This paper deals with the approximation of a magnetic Schrödinger operator with a singular <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-potential that is formally given by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(i \nabla + A)^2 + Q + \alpha \delta _\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mi>Q</mi> <mo>+</mo> <mi>α</mi> <msub> <mi>δ</mi> <mi mathvariant="normal">Σ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> by Schrödinger operators with regular potentials in the norm resolvent sense. This is done for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> being the finite union of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-hypersurfaces, for coefficients A, Q, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> under almost minimal assumptions such that the associated quadratic forms are closed and sectorial, and Q and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> are allowed to be complex-valued functions. In particular, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> can be a graph in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> or the boundary of a piecewise <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-domain. Moreover, spectral implications of the mentioned convergence result are discussed.

Approximation of magnetic Schrödinger operators with $\delta $-interactions supported on networks

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2026年2月28日

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Markus Holzmann

摘要

This paper deals with the approximation of a magnetic Schrödinger operator with a singular $\delta $ $δ$ -potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta _\Sigma $ ${(i \nabla + A)}^{2} + Q + α δ_{Σ}$ by Schrödinger operators with regular potentials in the norm resolvent sense. This is done for $\Sigma $ $Σ$ being the finite union of $C^2$ $C^{2}$ -hypersurfaces, for coefficients A, Q, and $\alpha $ $α$ under almost minimal assumptions such that the associated quadratic forms are closed and sectorial, and Q and $\alpha $ $α$ are allowed to be complex-valued functions. In particular, $\Sigma $ $Σ$ can be a graph in $\mathbb {R}^2$ $R^{2}$ or the boundary of a piecewise $C^2$ $C^{2}$ -domain. Moreover, spectral implications of the mentioned convergence result are discussed.

Approximation of magnetic Schrödinger operators with \(\delta \)-interactions supported on networks

摘要