odinger type operator associated with the branching rate measure and branching mechanism. We assume that the branching rate measure is small enough at infinity, and can be singular with respect to the Lebesgue measure. We finally apply our results to several concrete models. - Article
- Dec 2017

It is known that convergence of l.s.b. closed symmetric sesquilinear forms implies norm resolvent convergence of the associated self-adjoint operators and this in turn convergence of discrete spectra. In this paper in both cases sharp estimates for the rate of convergence are derived. An algorithm for the numerical computation of eigenvalues of generalized Schr\"{o}dinger operators in $L^2(\mathbb{R})$ is presented and illustrated by explicit examples; the mentioned general results on the rate of convergence are applied in order to obtain error estimates for these computations. An extension of the results to Schr\"{o}dinger operators on metric graphs is sketched. - We consider a class of two-dimensional Schr\"odinger operator with a singular interaction of the $\delta$ type and a fixed strength $\beta$ supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov-Bohm flux $\alpha\in [0,\frac12]$ in the center. It is shown that if $\beta\ne 0$, there is a critical value $\alpha_\mathrm{crit} \in(0,\frac12)$ such that the discrete spectrum has an accumulation point when $\alpha<\alpha_\mathrm{crit} $, while for $\alpha\ge\alpha_\mathrm{crit} $ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $\alpha\in (0,\frac12)$ and $|\beta|$ small enough.
- Article
- Nov 2017
- MATH NOTES+

Let Γ be a simply connected unbounded C2-hypersurface in ℝn such that Γ divides ℝn into two unbounded domains D±. We consider the essential spectrum of Schrödinger operators on ℝn with surface δΓ-interactions which can be written formally as $${H_\Gamma } = - \Delta + W - {\alpha _\Gamma }{\delta _{\Gamma ,}}$$, where −Δ is the nonnegative Laplacian in ℝn, W ∈ L∞(ℝn) is a real-valued electric potential, δΓ is the Dirac δ-function with the support on the hypersurface Γ and αΓ ∈ L∞(Γ) is a real-valued coupling coefficient depending of the points of Γ. We realize HΓ as an unbounded operator AΓ in L2(ℝn) generated by the Schrödinger operator $${H_\Gamma } = - \Delta + Won{\mathbb{R}^n}\backslash \Gamma $$ and Robin-type transmission conditions on the hypersurface Γ. We give a complete description of the essential spectrum of AΓ in terms of the limit operators generated by AΓ and the Robin transmission conditions. - Article
- Jan 2018
- APPL ANAL

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive -interaction supported on an open arc with two free endpoints. Under a constraint of fixed length of the arc, we prove that the maximizer is a line segment, the respective spectral isoperimetric inequality being strict. We also show that in the optimization problem for the same spectral quantity, but with the constraint of fixed endpoints, the optimizer is the line segment connecting them. As a consequence of the result for -interaction, we obtain that a line segment is also the maximizer in the optimization problem for the lowest eigenvalue of the Robin Laplacian on a plane with a slit along an open arc of fixed length. - ArticleFull-text available
- Jan 2018

The paper analyzes the possibility of applying the photometric method to determine the size of the vesicle of niosomal dispersion based on the determination of turbidity in disperse media. The size of the niosomes determined by this method did not differ significantly from the results of scanning electron microscopy. © 2018 Stavropol State Medical University. All rights reserved. - PreprintFull-text available
- May 2018

In this paper we review recent work that has been done on quantum many-particle systems on metric graphs. Topics include the implementation of singular interactions, Bose-Einstein condensation, sovable models and spectral properties of some simple models in connection with superconductivity in wires. - Article
- Jun 2018
- MATH METHOD APPL SCI

We consider Schrödinger operators H = −Δ + W + Ws on with regular potentials and singular potentials with supports on unbounded enough smooth hypersurfaces Γ. In particular, we consider singular potentials that are linear combinations of δ−functions on Γ and its normal derivatives. We consider extensions of H as symmetric operators in with domain to self‐adjoint operators in . These extensions are realized as operators of transmission problems for −Δ + W in the space with some transmission conditions on Γ. Applying this approach, we obtain an effective description of essential spectra of the described Schrödinger operators. - Preprint
- Jan 2019

