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2,502 results on '"Christiansen T"'

1. From resolvent expansions at zero to long time wave expansions

Author

Christiansen, T. J., Datchev, K., and Yang, M.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Spectral Theory
Abstract

We prove a general abstract theorem deducing wave expansions as time goes to infinity from resolvent expansions as energy goes to zero, under an assumption of polynomial boundedness of the resolvent at high energy. We give applications to obstacle scattering, to Aharonov--Bohm Hamiltonians, to scattering in a sector, and to scattering by a compactly supported potential., Comment: 13 pages, 1 figure

Published
2024

2. Low energy resolvent asymptotics of the multipole Aharonov--Bohm Hamiltonian

Author

Christiansen, T. J., Datchev, K., and Yang, M.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Spectral Theory
Abstract

We compute low energy asymptotics for the resolvent of the Aharonov--Bohm Hamiltonian with multiple poles for both integer and non-integer total fluxes. For integral total flux we reduce to prior results in black-box scattering while for non-integral total flux we build on the corresponding techniques using an appropriately chosen model resolvent. The resolvent expansion can be used to obtain long-time wave asymptotics for the Aharonov--Bohm Hamiltonian with multiple poles. An interesting phenomenon is that if the total flux is an integer then the scattering resembles even-dimensional Euclidean scattering, while if it is half an odd integer then it resembles odd-dimensional Euclidean scattering. The behavior for other values of total flux thus provides an `interpolation' between these., Comment: 15 pages, 1 figure

Published
2024

3. Persistence and disappearance of negative eigenvalues in dimension two

Author

Christiansen, T. J., Datchev, K., and Griffin, C.

Subjects
Mathematics - Spectral Theory, Mathematical Physics, Mathematics - Analysis of PDEs
Abstract

We compute asymptotics of eigenvalues approaching the bottom of the continuous spectrum, and associated resonances, for Schr\"odinger operators in dimension two. We distinguish persistent eigenvalues, which have associated resonances, from disappearing ones, which do not. We illustrate the significance of this distinction by computing corresponding scattering phase asymptotics and numerical Breit--Wigner peaks. We prove all of our results for circular wells, and extend some of them to more general problems using recent resolvent techniques., Comment: 29 pages, 6 figures

Published
2024

4. Low energy resolvent expansions in dimension two

Author

Christiansen, T. J. and Datchev, K.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Spectral Theory
Abstract

We prove resolvent expansions near zero energy for compactly supported perturbations of the Laplacian on $\mathbb R^2$. We obtain precise results for general self-adjoint black box perturbations, in the sense of Sj\"ostrand and Zworski, and also for some non-self-adjoint ones. We compute the most singular terms, relating them to spaces of zero eigenvalues and resonances. Our methods include resolvent identity arguments following Vodev and boundary pairing arguments following Melrose., Comment: 28 pages

Published
2023

5. Singularities and asymptotic distribution of resonances for Schr\'odinger operators in one dimension

Author

Christiansen, T. J. and Cunningham, T.

Subjects
Mathematical Physics, Mathematics - Spectral Theory, 35P25, 81U24, 58J50, 47A40
Abstract

We obtain new results about the high-energy distribution of resonances for the one-dimensional Schr\"odinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular support of the potential. We also prove results about the distribution of resonances in sectors away from the real axis, and construct a class of potentials producing multiple sequences of resonances along distinct logarithmic curves, explicitly calculating the asymptotic location of these resonances. The results are unified by the use of an integral representation of the reflection coefficients., Comment: 22 pages

Published
2023

6. Surface engineering of titanium by multi-interstitial diffusion using plasma processing

Author

Drouet M., Pichon L., Dubois J.B., Le Bourhis E., and Christiansen T. L.

Subjects
Engineering (General). Civil engineering (General), TA1-2040
Abstract

Titanium and its alloys possess a range of highly interesting properties such as excellent corrosion resistance, high specific strength and biocompatibility, but suffers from poor wear resistance. The present work addresses plasma assisted surface treatment of CP 2 titanium using various combinations of oxygen and nitrogen, i.e. mixed interstitials. The sequence of controlled plasma nitriding and oxidizing treatments plays a significant role for the evolution of the hardness depth profiles and the development of the surface compound layer and the underlying diffusion/transition zone. Composition profiles of oxygen and nitrogen are obtained by GDOES; Mixed interstitial solubility of nitrogen and oxygen is found in both h.c.p. α titanium and in the compound layer. The combination of interstitials leads to larger case depth, in particular for the diffusion zone (expanded h.c.p. α titanium). Therefore, it highlights the advantages of combined nitriding and oxidizing compared to single nitriding treatments on the mechanical properties.

Published
2020
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7. Low energy scattering asymptotics for planar obstacles

Author

Christiansen, T. J. and Datchev, K.

Subjects
Mathematics - Analysis of PDEs, Mathematics - Spectral Theory
Abstract

We compute low energy asymptotics for the resolvent of a planar obstacle, and deduce asymptotics for the corresponding scattering matrix, scattering phase, and exterior Dirichlet-to-Neumann operator. We use an identity of Vodev to relate the obstacle resolvent to the free resolvent and an identity of Petkov and Zworski to relate the scattering matrix to the resolvent. The leading singularities are given in terms of the obstacle's logarithmic capacity or Robin constant. We expect these results to hold for more general compactly supported perturbations of the Laplacian on $\mathbb R^2$, with the definition of the Robin constant suitably modified, under a generic assumption that the spectrum is regular at zero., Comment: 26 pages, 1 figure

Published
2022
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8. The semiclassical structure of the scattering matrix for a manifold with infinite cylindrical end

Author

Christiansen, T. J. and Uribe, A.

Subjects
Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, 58J50, 58J40, 35P25
Abstract

We study the microlocal properties of the scattering matrix associated to the semiclassical Schr\"odinger operator $P=h^2\Delta_X+V$ on a Riemannian manifold with an infinite cylindrical end. The scattering matrix at $E=1$ is a linear operator $S=S_h$ defined on a Hilbert subspace of $L^2(Y)$ that parameterizes the continuous spectrum of $P$ at energy $1$. Here $Y$ is the cross section of the end of $X$, which is not necessarily connected. We show that, under certain assumptions, microlocally $S$ is a Fourier integral operator associated to the graph of the scattering map $\kappa:\mathcal{D}_{\kappa}\to T^*Y$, with $\mathcal{D}_\kappa\subset T^*Y$. The scattering map $\kappa$ and its domain $\mathcal{D}_\kappa$ are determined by the Hamilton flow of the principal symbol of $P$. As an application we prove that, under additional hypotheses on the scattering map, the eigenvalues of the associated unitary scattering matrix are equidistributed on the unit circle., Comment: Version 2 has an additional subsection in the appendix, in which we compute the scattering map for a certain class of surfaces of revolution. Version 2 has 34 pages, 3 figures

Published
2021

9. Resonances for Schr\'odinger operators on infinite cylinders and other products

Author

Christiansen, T. J.

