On the emptiness criteria for the discrete spectrum of the Dirichlet problem for the equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 2, pp. 288-292 Cet article a éte moissonné depuis la source Math-Net.Ru

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An improved Rellich emptiness criterion is proposed for the discrete spectrum of the Dirichlet problem for the equation $\Delta v+\lambda v=0$. Examples of its application are given.
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M. D. Malykh. On the emptiness criteria for the discrete spectrum of the Dirichlet problem for the equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 2, pp. 288-292. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a6/

[1] Bogolyubov A. N., Malykh M. D., “O lovushechnykh modakh elektromagnitnogo volnovoda s neodnorodnym zapolneniem”, Radiotekhn. i elektronika, 50:2 (2005), 218–222

[2] Linton C. M., Mclver P., Embdedded trapped modes in water waves and acoustics, Preprint no 06-24, Loughborough Univ., Leicestershire, 2006

[3] Rellich F., “Das Eigenwertproblem von $\Delta u+\lambda u=0$ in Hairöhren”, Studies and Essays Presented to R. Courant, New York, 1948, 329–344 | MR | Zbl

[4] Krejciric D., Kriz J., “On the spectrum of curved planar waveguide”, Publ. Res. Inst. Math. Sei. Kyoto Univ., 41 (2005), 757–791 | DOI | MR

[5] Krejciric D., Guides d'ondes quantiques bidimensionnels, These de Doct., Univ. de Toulon et du Var, 2001

[6] Exner P., Šeba P., Tater M., Vaňek D., “Bound states and scattering in quantum waveguides coupled laterally through a boundary window”, J. Math. Phys., 37 (1996), 4867–4887 | DOI | MR | Zbl

[7] Bulla W., Gesztesy F., Renger W., Simon B., “Weakly coupled bound states in quantum waveguides”, Proc. Amer. Math. Soc., 127 (1997), 1487–1495 | DOI | MR

[8] Borisov D., Exner P., Gadylshin R., Krejciric D., “Bound states in weakly deformed strips and layers”, Ann. Henri Poincare, 2:3 (2001), 553–572 | DOI | MR | Zbl

[9] Bogolyubov A. H., Delitsyn A. L., Malykh M. D., “Ob odnom primere lovushechnykh mod v neregulyarnom volnovode”, Zh. vychisl. matem. i matem. fiz., 43:1 (2003), 127–132 | MR

[10] Hellwig G., Differentialoperatoren der mathematischen Physik, Springer, Berlin etc., 1964 | MR | Zbl

[11] Reilich Fr., “Darstellung der Eigenwerte von $\Delta u+\lambda u=0$ durch ein Randintegral”, Math. Z., 46 (1939), 635–636 | DOI | MR

[12] Reilich Fr., “Über das asympotysche Verhalten der Lösungen von $\Delta u+\lambda u=0$ in unendlichen Gebieten”, Jahresbericht der deutschen Mathematiker-Vereinigung, 53 (1943), 57–65 | MR

[13] Glazman I. M., Pryamye metody kachestvennogo spektralnogo analiza singulyarnykh differentsialnykh operatorov, Fizmatgiz, M., 1963 | MR

[14] Bernshtein C. H., Analiticheskaya priroda reshenii differentsialnykh uravnenii ellipticheskogo tipa, Izd-vo KhGU, Kharkov, 1956

[15] Werner P., “Resonanzphänomene in akustischen und elektromagnetischen Wellenleitern”, Z. Angew. Math. und Mech., 67:4 (1987), 43–54 | DOI | MR | Zbl

[16] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, T. 4, Mir, M., 1982