Eigenvalue theory for time scale symplectic systems depending nonlinearly on spectral parameter

Hilscher, Roman Šimon

doi:10.1016/j.amc.2012.08.026

Cited by 7 publications

(14 citation statements)

References 20 publications

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“…Theorem Assume that and hold and that

error R^{T} (λ) \subseteq error {\underset{̲}{R}}^{T} (λ), S (λ) \geq \underset{̲}{S} (λ), λ \in R,

the functional \underset{̲}{scriptG} (\cdot, λ_{0}) is positive definite for some λ_{0} < 0 .

n_{2} (λ)

and

{\underset{̲}{n}}_{2} (λ)

denote, respectively, the number of finite eigenvalues of (E j ) and

((), {\underset{̲}{E}}_{normalj})

(- \infty, λ]

, then

n_{2} (λ) \leq {\underset{̲}{n}}_{2} (λ) < \infty for all λ \in R .

Proof The proof is based on a transformation of the eigenvalue problems (E j ) and

((), {\underset{̲}{E}}_{normalj})

into auxiliary augmented eigenvalue problems in dimension 2 n with separated boundary conditions, to which Theorem 4.1 is applied. Such a transformation is presented in [, Section 6.3]. We shall not repeat these technical arguments and the details are therefore omitted. Remark Similarly as in Remark 4.4 we can show that

n_{2} (λ) \leq {\underset{̲}{n}}_{2} (...

…”

Section: Comparison Of Finite Eigenvaluesmentioning

confidence: 88%

“…The conditions below are slightly complicated as a price for allowing the (nonlinear) dependence on λ in the matrices representing the boundary conditions. The form of these conditions comes from the “symplectic structure” of linear Hamiltonian systems, as discussed in [, Section 6.1]. Let

R_{b} (λ) = Q_{b} (λ) D_{b} (λ)

be a polar decomposition of

R_{b} (λ)

, i.e.,

Q_{b} (λ)

is orthogonal and

D_{b} (λ) \geq 0

.…”

Section: Oscillation and Spectral Properties Of Linear Hamiltonian Symentioning

confidence: 99%

“…Therefore, its behavior at the right endpoint b determines the finite eigenvalues of (E). Define the matrices

(}, \begin{matrix} Λ (λ) : = R_{b}^{*} (λ) \bar{X} (b, λ) + R_{b} (λ) \bar{U} (b, λ), & M (λ) : = [I - Λ (λ) Λ^{†} (λ)] R_{b} (λ), \\ T (λ) : = I - M^{†} (λ) M (λ), & P (λ) : = T (λ) \bar{X} (b, λ) Λ^{†} (λ) R_{b} (λ) T (λ) . \end{matrix})

According to [, Definition 6.9], a number

λ_{0} \in R

is a finite eigenvalue of (E) if

θ_{s} (λ_{0}) : = error Λ (λ_{0}^{-}) - error Λ (λ_{0}) \geq 1 .

The subscript s refers to the separated boundary conditions in (E). Note that the rank of

Λ (λ)

is piecewise constant on

double-struckR

as in Proposition 2.1, since

Q_{b} (λ) Λ (λ)

can be identified with the value of

\bar{X} (b + 1, λ)

…”

Section: Oscillation and Spectral Properties Of Linear Hamiltonian Symentioning

confidence: 99%

“…Applying this inequality to

λ = λ_{k + n}

we obtain

k + n = n_{2} (λ_{k + n}) \leq {\underset{̲}{n}}_{2} (λ_{k + n}) + n

, i.e.,

{\underset{̲}{n}}_{2} (λ_{k + n}) \geq k

. Therefore, there is at least k finite eigenvalues of (

\underset{̲}{E}

) in the interval

(- \infty, λ_{k + n}]

, which shows inequality . Remark Without assumption , i.e., without assuming

\underset{̲}{scriptF} (\cdot, λ_{0}) > 0

for some

λ_{0} < 0

, we may apply a slightly more general oscillation theorem [, Theorems 6.10 and 6.16] in the proof of Theorem 4.1. In this case, inequality is generalized to

n_{2} (λ) + ℓ \leq {\underset{̲}{n}}_{2} (λ) + \underset{̲}{ℓ} for all λ \in R,

where

ℓ : = \underset{λ \to - \infty}{trueprefixlim} [n_{1} (λ) + s (λ)], \underset{̲}{ℓ} : = \underset{λ \to - \infty}{trueprefixlim} [{\underset{̲}{n}}_{1} (λ) + \underset{̲}{s} (λ)] .

