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Yoshio Tanigawa - One of the best experts on this subject based on the ideXlab platform.
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Titchmarsh s method for the approximate functional equations for unicode stix x1d701 prime s 2 unicode stix x1d701 s unicode stix x1d701 prime prime s and unicode stix x1d701 prime s unicode stix x1d701 prime prime s
Canadian Journal of Mathematics, 2018Co-Authors: Jun Furuya, Makoto T Minamide, Yoshio TanigawaAbstract:Let $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$ , where $-1/2<\unicode[STIX]{x1D70E}<3/2,x,y\geqslant 1,xy=(|t|/2\unicode[STIX]{x1D70B})^{2}$ . Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$ . In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ , and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0<\unicode[STIX]{x1D70E}<1$ ) whose error terms contain the factor $((x+y)/|t|)^{1/4}$ . In this paper we remove this factor from these three error terms by using the method of Titchmarsh.
Jun Furuya - One of the best experts on this subject based on the ideXlab platform.
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Titchmarsh s method for the approximate functional equations for unicode stix x1d701 prime s 2 unicode stix x1d701 s unicode stix x1d701 prime prime s and unicode stix x1d701 prime s unicode stix x1d701 prime prime s
Canadian Journal of Mathematics, 2018Co-Authors: Jun Furuya, Makoto T Minamide, Yoshio TanigawaAbstract:Let $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$ , where $-1/2<\unicode[STIX]{x1D70E}<3/2,x,y\geqslant 1,xy=(|t|/2\unicode[STIX]{x1D70B})^{2}$ . Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$ . In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ , and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0<\unicode[STIX]{x1D70E}<1$ ) whose error terms contain the factor $((x+y)/|t|)^{1/4}$ . In this paper we remove this factor from these three error terms by using the method of Titchmarsh.
Fritz Gesztesy - One of the best experts on this subject based on the ideXlab platform.
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dirichlet to neumann maps abstract weyl Titchmarsh m functions and a generalized index of unbounded meromorphic operator valued functions
Journal of Differential Equations, 2016Co-Authors: Jussi Behrndt, Fritz Gesztesy, Helge Holden, Roger NicholsAbstract:Abstract We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrodinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl–Titchmarsh M-functions and Donoghue-type M-functions corresponding to closed extensions of symmetric operators belong to it. The principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of Robin realizations of non-self-adjoint Schrodinger operators, and more abstract pairs of closed operators in Hilbert spaces with the generalized index of the corresponding energy dependent Dirichlet-to-Neumann maps and abstract Weyl–Titchmarsh M-functions, respectively.
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Inverse spectral problems for Schrödinger-type operators with distributional matrix-valued potentials
2016Co-Authors: Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, Er Sakhnovich, Gerald TeschlAbstract:Abstract. The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line from the underlying Weyl–Titchmarsh function. 1
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Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials
Operators and Matrices, 2013Co-Authors: Fritz Gesztesy, Rudi Weikard, Maxim ZinchenkoAbstract:We develop Weyl-Titchmarsh theory for self-adjoint Schrodinger operators Hα in L 2 ((a,b);dx;H ) associated with the operator-valued differential expression τ = −(d 2 /dx 2 ) + V(� ), with V : (a,b) → B(H ), and H a complex, separable Hilbert space. We assume reg- ularity of the left endpoint a and the limit point case at the right endpoint b. In addition, the bounded self-adjoint operator α = α ∗ ∈ B(H ) is used to parametrize the self-adjoint boundary condition at the left endpoint a of the type sin(α)u ' (a) +cos(α)u(a) = 0, with u lying in the domain of the underlying maximal operator Hmax in L 2 ((a,b);dx;H ) as- sociated with τ . More precisely, we establish the existence of the Weyl-Titchmarsh solution of Hα , the corresponding Weyl-Titchmarsh m-function mα and its Herglotz property, and deter- mine the structure of the Green's function of Hα . Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) so- lutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient V , ( −y '' + (V −z)y = f on (a,b), y(x0) = h0, y ' (x0) = h1,
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Initial Value Problems and Weyl--Titchmarsh Theory for Schr\
arXiv: Spectral Theory, 2011Co-Authors: Fritz Gesztesy, Rudi Weikard, Maxim ZinchenkoAbstract:We develop Weyl-Titchmarsh theory for self-adjoint Schr\"odinger operators $H_{\alpha}$ in $L^2((a,b);dx;\cH)$ associated with the operator-valued differential expression $\tau =-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $\alpha= \alpha^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(\alpha)u'(a)+\cos(\alpha)u(a)=0, $$ with $u$ lying in the domain of the underlying maximal operator $H_{\max}$ in $L^2((a,b);dx;\cH)$ associated with $\tau$. More precisely, we establish the existence of the Weyl-Titchmarsh solution of $H_{\alpha}$, the corresponding Weyl-Titchmarsh $m$-function $m_{\alpha}$ and its Herglotz property, and determine the structure of the Green's function of $H_{\alpha}$. Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient $V$, -y" + (V - z) y = f \, \text{on} \, (a,b), y(x_0) = h_0, \; y'(x_0) = h_1, under the following general assumptions: $(a,b)\subseteq\bbR$ is a finite or infinite interval, $x_0\in(a,b)$, $z\in\bbC$, $V:(a,b)\to\cB(\cH)$ is a weakly measurable operator-valued function with $\|V(\cdot)\|_{\cB(\cH)}\in L^1_\loc((a,b);dx)$, and $f\in L^1_{\loc}((a,b);dx;\cH)$, with $\cH$ a complex, separable Hilbert space. We also study the analog of this initial value problem with $y$ and $f$ replaced by operator-valued functions $Y, F \in \cB(\cH)$. Our hypotheses on the local behavior of $V$ appear to be the most general ones to date.
