Spectral Properties of the Differential Operators of the Fourth-Order with Eigenvalue Parameter Dependent Boundary Condition

Abstract

We consider the fourth-order spectral problem <svg style="vertical-align:-2.29482pt;width:254.1125px;" id="M1" height="16.637501" version="1.1" viewBox="0 0 254.1125 16.637501" width="254.1125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.6375)"> <g transform="translate(72,-58.69)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑦</tspan> </text> <text transform="matrix(1,0,0,-1,-65.75,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0">(</tspan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0">4</tspan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0">)</tspan> </text> <text transform="matrix(1,0,0,-1,-55.05,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑥</tspan> <tspan style="font-size: 12.50px; " x="11.040149" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="17.979315" y="0">−</tspan> <tspan style="font-size: 12.50px; " x="29.319534" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="33.483032" y="0">𝑞</tspan> <tspan style="font-size: 12.50px; " x="39.209408" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="43.372906" y="0">𝑥</tspan> <tspan style="font-size: 12.50px; " x="50.249557" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="54.413055" y="0">𝑦</tspan> <tspan style="font-size: 12.50px; " x="60.614544" y="0">′</tspan> <tspan style="font-size: 12.50px; " x="64.227913" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="68.391411" y="0">𝑥</tspan> <tspan style="font-size: 12.50px; " x="75.268059" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="79.431557" y="0">)</tspan> </text> <text transform="matrix(1,0,0,-1,28.56,66.23)"> <tspan style="font-size: 8.75px; " x="0" y="0"></tspan> </text> <text transform="matrix(1,0,0,-1,35.5,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0">=</tspan> <tspan style="font-size: 12.50px; " x="12.040389" y="0">𝜆</tspan> <tspan style="font-size: 12.50px; " x="18.804512" y="0">𝑦</tspan> <tspan style="font-size: 12.50px; " x="25.006001" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="29.169498" y="0">𝑥</tspan> <tspan style="font-size: 12.50px; " x="36.04615" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="40.209648" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="49.586899" y="0">𝑥</tspan> <tspan style="font-size: 12.50px; " x="59.92688" y="0">∈</tspan> <tspan style="font-size: 12.50px; " x="71.96727" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="76.130768" y="0">0</tspan> <tspan style="font-size: 12.50px; " x="82.382271" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="87.596016" y="0">𝑙</tspan> <tspan style="font-size: 12.50px; " x="91.571976" y="0">)</tspan> </text> </g> </g> </svg> with spectral parameter in the boundary condition. We associate this problem with a selfadjoint operator in Hilbert or Pontryagin space. Using this operator-theoretic formulation and analytic methods, we investigate locations (in complex plane) and multiplicities of the eigenvalues, the oscillation properties of the eigenfunctions, the basis properties in <svg style="vertical-align:-4.74141pt;width:46.862499px;" id="M2" height="16.6" version="1.1" viewBox="0 0 46.862499 16.6" width="46.862499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.6)"> <g transform="translate(72,-58.72)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan> </text> <text transform="matrix(1,0,0,-1,-63.1,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-58.32,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">0</tspan> <tspan style="font-size: 12.50px; " x="10.414999" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="15.616247" y="0">𝑙</tspan> <tspan style="font-size: 12.50px; " x="19.592201" y="0">)</tspan> </text> </g> </g> </svg>, <svg style="vertical-align:-2.29482pt;width:66.362503px;" id="M3" height="13.55" version="1.1" viewBox="0 0 66.362503 13.55" width="66.362503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑝</tspan> <tspan style="font-size: 12.50px; " x="9.5772982" y="0">∈</tspan> <tspan style="font-size: 12.50px; " x="21.617687" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="25.781185" y="0">1</tspan> <tspan style="font-size: 12.50px; " x="32.032684" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="37.246437" y="0">∞</tspan> <tspan style="font-size: 12.50px; " x="48.824215" y="0">)</tspan> </text> </g> </g> </svg>, of the system of root functions of this problem.