…”

Section: Comparison Of Finite Eigenvaluesmentioning

confidence: 99%

“…Namely, we have in this case that holds, where ℓ and

\underset{̲}{ℓ}

are defined by , in which the quantities

n_{1} (λ)

and

{\underset{̲}{n}}_{1} (λ)

denote the number of proper focal points in

(a, b]

of the principal solutions

(() \hat{X} (\cdot, λ), \hat{U} (\cdot, λ))

and

(() \underset{̲}{\overset{X}{true}} (\cdot, λ), \underset{̲}{\overset{U}{true}} (\cdot, λ))

of (H λ ) and

({\underset{̲}{H}}_{λ})

, respectively, and

s (λ) : = error M_{j} (λ) + error P_{j} (λ), \underset{̲}{s} (λ) : = error {\underset{̲}{M}}_{j} (λ) + error {\underset{̲}{P}}_{j} (λ) .

This can be seen from the oscillation theorem for linear Hamiltonian systems with jointly varying endpoints in [, Theorems 6.13 and 6.16]. Remark The result in Theorem 4.6 applies also to eigenvalue problems with periodic (or antiperiodic) boundary conditions. For example, following [, Remark 6.15] we get the periodic boundary conditions

x (a) = x (b)

and

u (a) = u (b)

in (E j ) and

((), {\underset{̲}{E}}_{normalj})

by taking

\begin{matrix} true \\ right R (λ) & ... \end{matrix}

…”

Section: Comparison Of Finite Eigenvaluesmentioning

confidence: 99%

See 4 more Smart Citations

Comparison theorems for self-adjoint linear Hamiltonian eigenvalue problems

Hilscher

2013

Math. Nachr.

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In this work we derive new comparison results for (finite) eigenvalues of two self‐adjoint linear Hamiltonian eigenvalue problems. The coefficient matrices depend on the spectral parameter nonlinearly and the spectral parameter is present also in the boundary conditions. We do not impose any controllability or strict normality assumptions. Our method is based on a generalization of the Sturmian comparison theorem for such systems. The results are new even for the Dirichlet boundary conditions, for linear Hamiltonian systems depending linearly on the spectral parameter, and for Sturm–Liouville eigenvalue problems with nonlinear dependence on the spectral parameter.

show abstract

“…Theorem Assume that and hold and that

error R^{T} (λ) \subseteq error {\underset{̲}{R}}^{T} (λ), S (λ) \geq \underset{̲}{S} (λ), λ \in R,

the functional \underset{̲}{scriptG} (\cdot, λ_{0}) is positive definite for some λ_{0} < 0 .

n_{2} (λ)

and

{\underset{̲}{n}}_{2} (λ)

denote, respectively, the number of finite eigenvalues of (E j ) and

((), {\underset{̲}{E}}_{normalj})

(- \infty, λ]

, then

n_{2} (λ) \leq {\underset{̲}{n}}_{2} (λ) < \infty for all λ \in R .

Proof The proof is based on a transformation of the eigenvalue problems (E j ) and

((), {\underset{̲}{E}}_{normalj})

n_{2} (λ) \leq {\underset{̲}{n}}_{2} (...