Subjects
Mathematics - Spectral Theory, Mathematical Physics, 58J50, 35P25
Abstract

We study the resonances of Schr\"odinger operators on the infinite product $X=\mathbb{R}^d\times \mathbb{S}^1$, where $d$ is odd, $\mathbb{S}^1$ is the unit circle, and the potential $V\in L^\infty_c(X)$. This paper shows that at high energy, resonances of the Schr\"odinger operator $-\Delta +V$ on $X=\mathbb{R}^d\times \mathbb{S}^1$ which are near the continuous spectrum are approximated by the resonances of $-\Delta +V_0$ on $X$, where the potential $V_0$ given by averaging $V$ over the unit circle. These resonances are, in turn, given in terms of the resonances of a Schr\"odinger operator on $\mathbb{R}^d$ which lie in a bounded set. If the potential is smooth, we obtain improved localization of the resonances, particularly in the case of simple, rank one poles of the corresponding scattering resolvent on $\mathbb{R}^d$. In that case, we obtain the leading order correction for the location of the corresponding high energy resonances. In addition to direct results about the location of resonances, we show that at high energies away from the resonances, the resolvent of the model operator $-\Delta+V_0$ on $X$ approximates that of $-\Delta+V$ on $X$. If $d=1$, in certain cases this implies the existence of an asymptotic expansion of solutions of the wave equation. Again for the special case of $d=1$, we obtain a resonant rigidity type result for the zero potential among all real-valued potentials., Comment: 46 pages; v. 2 is attempt to fix uploading error

Published
2020
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10. Manifolds with cylindrical ends having a finite and positive number of embedded eigenvalues

Author

Christiansen, T. J. and Datchev, K.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Differential Geometry, Mathematics - Spectral Theory
Abstract

We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge, this is the first example of a manifold with a cylindrical end whose number of eigenvalues is known to be finite and nonzero. The construction can be varied to give examples with arbitrary genus and with an arbitrarily large finite number of eigenvalues. The constructed surfaces also have resonance-free regions near the continuous spectrum and long-time asymptotic expansions of solutions to the wave equation., Comment: 7 pages, 3 figures

Published
2020
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11. Resolvent estimates, wave decay, and resonance-free regions for star-shaped waveguides

Author

Christiansen, T. J. and Datchev, K.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Spectral Theory
Abstract

Using coordinates $(x,y)\in \mathbb R\times \mathbb R^{d-1}$, we introduce the notion that an unbounded domain in $\mathbb R^d$ is star shaped with respect to $x=\pm \infty$. For such domains, we prove estimates on the resolvent of the Dirichlet Laplacian near the continuous spectrum. When the domain has infinite cylindrical ends, this has consequences for wave decay and resonance-free regions. Our results also cover examples beyond the star-shaped case, including scattering by a strictly convex obstacle inside a straight planar waveguide., Comment: 21 pages, 5 figures