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initial value problems and weyl Titchmarsh theory for schr odinger operators with operator valued potentials
arXiv: Spectral Theory, 2011Co-Authors: Fritz Gesztesy, Rudi Weikard, Maxim ZinchenkoAbstract:We develop Weyl-Titchmarsh theory for self-adjoint Schr\"odinger operators $H_{\alpha}$ in $L^2((a,b);dx;\cH)$ associated with the operator-valued differential expression $\tau =-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $\alpha= \alpha^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(\alpha)u'(a)+\cos(\alpha)u(a)=0, $$ with $u$ lying in the domain of the underlying maximal operator $H_{\max}$ in $L^2((a,b);dx;\cH)$ associated with $\tau$. More precisely, we establish the existence of the Weyl-Titchmarsh solution of $H_{\alpha}$, the corresponding Weyl-Titchmarsh $m$-function $m_{\alpha}$ and its Herglotz property, and determine the structure of the Green's function of $H_{\alpha}$. Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient $V$, -y" + (V - z) y = f \, \text{on} \, (a,b), y(x_0) = h_0, \; y'(x_0) = h_1, under the following general assumptions: $(a,b)\subseteq\bbR$ is a finite or infinite interval, $x_0\in(a,b)$, $z\in\bbC$, $V:(a,b)\to\cB(\cH)$ is a weakly measurable operator-valued function with $\|V(\cdot)\|_{\cB(\cH)}\in L^1_\loc((a,b);dx)$, and $f\in L^1_{\loc}((a,b);dx;\cH)$, with $\cH$ a complex, separable Hilbert space. We also study the analog of this initial value problem with $y$ and $f$ replaced by operator-valued functions $Y, F \in \cB(\cH)$. Our hypotheses on the local behavior of $V$ appear to be the most general ones to date.
Makoto T Minamide - One of the best experts on this subject based on the ideXlab platform.
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Titchmarsh s method for the approximate functional equations for unicode stix x1d701 prime s 2 unicode stix x1d701 s unicode stix x1d701 prime prime s and unicode stix x1d701 prime s unicode stix x1d701 prime prime s
Canadian Journal of Mathematics, 2018Co-Authors: Jun Furuya, Makoto T Minamide, Yoshio TanigawaAbstract:Let $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$ , where $-1/2<\unicode[STIX]{x1D70E}<3/2,x,y\geqslant 1,xy=(|t|/2\unicode[STIX]{x1D70B})^{2}$ . Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$ . In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ , and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0<\unicode[STIX]{x1D70E}<1$ ) whose error terms contain the factor $((x+y)/|t|)^{1/4}$ . In this paper we remove this factor from these three error terms by using the method of Titchmarsh.
Gerald Teschl - One of the best experts on this subject based on the ideXlab platform.
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Inverse spectral problems for Schrödinger-type operators with distributional matrix-valued potentials
2016Co-Authors: Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, Er Sakhnovich, Gerald TeschlAbstract:Abstract. The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line from the underlying Weyl–Titchmarsh function. 1
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Singular Weyl–Titchmarsh–Kodaira theory for Jacobi operators
TESCHL, 2016Co-Authors: Jonathan Eckhardt, Gerald TeschlAbstract:Abstract. We develop singular Weyl–Titchmarsh–Kodaira theory for Jacobi operators. In particular, we establish existence of a spectral transformation as well as local Borg–Marchenko and Hochstadt–Liebermann type uniqueness results. 1
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Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators
Monatshefte für Mathematik, 2014Co-Authors: Rainer Brunnhuber, Aleksey Kostenko, Jonathan Eckhardt, Gerald TeschlAbstract:We develop singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg–Marchenko and Hochstadt–Lieberman type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.
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singular weyl Titchmarsh kodaira theory for one dimensional dirac operators
Monatshefte für Mathematik, 2014Co-Authors: Rainer Brunnhuber, Aleksey Kostenko, Jonathan Eckhardt, Gerald TeschlAbstract:We develop singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg–Marchenko and Hochstadt–Lieberman type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.
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Singular Weyl-Titchmarsh-Kodaira theory for Jacobi operators
Operators and Matrices, 2013Co-Authors: Jonathan Eckhardt, Gerald TeschlAbstract:We develop singular Weyl–Titchmarsh–Kodaira theory for Jacobi operators. In particular, we establish existence of a spectral transformation as well as local Borg–Marchenko and Hochstadt–Liebermann type uniqueness results.