…”

Section: Comparison Of Finite Eigenvaluesmentioning

confidence: 88%

R_{b} (λ) = Q_{b} (λ) D_{b} (λ)

be a polar decomposition of

R_{b} (λ)

, i.e.,

Q_{b} (λ)

is orthogonal and

D_{b} (λ) \geq 0

.…”

Section: Oscillation and Spectral Properties Of Linear Hamiltonian Symentioning

confidence: 99%

“…Therefore, its behavior at the right endpoint b determines the finite eigenvalues of (E). Define the matrices

(}, \begin{matrix} Λ (λ) : = R_{b}^{*} (λ) \bar{X} (b, λ) + R_{b} (λ) \bar{U} (b, λ), & M (λ) : = [I - Λ (λ) Λ^{†} (λ)] R_{b} (λ), \\ T (λ) : = I - M^{†} (λ) M (λ), & P (λ) : = T (λ) \bar{X} (b, λ) Λ^{†} (λ) R_{b} (λ) T (λ) . \end{matrix})

According to [, Definition 6.9], a number

λ_{0} \in R

is a finite eigenvalue of (E) if

θ_{s} (λ_{0}) : = error Λ (λ_{0}^{-}) - error Λ (λ_{0}) \geq 1 .

The subscript s refers to the separated boundary conditions in (E). Note that the rank of

Λ (λ)

is piecewise constant on

double-struckR

as in Proposition 2.1, since

Q_{b} (λ) Λ (λ)

can be identified with the value of

\bar{X} (b + 1, λ)

…”

Section: Oscillation and Spectral Properties Of Linear Hamiltonian Symentioning

confidence: 99%

“…Applying this inequality to

λ = λ_{k + n}

we obtain

k + n = n_{2} (λ_{k + n}) \leq {\underset{̲}{n}}_{2} (λ_{k + n}) + n

, i.e.,

{\underset{̲}{n}}_{2} (λ_{k + n}) \geq k

. Therefore, there is at least k finite eigenvalues of (

\underset{̲}{E}

) in the interval

(- \infty, λ_{k + n}]

, which shows inequality . Remark Without assumption , i.e., without assuming

\underset{̲}{scriptF} (\cdot, λ_{0}) > 0

for some

λ_{0} < 0

, we may apply a slightly more general oscillation theorem [, Theorems 6.10 and 6.16] in the proof of Theorem 4.1. In this case, inequality is generalized to

n_{2} (λ) + ℓ \leq {\underset{̲}{n}}_{2} (λ) + \underset{̲}{ℓ} for all λ \in R,

where

ℓ : = \underset{λ \to - \infty}{trueprefixlim} [n_{1} (λ) + s (λ)], \underset{̲}{ℓ} : = \underset{λ \to - \infty}{trueprefixlim} [{\underset{̲}{n}}_{1} (λ) + \underset{̲}{s} (λ)] .

…”

Section: Comparison Of Finite Eigenvaluesmentioning

confidence: 99%

“…Namely, we have in this case that holds, where ℓ and

\underset{̲}{ℓ}

are defined by , in which the quantities

n_{1} (λ)

and

{\underset{̲}{n}}_{1} (λ)

denote the number of proper focal points in

(a, b]

of the principal solutions

(() \hat{X} (\cdot, λ), \hat{U} (\cdot, λ))

and

(() \underset{̲}{\overset{X}{true}} (\cdot, λ), \underset{̲}{\overset{U}{true}} (\cdot, λ))

of (H λ ) and

({\underset{̲}{H}}_{λ})

, respectively, and

s (λ) : = error M_{j} (λ) + error P_{j} (λ), \underset{̲}{s} (λ) : = error {\underset{̲}{M}}_{j} (λ) + error {\underset{̲}{P}}_{j} (λ) .

x (a) = x (b)

and

u (a) = u (b)

in (E j ) and

((), {\underset{̲}{E}}_{normalj})

by taking

\begin{matrix} true \\ right R (λ) & ... \end{matrix}

…”

Section: Comparison Of Finite Eigenvaluesmentioning

confidence: 99%

See 3 more Smart Citations

Comparison theorems for self-adjoint linear Hamiltonian eigenvalue problems

Hilscher

2013

Math. Nachr.

View full text Add to dashboard Buy / Rent full text

show abstract

Eigenvalue theory for time scale symplectic systems depending nonlinearly on spectral parameter

Cited by 7 publications

References 20 publications

Comparison theorems for self-adjoint linear Hamiltonian eigenvalue problems

Comparison theorems for self-adjoint linear Hamiltonian eigenvalue problems

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