Published
2019

12. Erratum to: Searches for long-lived charged particles in pp collisions at s = 7 and 8 TeV

Author

Chatrchyan, S., Khachatryan, V., Sirunyan, A. M., Tumasyan, A., Adam, W., Bergauer, T., Dragicevic, M., Erö, J., Fabjan, C., Friedl, M., Frühwirth, R., Ghete, V. M., Hörmann, N., Hrubec, J., Jeitler, M., Kiesenhofer, W., Knünz, V., Krammer, M., Krätschmer, I., Liko, D., Mikulec, I., Rabady, D., Rahbaran, B., Rohringer, C., Rohringer, H., Schöfbeck, R., Strauss, J., Taurok, A., Treberer-Treberspurg, W., Waltenberger, W., Wulz, C.-E., Mossolov, V., Shumeiko, N., Suarez Gonzalez, J., Alderweireldt, S., Bansal, M., Bansal, S., Cornelis, T., De Wolf, E. A., Janssen, X., Knutsson, A., Luyckx, S., Mucibello, L., Ochesanu, S., Roland, B., Rougny, R., Van Haevermaet, H., Van Mechelen, P., Van Remortel, N., Van Spilbeeck, A., Blekman, F., Blyweert, S., D’Hondt, J., Kalogeropoulos, A., Keaveney, J., Maes, M., Olbrechts, A., Tavernier, S., Van Doninck, W., Van Mulders, P., Van Onsem, G. P., Villella, I., Clerbaux, B., De Lentdecker, G., Favart, L., Gay, A. P. R., Hreus, T., Léonard, A., Marage, P. E., Mohammadi, A., Perniè, L., Reis, T., Seva, T., Thomas, L., Vander Velde, C., Vanlaer, P., Wang, J., Adler, V., Beernaert, K., Benucci, L., Cimmino, A., Costantini, S., Dildick, S., Garcia, G., Klein, B., Lellouch, J., Marinov, A., Mccartin, J., Ocampo Rios, A. A., Ryckbosch, D., Sigamani, M., Strobbe, N., Thyssen, F., Tytgat, M., Walsh, S., Yazgan, E., Zaganidis, N., Basegmez, S., Beluffi, C., Bruno, G., Castello, R., Caudron, A., Ceard, L., Delaere, C., du Pree, T., Favart, D., Forthomme, L., Giammanco, A., Hollar, J., Jez, P., Lemaitre, V., Liao, J., Militaru, O., Nuttens, C., Pagano, D., Pin, A., Piotrzkowski, K., Popov, A., Selvaggi, M., Vizan Garcia, J. M., Beliy, N., Caebergs, T., Daubie, E., Hammad, G. H., Alves, G. A., Correa Martins Junior, M., Martins, T., Pol, M. E., Souza, M. H. G., Aldá Júnior, W. L., Carvalho, W., Chinellato, J., Custódio, A., Da Costa, E. M., De Jesus Damiao, D., De Oliveira Martins, C., Fonseca De Souza, S., Malbouisson, H., Malek, M., Matos Figueiredo, D., Mundim, L., Nogima, H., Prado Da Silva, W. L., Santoro, A., Sznajder, A., Tonelli Manganote, E. J., Vilela Pereira, A., Bernardes, C. A., Dias, F. A., Fernandez Perez Tomei, T. R., Gregores, E. M., Lagana, C., Marinho, F., Mercadante, P. G., Novaes, S. F., Padula, Sandra S., Genchev, V., Iaydjiev, P., Piperov, S., Rodozov, M., Sultanov, G., Vutova, M., Dimitrov, A., Hadjiiska, R., Kozhuharov, V., Litov, L., Pavlov, B., Petkov, P., Bian, J. G., Chen, G. M., Chen, H. S., Jiang, C. H., Liang, D., Liang, S., Meng, X., Tao, J., Wang, J., Wang, X., Wang, Z., Xiao, H., Xu, M., Asawatangtrakuldee, C., Ban, Y., Guo, Y., Li, W., Liu, S., Mao, Y., Qian, S. J., Teng, H., Wang, D., Zhang, L., Zou, W., Avila, C., Carrillo Montoya, C. A., Gomez, J. P., Gomez Moreno, B., Sanabria, J. C., Godinovic, N., Lelas, D., Plestina, R., Polic, D., Puljak, I., Antunovic, Z., Kovac, M., Brigljevic, V., Duric, S., Kadija, K., Luetic, J., Mekterovic, D., Morovic, S., Tikvica, L., Attikis, A., Mavromanolakis, G., Mousa, J., Nicolaou, C., Ptochos, F., Razis, P. A., Finger, M., Finger, Jr., M., Assran, Y., Ellithi Kamel, A., Mahmoud, M. A., Mahrous, A., Radi, A., Kadastik, M., Müntel, M., Murumaa, M., Raidal, M., Rebane, L., Tiko, A., Eerola, P., Fedi, G., Voutilainen, M., Härkönen, J., Karimäki, V., Kinnunen, R., Kortelainen, M. J., Lampén, T., Lassila-Perini, K., Lehti, S., Lindén, T., Luukka, P., Mäenpää, T., Peltola, T., Tuominen, E., Tuominiemi, J., Tuovinen, E., Wendland, L., Korpela, A., Tuuva, T., Besancon, M., Choudhury, S., Couderc, F., Dejardin, M., Denegri, D., Fabbro, B., Faure, J. L., Ferri, F., Ganjour, S., Givernaud, A., Gras, P., Hamel de Monchenault, G., Jarry, P., Locci, E., Malcles, J., Millischer, L., Nayak, A., Rander, J., Rosowsky, A., Titov, M., Baffioni, S., Beaudette, F., Benhabib, L., Bianchini, L., Bluj, M., Busson, P., Charlot, C., Daci, N., Dahms, T., Dalchenko, M., Dobrzynski, L., Florent, A., Granier de Cassagnac, R., Haguenauer, M., Miné, P., Mironov, C., Naranjo, I. N., Nguyen, M., Ochando, C., Paganini, P., Sabes, D., Salerno, R., Sirois, Y., Veelken, C., Zabi, A., Agram, J.-L., Andrea, J., Bloch, D., Bodin, D., Brom, J.-M., Chabert, E. C., Collard, C., Conte, E., Drouhin, F., Fontaine, J.-C., Gelé, D., Goerlach, U., Goetzmann, C., Juillot, P., Le Bihan, A.-C., Van Hove, P., Gadrat, S., Beauceron, S., Beaupere, N., Boudoul, G., Brochet, S., Chasserat, J., Chierici, R., Contardo, D., Depasse, P., El Mamouni, H., Fay, J., Gascon, S., Gouzevitch, M., Ille, B., Kurca, T., Lethuillier, M., Mirabito, L., Perries, S., Sgandurra, L., Sordini, V., Tschudi, Y., Vander Donckt, M., Verdier, P., Viret, S., Tsamalaidze, Z., Autermann, C., Beranek, S., Calpas, B., Edelhoff, M., Feld, L., Heracleous, N., Hindrichs, O., Klein, K., Ostapchuk, A., Perieanu, A., Raupach, F., Sammet, J., Schael, S., Sprenger, D., Weber, H., Wittmer, B., Zhukov, V., Ata, M., Caudron, J., Dietz-Laursonn, E., Duchardt, D., Erdmann, M., Fischer, R., Güth, A., Hebbeker, T., Heidemann, C., Hoepfner, K., Klingebiel, D., Kreuzer, P., Merschmeyer, M., Meyer, A., Olschewski, M., Padeken, K., Papacz, P., Pieta, H., Reithler, H., Schmitz, S. A., Sonnenschein, L., Steggemann, J., Teyssier, D., Thüer, S., Weber, M., Cherepanov, V., Erdogan, Y., Flügge, G., Geenen, H., Geisler, M., Haj Ahmad, W., Hoehle, F., Kargoll, B., Kress, T., Kuessel, Y., Lingemann, J., Nowack, A., Nugent, I. M., Perchalla, L., Pooth, O., Stahl, A., Aldaya Martin, M., Asin, I., Bartosik, N., Behr, J., Behrenhoff, W., Behrens, U., Bergholz, M., Bethani, A., Borras, K., Burgmeier, A., Cakir, A., Calligaris, L., Campbell, A., Costanza, F., Diez Pardos, C., Dooling, S., Dorland, T., Eckerlin, G., Eckstein, D., Flucke, G., Geiser, A., Glushkov, I., Gunnellini, P., Habib, S., Hauk, J., Hellwig, G., Jung, H., Kasemann, M., Katsas, P., Kleinwort, C., Kluge, H., Krämer, M., Krücker, D., Kuznetsova, E., Lange, W., Leonard, J., Lipka, K., Lohmann, W., Lutz, B., Mankel, R., Marfin, I., Melzer-Pellmann, I.-A., Meyer, A. B., Mnich, J., Mussgiller, A., Naumann-Emme, S., Novgorodova, O., Nowak, F., Olzem, J., Perrey, H., Petrukhin, A., Pitzl, D., Placakyte, R., Raspereza, A., Ribeiro Cipriano, P. M., Riedl, C., Ron, E., Sahin, M. Ö., Salfeld-Nebgen, J., Schmidt, R., Schoerner-Sadenius, T., Sen, N., Stein, M., Walsh, R., Wissing, C., Blobel, V., Enderle, H., Erfle, J., Gebbert, U., Görner, M., Gosselink, M., Haller, J., Heine, K., Höing, R. S., Kaussen, G., Kirschenmann, H., Klanner, R., Kogler, R., Lange, J., Marchesini, I., Peiffer, T., Pietsch, N., Rathjens, D., Sander, C., Schettler, H., Schleper, P., Schlieckau, E., Schmidt, A., Schröder, M., Schum, T., Seidel, M., Sibille, J., Sola, V., Stadie, H., Steinbrück, G., Thomsen, J., Troendle, D., Vanelderen, L., Barth, C., Baus, C., Berger, J., Böser, C., Chwalek, T., De Boer, W., Descroix, A., Dierlamm, A., Feindt, M., Guthoff, M., Hackstein, C., Hartmann, F., Hauth, T., Heinrich, M., Held, H., Hoffmann, K. H., Husemann, U., Katkov, I., Komaragiri, J. R., Kornmayer, A., Lobelle Pardo, P., Martschei, D., Mueller, S., Müller, Th., Niegel, M., Nürnberg, A., Oberst, O., Ott, J., Quast, G., Rabbertz, K., Ratnikov, F., Röcker, S., Schilling, F.