In this article Dirac operators $A_{\eta, \tau}$ coupled with combinations of electrostatic and Lorentz scalar $\delta$-shell interactions of constant strength $\eta$ and $\tau$, respectively, supported on compact surfaces $\Sigma \subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $\delta$-potentials are modelled by coupling conditions at $\Sigma$. In the proof of the self-adjointness of $A_{\eta, \tau}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{\eta, \tau}$ is possible. In particular, the essential spectrum of $A_{\eta, \tau}$ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of $A_{\eta, \tau}$ is computed and it is discussed that for some special interaction strengths $A_{\eta, \tau}$ is decoupled to two operators acting in the domains with the common boundary $\Sigma$. - Preprint
- Feb 2019

In this survey we gather recent results on Dirac operators coupled with $\delta$-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterward we switch to an approximation question: can these operators be recovered as limits of Dirac operators coupled with squeezing potentials ? We also discuss spectral features of these models. Namely, we recall the main spectral consequences of a resolvent formula and conclude the survey by commenting a result of asymptotic nature for the eigenvalues in the gap of a Dirac operator coupled with a Lorentz-scalar interaction. - ArticleFull-text available
- Mar 2019

In this article, Dirac operators \(A_{\eta , \tau }\) coupled with combinations of electrostatic and Lorentz scalar \(\delta \)-shell interactions of constant strength \(\eta \) and \(\tau \), respectively, supported on compact surfaces \(\Sigma \subset \mathbb {R}^3\) are studied. In the rigorous definition of these operators, the \(\delta \)-potentials are modeled by coupling conditions at \(\Sigma \). In the proof of the self-adjointness of \(A_{\eta , \tau }\), a Krein-type resolvent formula and a Birman–Schwinger principle are obtained. With their help, a detailed study of the qualitative spectral properties of \(A_{\eta , \tau }\) is possible. In particular, the essential spectrum of \(A_{\eta , \tau }\) is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of \(A_{\eta , \tau }\) is computed and it is discussed that for some special interaction strengths, \(A_{\eta , \tau }\) is decoupled to two operators acting in the domains with the common boundary \(\Sigma \). - Preprint
- Mar 2019

We determine the long time behavior and the exact order of the tail probability for the maximal displacement of a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of the associated Schr\"odinger type operator. We also prove the existence of the Yaglom type limit for the distribution of the population outside the forefront. To establish our results, we show a sharp and locally uniform growth order of the Feynman-Kac semigroup - Chapter
- Jun 2019

We consider the scattering problem for a class of strongly singular Schrödinger operators in \(L^2({\mathbb R}^3)\) which can be formally written as \(H_{\alpha ,\varGamma }= -\varDelta + \delta _\alpha (x-\varGamma )\), where \(\alpha \in {\mathbb R}\) is the coupling parameter and \(\varGamma \) is an infinite curve which is a local smooth deformation of a straight line \(\varSigma \subset {\mathbb R}^3\). Using Kato–Birman method, we prove that the wave operators \(\varOmega _\pm (H_{\alpha ,\varGamma }, H_{\alpha ,\varSigma })\) exist and are complete. - Preprint
- Jun 2019

We discuss the spectral properties of singular Schr\"odinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case the discrete spectrum is nonempty if the star arms are long enough. Our main result concerns spectral optimization: we show that the principal eigenvalue is uniquely maxi\-mized when the arms are arranged in one of the known five sharp configurations known as solutions of the closely related Thomson problem. - Article
- Sep 2016

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schr\"odinger operator with an attractive $\delta$-interaction supported on an open arc with two free endpoints. Under a constraint of fixed length of the arc, we prove that the maximizer is a line segment, the respective spectral isoperimetric inequality being strict. We also show that in the optimization problem for the same spectral quantity, but with the constraint of fixed endpoints, the optimizer is the line segment connecting them. Furthermore, we prove that a line segment is also the maximizer in the optimization problem for the lowest eigenvalue of the Robin Laplacian on a plane with a slit along an open arc of fixed length.

