<p>We consider in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_2(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> an elliptic second-order differential operator <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A_{\varepsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, given by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A_{\varepsilon } = - \frac{d}{dx} g(x/\varepsilon ) \frac{d}{dx} + \varepsilon ^{-2} p({x}/\varepsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>ε</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mi>d</mi> <mrow> <mi mathvariant="italic">dx</mi> </mrow> </mfrac> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">/</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mi>d</mi> <mrow> <mi mathvariant="italic">dx</mi> </mrow> </mfrac> <mo>+</mo> <msup> <mi>ε</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">/</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with periodic coefficients. For small <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>, we study the behavior of the resolvent of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A_{\varepsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> at a regular point close to the edge of a spectral gap. We obtain an approximation of this resolvent in the “energy” norm with an error <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$O(\varepsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The approximation is described in terms of the spectral characteristics of the operator at the edge of the gap. Bibliography: 22 titles.</p>

HOMOGENIZATION OF A ONE-DIMENSIONAL PERIODIC ELLIPTIC OPERATOR AT THE EDGE OF A SPECTRAL GAP: OPERATOR ESTIMATES IN THE ENERGY NORM

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发表日期：

2026年3月20日

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A. A. Mishulovich,
V. A. Sloushch,
T. A. Suslina

摘要

We consider in $L_2(\mathbb {R})$ $L_{2} (R)$ an elliptic second-order differential operator $A_{\varepsilon }$ $A_{ε}$ , $\varepsilon >0$ $ε > 0$ , given by $A_{\varepsilon } = - \frac{d}{dx} g(x/\varepsilon ) \frac{d}{dx} + \varepsilon ^{-2} p({x}/\varepsilon )$ $A_{ε} = - \frac{d}{dx} g (x / ε) \frac{d}{dx} + ε^{- 2} p (x / ε)$ , with periodic coefficients. For small $\varepsilon $ $ε$ , we study the behavior of the resolvent of $A_{\varepsilon }$ $A_{ε}$ at a regular point close to the edge of a spectral gap. We obtain an approximation of this resolvent in the “energy” norm with an error $O(\varepsilon )$ $O (ε)$ . The approximation is described in terms of the spectral characteristics of the operator at the edge of the gap. Bibliography: 22 titles.