-P., Schott, G., Simonis, H. J., Stober, F. M., Ulrich, R., Wagner-Kuhr, J., Wayand, S., Weiler, T., Zeise, M., Anagnostou, G., Daskalakis, G., Geralis, T., Kesisoglou, S., Kyriakis, A., Loukas, D., Markou, A., Markou, C., Ntomari, E., Gouskos, L., Mertzimekis, T. J., Panagiotou, A., Saoulidou, N., Stiliaris, E., Aslanoglou, X., Evangelou, I., Flouris, G., Foudas, C., Kokkas, P., Manthos, N., Papadopoulos, I., Paradas, E., Bencze, G., Hajdu, C., Hidas, P., Horvath, D., Radics, B., Sikler, F., Veszpremi, V., Vesztergombi, G., Zsigmond, A. J., Beni, N., Czellar, S., Molnar, J., Palinkas, J., Szillasi, Z., Karancsi, J., Raics, P., Trocsanyi, Z. L., Ujvari, B., Beri, S. B., Bhatnagar, V., Dhingra, N., Gupta, R., Kaur, M., Mehta, M. Z., Mittal, M., Nishu, N., Saini, L. K., Sharma, A., Singh, J. B., Kumar, Ashok, Kumar, Arun, Ahuja, S., Bhardwaj, A., Choudhary, B. C., Malhotra, S., Naimuddin, M., Ranjan, K., Saxena, P., Sharma, V., Shivpuri, R. K., Banerjee, S., Bhattacharya, S., Chatterjee, K., Dutta, S., Gomber, B., Jain, Sa., Jain, Sh., Khurana, R., Modak, A., Mukherjee, S., Roy, D., Sarkar, S., Sharan, M., Abdulsalam, A., Dutta, D., Kailas, S., Kumar, V., Mohanty, A. K., Pant, L. M., Shukla, P., Topkar, A., Aziz, T., Chatterjee, R. M., Ganguly, S., Ghosh, S., Guchait, M., Gurtu, A., Kole, G., Kumar, S., Maity, M., Majumder, G., Mazumdar, K., Mohanty, G. B., Parida, B., Sudhakar, K., Wickramage, N., Banerjee, S., Dugad, S., Arfaei, H., Bakhshiansohi, H., Etesami, S. M., Fahim, A., Hesari, H., Jafari, A., Khakzad, M., Mohammadi Najafabadi, M., Paktinat Mehdiabadi, S., Safarzadeh, B., Zeinali, M., Grunewald, M., Abbrescia, M., Barbone, L., Calabria, C., Chhibra, S. S., Colaleo, A., Creanza, D., De Filippis, N., De Palma, M., Fiore, L., Iaselli, G., Maggi, G., Maggi, M., Marangelli, B., My, S., Nuzzo, S., Pacifico, N., Pompili, A., Pugliese, G., Selvaggi, G., Silvestris, L., Singh, G., Venditti, R., Verwilligen, P., Zito, G., Abbiendi, G., Benvenuti, A. C., Bonacorsi, D., Braibant-Giacomelli, S., Brigliadori, L., Campanini, R., Capiluppi, P., Castro, A., Cavallo, F. R., Cuffiani, M., Dallavalle, G. M., Fabbri, F., Fanfani, A., Fasanella, D., Giacomelli, P., Grandi, C., Guiducci, L., Marcellini, S., Masetti, G., Meneghelli, M., Montanari, A., Navarria, F. L., Odorici, F., Perrotta, A., Primavera, F., Rossi, A. M., Rovelli, T., Siroli, G. P., Tosi, N., Travaglini, R., Albergo, S., Chiorboli, M., Costa, S., Giordano, F., Potenza, R., Tricomi, A., Tuve, C., Barbagli, G., Ciulli, V., Civinini, C., D’Alessandro, R., Focardi, E., Frosali, S., Gallo, E., Gonzi, S., Gori, V., Lenzi, P., Meschini, M., Paoletti, S., Sguazzoni, G., Tropiano, A., Benussi, L., Bianco, S., Fabbri, F., Piccolo, D., Fabbricatore, P., Musenich, R., Tosi, S., Benaglia, A., De Guio, F., Di Matteo, L., Fiorendi, S., Gennai, S., Ghezzi, A., Govoni, P., Lucchini, M. T., Malvezzi, S., Manzoni, R. A., Martelli, A., Menasce, D., Moroni, L., Paganoni, M., Pedrini, D., Ragazzi, S., Redaelli, N., Tabarelli de Fatis, T., Buontempo, S., Cavallo, N., De Cosa, A., Fabozzi, F., Iorio, A. O. M., Lista, L., Meola, S., Merola, M., Paolucci, P., Azzi, P., Bacchetta, N., Bisello, D., Branca, A., Carlin, R., Checchia, P., Dorigo, T., Dosselli, U., Galanti, M., Gasparini, F., Gasparini, U., Giubilato, P., Gozzelino, A., Gulmini, M., Kanishchev, K., Lacaprara, S., Lazzizzera, I., Margoni, M., Maron, G., Meneguzzo, A. T., Pazzini, J., Pozzobon, N., Ronchese, P., Simonetto, F., Torassa, E., Tosi, M., Vanini, S., Zotto, P., Zucchetta, A., Zumerle, G., Gabusi, M., Ratti, S. P., Riccardi, C., Vitulo, P., Biasini, M., Bilei, G. M., Fanò, L., Lariccia, P., Mantovani, G., Menichelli, M., Nappi, A., Romeo, F., Saha, A., Santocchia, A., Spiezia, A., Androsov, K., Azzurri, P., Bagliesi, G., Boccali, T., Broccolo, G., Castaldi, R., D’Agnolo, R. T., Dell’Orso, R., Fiori, F., Foà, L., Giassi, A., Kraan, A., Ligabue, F., Lomtadze, T., Martini, L., Messineo, A., Palla, F., Rizzi, A., Serban, A. T., Spagnolo, P., Squillacioti, P., Tenchini, R., Tonelli, G., Venturi, A., Verdini, P. G., Vernieri, C., Barone, L., Cavallari, F., Del Re, D., Diemoz, M., Grassi, M., Longo, E., Margaroli, F., Meridiani, P., Micheli, F., Nourbakhsh, S., Organtini, G., Paramatti, R., Rahatlou, S., Soffi, L., Amapane, N., Arcidiacono, R., Argiro, S., Arneodo, M., Biino, C., Cartiglia, N., Casasso, S., Costa, M., Demaria, N., Mariotti, C., Maselli, S., Migliore, E., Monaco, V., Musich, M., Obertino, M. M., Ortona, G., Pastrone, N., Pelliccioni, M., Potenza, A., Romero, A., Ruspa, M., Sacchi, R., Solano, A., Staiano, A., Tamponi, U., Belforte, S., Candelise, V., Casarsa, M., Cossutti, F., Della Ricca, G., Gobbo, B., La Licata, C., Marone, M., Montanino, D., Penzo, A., Schizzi, A., Zanetti, A., Kim, T. Y., Nam, S. K., Chang, S., Kim, D. H., Kim, G. N., Kim, J. E., Kong, D. J., Oh, Y. D., Park, H., Son, D. C., Kim, J. Y., Kim, Zero J., Song, S., Choi, S., Gyun, D., Hong, B., Jo, M., Kim, H., Kim, T. J., Lee, K. S., Park, S. K., Roh, Y., Choi, M., Kim, J. H., Park, C., Park, I. 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M., Kroeger, R., Perera, L., Rahmat, R., Sanders, D. A., Summers, D., Avdeeva, E., Bloom, K., Bose, S., Claes, D. R., Dominguez, A., Eads, M., Gonzalez Suarez, R., Keller, J., Kravchenko, I., Lazo-Flores, J., Malik, S., Meier, F., Snow, G. R., Dolen, J., Godshalk, A., Iashvili, I., Jain, S., Kharchilava, A., Kumar, A., Rappoccio, S., Wan, Z., Alverson, G., Barberis, E., Baumgartel, D., Chasco, M., Haley, J., Massironi, A., Nash, D., Orimoto, T., Trocino, D., Wood, D., Zhang, J., Anastassov, A., Hahn, K. A., Kubik, A., Lusito, L., Mucia, N., Odell, N., Pollack, B., Pozdnyakov, A., Schmitt, M., Stoynev, S., Velasco, M., Won, S., Berry, D., Brinkerhoff, A., Chan, K. M., Hildreth, M., Jessop, C., Karmgard, D. J., Kolb, J., Lannon, K., Luo, W., Lynch, S., Marinelli, N., Morse, D. M., Pearson, T., Planer, M., Ruchti, R., Slaunwhite, J., Valls, N., Wayne, M., Wolf, M., Antonelli, L., Bylsma, B., Durkin, L. S., Hill, C., Hughes, R., Kotov, K., Ling, T. Y., Puigh, D., Rodenburg, M., Smith, G., Vuosalo, C., Williams, G., Winer, B. L., Wolfe, H., Berry, E., Elmer, P., Halyo, V., Hebda, P., Hegeman, J., Hunt, A., Jindal, P., Koay, S. A., Lopes Pegna, D., Lujan, P., Marlow, D., Medvedeva, T., Mooney, M., Olsen, J., Piroué, P., Quan, X., Raval, A., Saka, H., Stickland, D., Tully, C., Werner, J. S., Zenz, S. C., Zuranski, A., Brownson, E., Lopez, A., Mendez, H., Ramirez Vargas, J. E., Alagoz, E., Benedetti, D., Bolla, G., Bortoletto, D., De Mattia, M., Everett, A., Hu, Z., Jones, M., Jung, K., Koybasi, O., Kress, M., Leonardo, N., Maroussov, V., Merkel, P., Miller, D. H., Neumeister, N., Shipsey, I., Silvers, D., Svyatkovskiy, A., Vidal Marono, M., Wang, F., Xu, L., Yoo, H. D., Zablocki, J., Zheng, Y., Guragain, S., Parashar, N., Adair, A., Akgun, B., Ecklund, K. M., Geurts, F. J. M., Li, W., Padley, B. P., Redjimi, R., Roberts, J., Zabel, J., Betchart, B., Bodek, A., Covarelli, R., de Barbaro, P., Demina, R., Eshaq, Y., Ferbel, T., Garcia-Bellido, A., Goldenzweig, P., Han, J., Harel, A., Miner, D. C., Petrillo, G., Vishnevskiy, D., Zielinski, M., Bhatti, A., Ciesielski, R., Demortier, L., Goulianos, K., Lungu, G., Malik, S., Mesropian, C., Arora, S., Barker, A., Chou, J. P., Contreras-Campana, C., Contreras-Campana, E., Duggan, D., Ferencek, D., Gershtein, Y., Gray, R., Halkiadakis, E., Hidas, D., Lath, A., Panwalkar, S., Park, M., Patel, R., Rekovic, V., Robles, J., Rose, K., Salur, S., Schnetzer, S., Seitz, C., Somalwar, S., Stone, R., Thomas, S., Walker, M., Cerizza, G., Hollingsworth, M., Spanier, S., Yang, Z. C., York, A., Eusebi, R., Flanagan, W., Gilmore, J., Kamon, T., Khotilovich, V., Montalvo, R., Osipenkov, I., Pakhotin, Y., Perloff, A., Roe, J., Safonov, A., Sakuma, T., Suarez, I., Tatarinov, A., Toback, D., Akchurin, N., Damgov, J., Dragoiu, C., Dudero, P. R., Jeong, C., Kovitanggoon, K., Lee, S. W., Libeiro, T., Volobouev, I., Appelt, E., Delannoy, A. G., Greene, S., Gurrola, A., Johns, W., Maguire, C., Mao, Y., Melo, A., Sharma, M., Sheldon, P., Snook, B., Tuo, S., Velkovska, J., Arenton, M. W., Boutle, S., Cox, B., Francis, B., Goodell, J., Hirosky, R., Ledovskoy, A., Lin, C., Neu, C., Wood, J., Gollapinni, S., Harr, R., Karchin, P. E., Kottachchi Kankanamge Don, C., Lamichhane, P., Sakharov, A., Anderson, M., Belknap, D. A., Borrello, L., Carlsmith, D., Cepeda, M., Dasu, S., Friis, E., Grogg, K. S., Grothe, M., Hall-Wilton, R., Herndon, M., Hervé, A., Kaadze, K., Klabbers, P., Klukas, J., Lanaro, A., Lazaridis, C., Loveless, R., Mohapatra, A., Mozer, M. U., Ojalvo, I., Pierro, G. A., Ross, I., Savin, A., Smith, W. H., and Swanson, J.