- Article
- Dec 1989
- J MATH PHYS

Dirac operators with a contact interaction supported by a sphere are studied restricting attention to the operators that are rotationally and space‐reflection symmetric. The partial wave operators are constructed using the self‐adjoint extension theory, a particular attention being paid to those among them that can be interpreted as δ‐function shells of scalar and vector nature. The class of interactions for which the sphere becomes impenetrable is specified and spectral properties of the obtained Hamiltonians are discussed. - Article
- Jan 1992
- T AM MATH SOC

Potential scattering theory is a very well-developed and understood subject. Scattering for SchrOdinger operators represented formally by Δ-+V, where V is a generalized function such as a a-function, is less understood and requires form perturbation techniques. A general scattering theory for a large class of suchsingular perturbations of the Laplacian is developed. The theory has application to obstacle scattering. One considers an alternative mathematical model of an obstacle in Rn.Instead of representing the obstacle by deleting the region inhabited by the obstacle from Rn, the surface of the obstacle is treated as impenetrable. The impenetrable surface is understood to be the limiting case of a sequence of highly spiked potentials whosesupport converges to the surface of the obstacle and whose peaks grow without bound. The limiting case is identified as a a-function acting on the surface of the obstacle. Hamiltonians for the limiting case are constructed and the conditions governing the existence and completeness of the associated wave operators are determined through application of the general theory. - Relations between Schrödinger forms associated with Schrödinger operators in L2(Ω;dnx), Ω ⊂ Rn open, n≥1 and the corresponding Dirichlet forms are investigated. Various concrete examples are presented.
- Article
- Apr 1985
- J MATH PHYS

We study the Hamiltonians for nonrelativistic quantum mechanics in one dimension, in terms of energy forms ∫‖df/dx‖2 dx+∫‖ f ‖2 d( μ −ν), where μ and ν are positive, not necessarily finite measures on the real line. We cover, besides regular potentials, cases of very singular interactions (e.g., a particle interacting with an infinite number of fixed particles by ‘‘delta function potentials’’ of arbitrary strengths). We give conditions for lower semiboundedness and closability of the above energy forms, which are sufficient and, for certain classes of potentials (e.g., μ−ν a signed measure), also necessary. In contrast to the results in other approaches, no regularity conditions and no restrictions on the growth of the measures μ and ν at infinity are needed. - Article
- Mar 1988
- J MATH PHYS

Using the theory of self-adjoint extensions of symmetric operators the precise mathematical definition of the quantum Hamiltonian describing a finite number of δ interactions with supports on concentric spheres is given. Its resolvent is also derived, its spectral properties are described, and it is shown how this Hamiltonian can be obtained as a norm resolvent limit of a family of local scaled short-range Hamiltonians. - Article
- Oct 1988
- J MATH PHYS

Let be the closure of the restriction of the three‐dimensional Laplacian −Δ on the domain C∞0(R3\Σ), where Σ=∪Nj=1∂K(0,Rj) and ∼(K(0,Rj)) is a closed ball of radius Rj centered at the origin in R3. It is well known that is a closed symmetric operator with deficiency indices (∞,∞). In this paper all self‐adjoint (s.a.) extensions of are constructed; these extensions contain as particular cases the quantum Hamiltonian describing concentric δ‐ and δ′‐sphere interactions. It is also shown that the s.a. extensions of may be obtained as norm‐resolvent limits of momentum cutoff and scaled separable potentials. - In this paper, the free motion of a particle on a manifold that consists of a one‐dimensional and a two‐dimensional part connected in one point is discussed. The class of admissible Hamiltonians is found using the theory of self‐adjoint extensions. Particular attention is paid to those Hamiltonians that allow the particle to pass through the point singularity; the reflection coefficient and other quantities characterizing scattering on the connection point are calculated. A possible application is also discussed.
- Article
- Nov 1988
- J Phys Math Gen

The authors consider the motion of a charged quantum particle on a loop with two external leads which is placed into an electrostatic field. The loop Hamiltonian is chosen in the simplest possible way; in order to join it to the free Hamiltonians describing the leads, they employ a method based on self-adjoint extensions. Under a symmetry requirement, the resulting full Hamiltonian contains four free parameters; each junction is characterised by a pair of them. The system under consideration represents a model of metallic or semiconductor structure that can be fabricated by presently available technologies. Assuming the ballistic regime for electrons in such a structure, the authors calculate the resistance dependence on intensity of the external field. The results suggest the possibility of constructing quantum interference transistors whose size and switching voltage would be much smaller than in current microchips. - Article
- Jan 1989
- J MATH PHYS

Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with tα−1/Γ(α), α=1,2, respectively. Fractional diffusion and wave equations are obtained by letting α vary in (0,1) and (1,2), respectively. The corresponding Green’s functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited. In particular, it is shown that the Green’s function of fractional diffusion is a probability density. - Article
- May 1987
- LETT MATH PHYS

