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Generalized Robin Boundary Conditions, RobintoDirichlet Maps, and KreinType Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains
 IN PERSPECTIVES IN PARTIAL DIFFERENTIAL EQUATIONS, HARMONIC ANALYSIS AND APPLICATIONS, D. MITREA AND M. MITREA (EDS.), PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS, AMERICAN MATHEMATICAL SOCIETY
, 2008
"... We study generalized Robin boundary conditions, RobintoDirichlet maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2. We also discuss the case of bounded C 1,rdomains, (1/2) < r < 1. ..."
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Cited by 31 (11 self)
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We study generalized Robin boundary conditions, RobintoDirichlet maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2. We also discuss the case of bounded C 1,rdomains, (1/2) < r < 1.
RobintoRobin Maps and KreinType Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains
, 2008
"... We study RobintoRobin maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2, with generalized Robin boundary conditions. ..."
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Cited by 28 (10 self)
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We study RobintoRobin maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2, with generalized Robin boundary conditions.
Derivatives of (modified) Fredholm determinants and stability of standing and travelling waves
 J. Math. Pures Appl
, 2008
"... Abstract. Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman–Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency z ..."
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Cited by 19 (7 self)
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Abstract. Continuing a line of investigation initiated in [11] exploring the connections between Jost and Evans functions and (modified) Fredholm determinants of Birman–Schwinger type integral operators, we here examine the stability index, or sign of the first nonvanishing derivative at frequency zero of the characteristic determinant, an object that has found considerable use in the study by Evans function techniques of stability of standing and traveling wave solutions of partial differential equations (PDE) in one dimension. This leads us to the derivation of general perturbation expansions for analyticallyvarying modified Fredholm determinants of abstract operators. Our main conclusion, similarly in the analysis of the determinant itself, is that the derivative of the characteristic Fredholm determinant may be efficiently computed from first principles for integral operators with semiseparable integral kernels, which include in particular the general onedimensional case, and for sums thereof, which latter possibility appears to offer applications in the multidimensional case. A second main result is to show that the multidimensional characteristic Fredholm determinant is the renormalized limit of a sequence of Evans functions defined in [23] on successive Galerkin subspaces, giving a natural extension of the onedimensional results of [11] and answering a question of [27] whether this sequence might possibly converge (in general, no, but with renormalization, yes). Convergence is useful in practice for numerical error control and acceleration. 1.
Generalized Qfunctions and DirichlettoNeumann maps for elliptic differential operators
"... Abstract. The classical concept of Qfunctions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the DirichlettoNeumann map in the theory of elliptic differential equations can be interpreted as a generalized Qfunction. For couplings ..."
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Cited by 19 (8 self)
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Abstract. The classical concept of Qfunctions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the DirichlettoNeumann map in the theory of elliptic differential equations can be interpreted as a generalized Qfunction. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an H 2framework are obtained. 1.
Spectral theory for perturbed Krein Laplacians in nonsmooth domains
, 2010
"... We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all do ..."
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Cited by 16 (11 self)
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We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r> 1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula #{j ∈ N  λK,Ω,j ≤ λ} = (2π) −n vnΩ  λ n/2 + O ` λ (n−(1/2))/2 ´ as λ → ∞, where vn = πn/2 /Γ((n/2)+1) denotes the volume of the unit ball in Rn, and λK,Ω,j, j ∈ N, are the nonzero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − ∆ + V defined on C ∞ 0 (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exteriortype domains Ω = Rn \K, n ≥ 3, with K ⊂ Rn compact and vanishing
NONLOCAL ROBIN LAPLACIANS AND SOME REMARKS ON A PAPER BY FILONOV
, 2009
"... The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to selfadjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robintype boundary conditions on bounded Lipschitz domains Ω ⊂ R n, n ∈ N, n ≥ 2. Second, we e ..."
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Cited by 16 (5 self)
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The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to selfadjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robintype boundary conditions on bounded Lipschitz domains Ω ⊂ R n, n ∈ N, n ≥ 2. Second, we extend Friedlander’s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems.
A description of all selfadjoint extensions of the Laplacian and Krĕıntype resolvent formulas on nonsmooth domains
 J. Anal. Math
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Selfadjoint extensions of the Laplacian and Kreintype resolvent formulas in nonsmooth domains
, 2009
"... This paper has two main goals. First, we are concerned with the classification of selfadjoint extensions of the Laplacian − ∆ ˛ ˛ C ∞ 0 (Ω) in L2 (Ω; d n x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasiconvex domains), which contain all convex domai ..."
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Cited by 13 (9 self)
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This paper has two main goals. First, we are concerned with the classification of selfadjoint extensions of the Laplacian − ∆ ˛ ˛ C ∞ 0 (Ω) in L2 (Ω; d n x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasiconvex domains), which contain all convex domains, as well as all domains of class C 1,r, for r ∈ (1/2, 1). Second, we establish Kreintype formulas for the resolvents of the various selfadjoint extensions of the Laplacian in quasiconvex domains and study the properties of the corresponding Weyl–Titchmarsh operators (or energydependent DirichlettoNeumann maps). One significant technical innovation in this paper is an extension of the classical boundary trace theory for functions in spaces which lack Sobolev regularity in a traditional sense, but are suitably adapted to the
Trace formulas for Schrödinger operators in connection with scattering theory for finitegap backgrounds
 in Spectral Theory and Analysis
, 2011
"... Abstract. We investigate trace formulas for onedimensional Schrödinger operators which are trace class perturbations of quasiperiodic finitegap operators using Krein’s spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg–de Vries hierarchy ..."
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Cited by 7 (5 self)
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Abstract. We investigate trace formulas for onedimensional Schrödinger operators which are trace class perturbations of quasiperiodic finitegap operators using Krein’s spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg–de Vries hierarchy in this class and relate them to the reflection coefficients via Abelian integrals on the underlying hyperelliptic Riemann surface. 1.
Spectral boundary value problems and their linear operators
, 2009
"... The paper offers a selfconsistent account of the spectral boundary value problems developed from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary condition is introduced and results on its solvabil ..."
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Cited by 7 (1 self)
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The paper offers a selfconsistent account of the spectral boundary value problems developed from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary condition is introduced and results on its solvability complemented by representations of weak and strong solutions are obtained. The question of existence of a closed linear operator defined by a given boundary condition and description of its domain is studied in detail. This question is addressed on the basis of a version of Krein’s resolvent formula derived from the obtained representations for solutions. Usual resolvent identities for two operators associated with two different boundary conditions are written in terms of the so called Moperator and closed linear operators defining these conditions. Two examples illustrate the abstract core of the paper. Other applications to the theory of partial differential operators and to the mathematical physics are outlined.