Published
2022
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13. Resonant rigidity for Schr\'odinger operators in even dimensions

Author

Christiansen, T. J.

Subjects
Mathematics - Spectral Theory, Mathematical Physics, 35P25, 58J50, 81U40
Abstract

This paper studies the resonances of Schr\"odinger operators with bounded, compactly supported, real-valued potentials on d-dimensional Euclidean space, where d is even. If the potential V is non-trivial and d is not 4 then the meromorphic continuation of the resolvent of the Schr\"odinger operator has infinitely many poles, with a quantitative lower bound on their density. A somewhat weaker statement holds if d =4. We prove several inverse-type results. If the meromorphic continuations of the resolvents of two Schr\"odinger operators $-\Delta +V_1$ and $-\Delta +V_2$ have the same poles, with both potentials bounded, compactly supported and real-valued, if k is a natural number and if $V_1\in H^k({\mathbb R}^d; {\mathbb R})$, then $V_2\in H^k$ as well. Moreover, we prove that certain sets of isoresonant potentials are compact. We also show that the poles of the resolvent for a smooth potential determine the heat coefficients and that the (resolvent) resonance sets of two bounded, real-valued potentials with compact support cannot differ by a nonzero finite number of elements away from $0$., Comment: 37 pages

Published
2017

14. Wave asymptotics for waveguides and manifolds with infinite cylindrical ends

Author

Christiansen, T. J. and Datchev, K.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Spectral Theory
Abstract

We describe wave decay rates associated to embedded resonances and spectral thresholds for waveguides and manifolds with infinite cylindrical ends. We show that if the cut-off resolvent is polynomially bounded at high energies, as is the case in certain favorable geometries, then there is an associated asymptotic expansion, up to a $O(t^{-k_0})$ remainder, of solutions of the wave equation on compact sets as $t \to \infty$. In the most general such case we have $k_0=1$, and under an additional assumption on the infinite ends we have $k_0 = \infty$. If we localize the solutions to the wave equation in frequency as well as in space, then our results hold for quite general waveguides and manifolds with infinite cylindrical ends. To treat problems with and without boundary in a unified way, we introduce a black box framework analogous to the Euclidean one of Sj\"ostrand and Zworski. We study the resolvent, generalized eigenfunctions, spectral measure, and spectral thresholds in this framework, providing a new approach to some mostly well-known results in the scattering theory of manifolds with cylindrical ends., Comment: In this revision we work in a more general black box setting than in the first version of the paper. In particular, we allow a boundary extending to infinity. The changes to the proofs of the main theorems are minor, but the presentation of the needed basic material from scattering theory is substantially expanded. New examples are included, both for the main results and for the black box setting