Two new analytically solvable models of relativistic point interactions in one dimension (being natural extensions of the nonrelativistic , resp. , interactions) are considered. Their spectral properties in the case of finitely many point interactions as well as in the periodic case are fully analyzed. Moreover, we explicitly determine the spectrum in the case of independent, identically distributed random coupling constants and derive the analog of the Saxon and Hutner conjecture concerning gaps in the energy spectrum of such systems. - Article
- Oct 1988
- Czech J Phys

We treat a free spinless quantum particle moving on a configuration manifold which consists of two identical parts connected in one point. Most attention is paid to the three-dimensional case when these parts are halfspaces with Neumann condition on the boundary; we also discuss briefly a more general boundary conditions. The class of admissible Hamiltonians is constructed by means of the theory of self-adjoint extensions. Among them, particularly important is a two-parameter family whose elements are invariant with respect to exchange of the halfspaces; we compute the transmission coefficient for each of these extensions. We discuss also the motion on two planes considered in our recent paper, obtaining another characterization of the admissible Hamiltonians. In conclusion, the two situations are compared as models for point-contact spectroscopical experiments in thin metal films. - Article
- Jan 1986
- LETT MATH PHYS

We study the free motion of a particle on the manifold which consists of two planes connected at one point. The four-parameter family of admissible Hamiltonians is constructed by self-adjoint extensions of the free Hamiltonian with the singular point removed. The probability of penetration between the two parts of the configuration manifold is calculated. The results can be used as a model for quantum point-contact spectroscopy. - Article
- Apr 1988
- Il Nuovo Cimento

We study the Schrödinger Hamiltonian corresponding to a finite number of δ’-interactions supported by concentric spheres. In particular we discuss the self-adjointness of the Hamiltonian, derive the resolvent equation and study the spectral properties. Si studia l’hamiltoniana di Schrödinger che corrisponde a un numero fïnito di interazioni δ’ sostenute da sfere concentriche. In particolare si discute la capacità dell’Hamiltoniana di essere autoaggiunta, si deriva l’equazione risolutiva e si studiano le proprietà spettrali. Мы исследуем Гамильт ониан Шредингера, соответствующий конечному числу σ′-вз аимодействий, опираю щихся на концентрические сфе ры. В частности, мы обсужда ем самосопряженный Гамильтониан, выводи м уравнение резольвенты и исслед уем спектральные сво йства. - We sketch here two mathematical models intended to describe the point-contact spectroscopical experiments. A new item is added to the list of recently discovered applications of the self-adjoint extensions theory.
- Article
- Aug 1989
- REP MATH PHYS

We consider the free motion of a quantum particle on the graph consisting of three half-lines whose ends are connected. It is shown that the time evolution can be described by a Hamiltonian and the class of all admissible Hamiltonians is constructed using the theory of self-adjoint extensions. Three subclasses are discussed in detail: (a) the one-parameter family of Hamiltonians whose domains contain functions continuous at the junction, (b) the wider four-parameter family with the wavefunctions continuous between two branches of the graph only, (c) the Hamiltonian invariant under permutations of the branches. For the class (c), generalization to the graphs consisting of n half-lines is given. The scattering problem of such a branching graph is also discussed. - Article
- Oct 1977
- ANN PHYS-NEW YORK

Under certain conditions on the potential a one-dimensional Schrödinger operator has a unique bound state in the limit of weak coupling while under other conditions no bound state is present in this limit. This question is investigated for potentials obeying ∫ (1 + |χ|) | V(χ)|dχ<∞. An asymptotic formula for the bound state is proven. - Article
- May 1990
- J MATH PHYS

The free Dirac operator defined on composite one‐dimensional structures consisting of finitely many half‐lines and intervals is investigated. The influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self‐adjoint extensions of the Dirac operator. General relations between the parameters of the extensions and the eigenvaluesr e s p. the scattering coefficients are derived and then applied to the cases of a bundle of half‐lines, a point defect, a branching line, and an eye‐shaped structure. - Singular perturbations of — Δ in L²(R³) supported by points, regular curves and regular surfaces are considered. Using a renormalization technique the corresponding quadratic forms are constructed and a complete characterization of the domain and the action of the operators is given, together with explicit expressions for the resolvent. © 1990, Research Institute forMathematical Sciences. All rights reserved.