Published
2017

15. Resolvent estimates on asymptotically cylindrical manifolds and on the half line

Author

Christiansen, T. J. and Datchev, K.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Spectral Theory
Abstract

Manifolds with infinite cylindrical ends have continuous spectrum of increasing multiplicity as energy grows, and in general embedded resonances (resonances on the real line, embedded in the continuous spectrum) and embedded eigenvalues can accumulate at infinity. However, we prove that if geodesic trapping is sufficiently mild, then the number of embedded resonances and eigenvalues is finite, and moreover the cutoff resolvent is uniformly bounded at high energies. We obtain as a corollary the existence of resonance free regions near the continuous spectrum. We also obtain improved estimates when the resolvent is cut off away from part of the trapping, and along the way we prove some resolvent estimates for repulsive potentials on the half line which may be of independent interest., Comment: This paper is a companion to the paper `Wave asymptotics for manifolds with infinite cylindrical ends' by the same authors, but each paper can be read independently of the other

Published
2017

16. A sharp lower bound for a resonance-counting function in even dimensions

Author

Christiansen, T. J.

Subjects
Mathematics - Spectral Theory, Mathematical Physics, 35P25, 58J50
Abstract

This paper proves sharp lower bounds on a resonance counting function for obstacle scattering in even-dimensional Euclidean space without a need for trapping assumptions. Similar lower bounds are proved for some other compactly supported perturbations of the Laplacian on even-dimensional Euclidean space, for example, for the Laplacian for certain metric perturbations. The proof uses a Poisson formula for resonances, complementary to one proved by Zworski in even dimensions., Comment: 21 pages

Published
2015

17. Lower bounds for resonance counting functions for obstacle scattering in even dimensions

Author

Christiansen, T. J.

Subjects
Mathematical Physics, Mathematics - Spectral Theory, 35P25, 47A40, 58J50
Abstract

In even dimensional Euclidean scattering, the resonances lie on the logarithmic cover of the complex plane. This paper studies resonances for obstacle scattering in ${\mathbb R}^d$ with Dirchlet or admissable Robin boundary conditions, when $d$ is even. Set $n_m(r)$ to be the number of resonances with norm at most $r$ and argument between $m\pi$ and $(m+1)\pi$. Then $\lim\sup _{r\rightarrow \infty}\frac{\log n_m(r)}{\log r}=d$ if $m\in {\mathbb Z}\setminus \{ 0\}$.

Published
2014

19. Lower bounds for resonance counting functions for Schr\'odinger operators with fixed sign potentials in even dimensions

Author

Christiansen, T. J.

Subjects
Mathematics - Spectral Theory, Mathematical Physics, 35P25, 58J50, 81U05
Abstract

If the dimension $d$ is even, the resonances of the Schr\"odinger operator $-\Delta +V$ on ${\mathbb R}^d$ with $V$ bounded and compactly supported are points on $\Lambda$, the logarithmic cover of ${\mathbb C} \setminus \{0\}$. We show that for fixed sign potentials $V$ and for nonzero integers $m$, the resonance counting function for the $m$th sheet of $\Lambda$ has maximal order of growth., Comment: 21 pages

Published
2013

20. Some remarks on resonances in even-dimensional Euclidean scattering

Author

Christiansen, T. J. and Hislop, P. D.

Subjects
Mathematical Physics, Mathematics - Spectral Theory, 35P25, 81U05
Abstract

The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions $d \geq 2$. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto $\Lambda$, the Riemann surface of the logarithm function. We relate the multiplicities of the poles of the continued scattering matrix to the multiplicities of the poles of the resolvent. Moreover, we show that the poles of the scattering matrix on the $m$th sheet of $\Lambda$ are related to the zeros of a scalar function defined on the physical sheet. This paper contains a number of results about "pure imaginary" resonances. As an example, in contrast with the odd-dimensional case, we show that in even dimensions there are no "purely imaginary" resonances on any sheet of $\Lambda$ for Schr\"odinger operators with potentials $0 \leq V \in L_0^\infty (\R^d)$., Comment: 27 pages

Published
2013

22. Schrodinger operators and the distribution of resonances in sectors

Author

Christiansen, T. J.

Subjects
Mathematics - Spectral Theory, Mathematical Physics, 35P25, 81U05, 35J10, 58J50
Abstract

The purpose of this paper is to give some refined results about the distribution of resonances in potential scattering. We use techniques and results from one and several complex variables, including properties of functions of completely regular growth. This enables us to find asymptotics for the distribution of resonances in sectors for certain potentials and for certain families of potentials., Comment: typos corrected; elaboration of some proofs

Published
2010

23. Resonances for manifolds hyperbolic at infinity: optimal lower bounds on order of growth

Author

Borthwick, D., Christiansen, T. J., Hislop, P. D., and Perry, P. A.

Subjects
Mathematics - Spectral Theory, Mathematics - Differential Geometry, 58J50
Abstract

Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of growth $(n+1)$ generically for such manifolds., Comment: 28 pages, 1 figure

Published
2010

24. Probabilistic Weyl laws for quantized tori

Author

Christiansen, T. J. and Zworski, M.

Subjects
Mathematics - Spectral Theory, 15A18, 35S05, 47A10, 15B52
Abstract

For the Toeplitz quantization of complex-valued functions on a $2n$-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law. In numerical experiments the same Weyl law also holds for ``false'' eigenvalues created by pseudospectral effects., Comment: 33 pages, 3 figures, v2 corrected listed title

Published
2009
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25. A mathematical formulation of the Mahaux-Weidenm\'uller formula for the scattering matrix

Author

Christiansen, T. J. and Zworski, M.

Subjects
Mathematical Physics, Mathematics - Spectral Theory, 81U20, 35P25, 58J50
Abstract

The purpose of this note is to give a mathematical explanation of a formula for the scattering matrix for a manifold with infinite cylindrical ends or a waveguide. This formula, which is well known in the physics literature, is sometimes referred to as the Mahaux-Weidenm\"uller formula. We show that a version of this formula given below gives the standard scattering matrix used in the mathematics literature. We also show that the finite rank approximation of the interaction matrix gives an approximation of the scattering matrix with errors inversely proportional to a fixed dimension-dependent power of the rank. A simple example shows that this estimate is optimal., Comment: 24 pages, 2 figures; Correction to Lemma 4.3, additional references; Elaborations in section 3; typos corrected

Published
2009

26. Resonances for Schrodinger operators with compactly supported potentials

Author

Christiansen, T. J. and Hislop, P. D.

Subjects
Mathematical Physics, 35P25, 47A10, 47A40, 81U20
Abstract

We describe the generic behavior of the resonance counting function for a Schr\"odinger operator with a bounded, compactly-supported real or complex valued potential in $d \geq 1$ dimensions. This note contains a sketch of the proof of our main results \cite{ch-hi1,ch-hi2} that generically the order of growth of the resonance counting function is the maximal value $d$ in the odd dimensional case, and that it is the maximal value $d$ on each nonphysical sheet of the logarithmic Riemann surface in the even dimensional case. We include a review of previous results concerning the resonance counting functions for Schr\"odinger operators with compactly-supported potentials., Comment: 19 pages

Published
2009

29. Maximal order of growth for the resonance counting functions for generic potentials in even dimensions

Author

Christiansen, T. J. and Hislop, P. D.

Subjects
Mathematical Physics, 35P25, 47A10, 47A40, 81U20
Abstract

We prove that the resonance counting functions for Schr\"odinger operators $H_V = - \Delta + V$ on $L^2 (\R^d)$, for $d \geq 2$ {\it even}, with generic, compactly-supported, real- or complex-valued potentials $V$, have the maximal order of growth $d$ on each sheet $\Lambda_m$, $m \in \Z \backslash \{0 \}$, of the logarithmic Riemann surface. We obtain this result by constructing, for each $m \in \Z \backslash \{0 \}$, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet $\Lambda_0$ determine the poles on $\Lambda_m$. We prove that the order of growth of the counting function is related to a suitable estimate on this function that we establish for generic potentials. We also show that for a potential that is the characteristic function of a ball, the resonance counting function is bounded below by $C_m r^d$ on each sheet $\Lambda_m$, $m \in \Z \backslash \{0\}$., Comment: 33 pages and 1 figure

Published
2008

30. Commissioning of CMS and early standard model measurements with jets, missing transverse energy and photons at the LHC

Author

Christiansen, T.

Subjects
High Energy Physics - Experiment
Abstract

We report on the status and history of the CMS commissioning, together with selected results from cosmic-ray muon data. The second part focuses on strategies for optimizing the reconstruction of jets, missing transverse energy and photons for early standard model measurements at ATLAS and CMS with the first collision data from the Large Hadron Collider at CERN., Comment: Prepared for the 43rd Rencontres de Moriond session devoted to Electroweak Interactions and Unified Theories, March 1 - 8, 2008, La Thuile, Italy, 6 pages, 10 figures

Published
2008

31. Resonances and balls in obstacle scattering with Neumann boundary conditions

Author

Christiansen, T. J.

Subjects
Mathematical Physics, Mathematics - Spectral Theory, 35P25, 58J50, 35J25
Abstract

We consider scattering by an obstacle in $\Real^d$, $d\geq 3 $ odd. We show that for the Neumann Laplacian if an obstacle has the same resonances as the ball of radius $\rho$ does, then the obstacle is a ball of radius $\rho$. We give related results for obstacles which are disjoint unions of several balls of the same radius.

Published
2008

32. CHAPTER TWENTY-FOUR. Conservation of Greater Sage-Grouse: A SYNTHESIS OF CURRENT TRENDS AND FUTURE MANAGEMENT

Author

Connelly, J. W., primary, Knick, S. T., additional, Braun, C. E., additional, Baker, W. L., additional, Beever, E. A., additional, Christiansen, T., additional, Doherty, K. E., additional, Garton, E. O., additional, Hanser, S. E., additional, Johnson, D. H., additional, Leu, M., additional, Miller, R. F., additional, Naugle, D. E., additional, Oyler-McCance, S. J., additional, Pyke, D. A., additional, Reese, K. P., additional, Schroeder, M. A., additional, Stiver, S. J., additional, Walker, B. L., additional, and Wisdom, M. J., additional

Published
2019
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33. The resonance counting function for Schr\'odinger operators with generic potentials

Author

Christiansen, T. and Hislop, P. D.

Subjects
Mathematical Physics, Mathematics - Spectral Theory, 81U05, 35P25, 47A40
Abstract

We show that the resonance counting function for a Schr\"odinger operator has maximal order of growth for generic sets of real-valued, or complex-valued, $L^\infty$-compactly supported potentials.

Published
2005

34. Schr\'odinger operators with complex-valued potentials and no resonances

Author

Christiansen, T.

Subjects
Mathematical Physics, Mathematics - Spectral Theory, 35P25, 47A40, 81U05, 58J50
Abstract

In dimension $d\geq 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schr\"odinger operators have no resonances. If $d=2$, we show that there are potentials with no resonances away from the origin. These Schr\"odinger operators are isophasal and have the same scattering phase as the Laplacian on $\Real^d$. In odd dimensions $d\geq 3$ we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is super-exponentially decaying in time., Comment: 9 pages

Published
2004

35. Several complex variables and the distribution of resonances in potential scattering

Author

Christiansen, T.

Subjects
Mathematics - Spectral Theory, Mathematical Physics, 35P25, 81U05
Abstract

We study resonances associated to Schr\"odinger operators with compactly supported potentials on ${\mathbb R}^d$, $d\geq3$, odd. We consider compactly supported potentials depending holomorphically on a complex parameter $z$. For certain such families, for all $z$ except those in a pluripolar set, the associated resonance-counting function has order of growth $d$. Our proofs use some results from several complex variables., Comment: 18 pages

Published
2004
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36. Asymptotics for a resonance-counting function for potential scattering on cylinders

Author

Christiansen, T.

Subjects
Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, 58J50 (Primary), 35P25 (Secondary)
Abstract

We study certain resonance-counting functions for potential scattering on infinite cylinders or half-cylinders. Under certain conditions on the potential, we obtain asymptotics of the counting functions, with an explicit formula for the constant appearing in the leading term.

Published
2003

37. Resonances for steplike potentials: forward and inverse results

Author

Christiansen, T.

Subjects
Mathematics - Spectral Theory, Mathematical Physics, 34L25, 34A55, 81U40
Abstract

We consider resonances associated to the operator $-\frac{d^2}{dx^2}+V(x)$, where $V(x)=V_+$ if $x>x_M$ and $V(x)=V_-$ if $x<-x_M$, with $V_+\not = V_-$. We obtain asymptotics of the resonance-counting function in several regions. Moreover, we show that in several situations, the resonances, $V_+$, and $V_-$ determine $V$ uniquely up to translation., Comment: 16 pages; submitted

Published
2003

41. Scattering on stratified media: the micro-local properties of the scattering matrix and recovering asymptotics of perturbations

Author

Christiansen, T. J. and Joshi, M. S.

Subjects
Mathematics - Spectral Theory, Mathematics - Analysis of PDEs
Abstract

The fixed energy scattering matrix is defined on a perturbed stratified medium, and for a class of perturbations, its main part is shown to be a Fourier integral operator on the sphere at infinity. This is facilitated by developing a refined limiting absorption principle. The symbol of the scattering matrix is shown to determine the asymptotics of a large class of perturbations.

Published
2000

42. Higher order scattering on asymptotically Euclidean Manifolds

Author

Christiansen, T. J. and Joshi, M. S.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Spectral Theory, 58J40, 58J20, 35P25
Abstract

We develop a scattering theory for perturbations of powers of the Laplacian on asymptotically Euclidean manifolds. The (absolute) scattering matrix is shown to be a Fourier integral operator associated to the geodesic flow at time \pi on the boundary. Furthermore, it is shown that on \Real^n the asymptotics of certain short-range perturbations of \Delta^k can be recovered from the scattering matrix at a finite number of energies., Comment: To appear in the Canadian Journal of Mathematics; 26 pages

Published
2000

43. Probabilistic Weyl Laws for Quantized Tori

Author

Christiansen, T. J. and Zworski, M.

Subjects
Physics, Classical and Quantum Gravitation, Relativity Theory, Statistical Physics, Dynamical Systems and Complexity, Quantum Physics, Theoretical, Mathematical and Computational Physics
Abstract

For the Toeplitz quantization of complex-valued functions on a 2n-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law (1.3). In numerical experiments the same Weyl law also holds for “false” eigenvalues created by pseudospectral effects.

Published
2010

46. Influence of Laser Marking on Microstructure and Corrosion Performance of Martensitic Stainless Steel Surfaces for Biomedical Applications

Author

Henriksen, N. G., Andersen, O. Z., Jellesen, M. S., Christiansen, T. L., and Somers, M. A. J.

Subjects
Corrosion, Surface treatment, Materials Chemistry, Metals and Alloys, Laser marking, Martensitic stainless steel, Heat affected zone, Industrial and Manufacturing Engineering, Manganese sulfide
Abstract

The medical device industry demands unique device identification (UDI) tags on metallic components applied via laser marking. A common issue is that the visual appearance of the marking becomes poorly legible over time due to loss of contrast. Nanosecond pulsed laser irradiation was used to grow an oxide layer on two different martensitic stainless steels AISI 420F mod and 420B to compare the influences of the chemical composition of the steel (with and without S), power density, and energy input. The corrosion behavior was found to depend strongly on laser energy input. The presence of sulfur negatively affected the corrosion resistance and narrowed the applicable window for the laser processing parameters significantly. For the sulfur-containing AISI 420F steel, 3‒5 μm wide craters formed on the surface after laser marking, which is interpreted as thermal degradation of protruding MnS inclusions resulting from the laser marking process. Also, substantial cracking in the oxide layer was observed. The marked specimens suffered from corrosion in a thin zone below the formed oxide layer. This behavior is attributed to Cr-depletion in the zone adjacent to the oxide layer, resulting from providing Cr to the growing oxide layer.

Published
2022

50. Erratum to: Searches for long-lived charged particles in pp collisions at <math> <msqrt> <mi>s</mi> </msqrt> </math> $$ \sqrt{\textrm{s}} $$ = 7 and 8 TeV

Author

Chatrchyan, S., Khachatryan, V., Sirunyan, A., Tumasyan, A., Adam, W., Bergauer, T., Dragicevic, M., Erö, J., Fabjan, C., Friedl, M., Frühwirth, R., Ghete, V., Hörmann, N., Hrubec, J., Jeitler, M., Kiesenhofer, W., Knünz, V., Krammer, M., Krätschmer, I., Liko, D., Mikulec, I., Rabady, D., Rahbaran, B., Rohringer, C., Rohringer, H., Schöfbeck, R., Strauss, J., Taurok, A., Treberer-Treberspurg, W., Waltenberger, W., Wulz, C.E., Mossolov, V., Shumeiko, N., Suarez Gonzalez, J., Alderweireldt, S., Bansal, M., Bansal, S., Cornelis, T., Wolf, E., Janssen, X., Knutsson, A., Luyckx, S., Mucibello, L., Ochesanu, S., Roland, B., Rougny, R., Haevermaet, H., Mechelen, P., Remortel, N., Spilbeeck, A., Blekman, F., Blyweert, S., D’Hondt, J., Kalogeropoulos, A., Keaveney, J., Maes, M., Olbrechts, A., Tavernier, S., Doninck, W., Mulders, P., Onsem, G., Villella, I., Clerbaux, B., Lentdecker, G., Favart, L., Gay, A., Hreus, T., Léonard, A., Marage, P., Mohammadi, A., Perniè, L., Reis, T., Seva, T., Thomas, L., Vander Velde, C., Vanlaer, P., Wang, J., Adler, V., Beernaert, K., Benucci, L., Cimmino, A., Costantini, S., Dildick, S., Garcia, G., Klein, B., Lellouch, J., Marinov, A., Mccartin, J., Ocampo Rios, A., Ryckbosch, D., Sigamani, M., Strobbe, N., Thyssen, F., Tytgat, M., Walsh, S., Yazgan, E., Zaganidis, N., Basegmez, S., Beluffi, C., Bruno, G., Castello, R., Caudron, A., Ceard, L., Delaere, C., Pree, T., Favart, D., Forthomme, L., Giammanco, A., Hollar, J., Jez, P., Lemaitre, V., Liao, J., Militaru, O., Nuttens, C., Pagano, D., Pin, A., Piotrzkowski, K., Popov, A., Selvaggi, M., Vizan Garcia, J., Beliy, N., Caebergs, T., Daubie, E., Hammad, G., Alves, G., Correa Martins Junior, M., Martins, T., Pol, M., Souza, M., Aldá Júnior, W., Carvalho, W., Chinellato, J., Custódio, A., Costa, E., De Jesus Damiao, D., De Oliveira Martins, C., Fonseca De Souza, S., Malbouisson, H., Malek, M., Matos Figueiredo, D., Mundim, L., Nogima, H., Prado Da Silva, W., Santoro, A., Sznajder, A., Tonelli Manganote, E., Vilela Pereira, A., Bernardes, C., Dias, F., Fernandez Perez Tomei, T., Gregores, E., Lagana, C., Marinho, F., Mercadante, P., Novaes, S., Padula, Sandra, Genchev, V., Iaydjiev, P., Piperov, S., Rodozov, M., Sultanov, G., Vutova, M., Dimitrov, A., Hadjiiska, R., Kozhuharov, V., Litov, L., Pavlov, B., Petkov, P., Bian, J., Chen, G., Chen, H., Jiang, C., Liang, D., Liang, S., Meng, X., Tao, J., Wang, X., Wang, Z., Xiao, H., Xu, M., Asawatangtrakuldee, C., Ban, Y., Guo, Y., Li, W., Liu, S., Mao, Y., Qian, S., Teng, H., Wang, D., Zhang, L., Zou, W., Avila, C., Carrillo Montoya, C., Gomez, J., Gomez Moreno, B., Sanabria, J., Godinovic, N., Lelas, D., Plestina, R., Polic, D., Puljak, I., Antunovic, Z., Kovac, M., Brigljevic, V., Duric, S., Kadija, K., Luetic, J., Mekterovic, D., Morovic, S., Tikvica, L., Attikis, A., Mavromanolakis, G., Mousa, J., Nicolaou, C., Ptochos, F., Razis, P., Finger, M., Assran, Y., Ellithi Kamel, A., Mahmoud, M., Mahrous, A., Radi, A., 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Subjects
Nuclear and High Energy Physics
Published
2022

Catalog

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